8.6. Calculating the Length of a Path


We now consider paths and their lengths. This will finish the geometry section in our integral course.

Definition:  Let M k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@3B47@ be a subset of k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGRbaaaaaa@3879@ . If [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ is a closed interval in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375C@ we call any function

w=( w 1 , w k ):[a,b]M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Daiabg2da9iaacIcacaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaadEhadaWgaaWcbaGaam4AaaqabaGccaGGPaGaaiOoaiaacUfacaWGHbGaaiilaiaadkgacaGGDbGaeyOKH4Qaamytaaaa@46E2@
[8.6.1]

a path (in M). w(a) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWGHbGaaiykaaaa@3927@ is the initial point and w(b) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWGIbGaaiykaaaa@3928@ the final point of w. The path w is closed if its initial and final points coincide: w(a)=w(b) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWGHbGaaiykaiabg2da9iaadEhacaGGOaGaamOyaiaacMcaaaa@3D69@ . The range {w(t)|t[a,b]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadEhacaGGOaGaamiDaiaacMcacaGG8bGaamiDaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbGaaiyFaaaa@42F4@ is called the curve belonging to w and the path [8.6.1] is then regarded as a parametrization of its curve.

If all the coordinate functions w 1 ,, w k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaaaaa@3C73@ are

  • continuous then w is a continuous path (occasionally: C 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGimaaaaaaa@379B@ -path).

  • differentiable, w is a differentiable path ( D 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGymaaaaaaa@379D@ -path) and the function

w ( w 1 ,, w k ):[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4DayaafaGaeyypa0JaaiikaiqadEhagaqbamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGabm4DayaafaWaaSbaaSqaaiaadUgaaeqaaOGaaiykaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@4971@
[8.6.2]

is called the derivative of w. w is a regular path if w (t)0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4DayaafaGaaiikaiaadshacaGGPaGaeyiyIKRaaGimaaaa@3BC7@ for all t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@3CA6@ .

Repeatedly differentiable paths and higher derivatives are introduced analogously.

  • continously differentiable, w is called a smooth path ( C 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGymaaaaaaa@379C@ -path).

  • integrable, then w is an integrable path and for any r,s[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacYcacaWGZbGaeyicI4Saai4waiaadggacaGGSaGaamOyaiaac2faaaa@3E4C@ the vector

r s w ( r s w 1 ,, r s w k ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWG3baaleaacaWGYbaabaGaam4CaaqdcqGHRiI8aOGaeyypa0JaaiikamaapehabaGaam4DamaaBaaaleaacaaIXaaabeaaaeaacaWGYbaabaGaam4CaaqdcqGHRiI8aOGaaiilaiablAciljaacYcadaWdXbqaaiaadEhadaWgaaWcbaGaam4AaaqabaaabaGaamOCaaqaaiaadohaa0Gaey4kIipakiaacMcaaaa@4CD4@
[8.6.3]

is read as the integral of w from r to s.

Consider:

  • If f:[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHcaa@3F2F@ is a common real-valued function, its graph is the curve belonging to the path

    w:t(t,f(t)),t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaWG0bGaeSOPHeMaaiikaiaadshacaGGSaGaamOzaiaacIcacaWG0bGaaiykaiaacMcacaGGSaGaaGzbVlaadshacqGHiiIZcaGGBbGaamyyaiaacYcacaWGIbGaaiyxaaaa@498F@ .

    As X is arbitrary often differentiable (thus continuous and integrable as well), w has the same qualities as f has.

  • The path concept is, among others, designed to describe movement. In this case it is the movement of a particle along its way, with the interval [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ being the section of time in which the movement is monitored. The notation

    t( w 1 (t),, w k (t)),t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaacIcacaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadshacaGGPaGaaiilaiablAciljaacYcacaWG3bWaaSbaaSqaaiaadUgaaeqaaOGaaiikaiaadshacaGGPaGaaiykaiaacYcacaaMf8UaamiDaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@4E24@

    (instead of x( w 1 (x),, w k (x)),x[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiablAAiHjaacIcacaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadIhacaGGPaGaaiilaiablAciljaacYcacaWG3bWaaSbaaSqaaiaadUgaaeqaaOGaaiikaiaadIhacaGGPaGaaiykaiaacYcacaaMf8UaamiEaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@4E34@ ) supports this perception. If w is D 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGymaaaaaaa@379D@ we interpret the derivative w (t) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4DayaafaGaaiikaiaadshacaGGPaaaaa@3946@ as the speed of the particle at time t.

  • There is a clear difference between a path and the associated curve. A single curve may belong to several different paths. For an example consider the paths

    w:t(t,0),t[0,1] v:t(1t,0),t[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiaadEhacaGG6aGaamiDaiablAAiHjaacIcacaWG0bGaaiilaiaaicdacaGGPaGaaiilaiaaywW7caWG0bGaeyicI4Saai4waiaaicdacaGGSaGaaGymaiaac2faaeaacaWG2bGaaiOoaiaadshacqWIMgsycaGGOaGaaGymaiabgkHiTiaadshacaGGSaGaaGimaiaacMcacaGGSaGaaGzbVlaadshacqGHiiIZcaGGBbGaaGimaiaacYcacaaIXaGaaiyxaaaaaaa@592F@

    Both of them produce the line segment between (0,0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaicdacaGGSaGaaGimaiaacMcaaaa@3969@ and (1,0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacaGGSaGaaGimaiaacMcaaaa@396A@ , but they differ in their running direction: Whereas w generates the line segment from left to right, v does it the reverse way.

    Also, running through a curve repeatedly will change the path but not the curve. If we substitute in the following example the interval [0,2π] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGOmaiabec8aWjaac2faaaa@3B8F@ by [0,2kπ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGOmaiaadUgacqaHapaCcaGGDbaaaa@3C7F@ the ellipse will be cycled k-times. But this of course won't alter its shape.

    Finally we note that cutting constant segments (i.e. w (t)=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4DayaafaGaaiikaiaadshacaGGPaGaeyypa0JaaGimaaaa@3B06@ for all t[r,s][a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgIGiolaacUfacaWGYbGaaiilaiaadohacaGGDbGaeyOGIWSaai4waiaadggacaGGSaGaamOyaiaac2faaaa@4301@ ) will leave the curve untouched, but not the path.

     

Example:  

  • With a,b>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyOpa4JaaGimaaaa@3A2B@ the ellipse

    t(acost,bsint),t[0,2π] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaacIcacaWGHbGaeyyXICTaci4yaiaac+gacaGGZbGaamiDaiaacYcacaWGIbGaeyyXICTaci4CaiaacMgacaGGUbGaamiDaiaacMcacaGGSaGaaGzbVlaadshacqGHiiIZcaGGBbGaaGimaiaacYcacaaIYaGaeqiWdaNaaiyxaaaa@5303@

    is a closed C MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaeyOhIukaaaaa@3852@ -path in 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3845@ . The sketch shows the associated curve for a=2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaaikdaaaa@3894@ and b=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2da9iaaigdaaaa@3894@ .

     

  • The C MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaeyOhIukaaaaa@3852@ -path

    t( t 2 1, t 3 t),t[ 2 , 2 ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaacIcacaWG0bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGymaiaacYcacaWG0bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamiDaiaacMcacaGGSaGaaGzbVlaadshacqGHiiIZcaGGBbGaeyOeI0YaaOaaaeaacaaIYaaaleqaaOGaaiilamaakaaabaGaaGOmaaWcbeaakiaac2faaaa@4BE8@

    is not closed and as 1(0,0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiablAAiHjaacIcacaaIWaGaaiilaiaaicdacaGGPaaaaa@3BDD@ and 1(0,0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaGymaiablAAiHjaacIcacaaIWaGaaiilaiaaicdacaGGPaaaaa@3CCA@ the curve runs through (0,0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaicdacaGGSaGaaGimaiaacMcaaaa@3969@ twice.

     

  • Lissajous-curves are generated by the following C MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaeyOhIukaaaaa@3852@ -paths:

    t(asin(nt+c),bsint),t[0,2π] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaacIcacaWGHbGaeyyXICTaci4CaiaacMgacaGGUbGaaiikaiaad6gacaWG0bGaey4kaSIaam4yaiaacMcacaGGSaGaamOyaiabgwSixlGacohacaGGPbGaaiOBaiaadshacaGGPaGaaiilaiaaywW7caWG0bGaeyicI4Saai4waiaaicdacaGGSaGaaGOmaiabec8aWjaac2faaaa@571E@

    The sketch shows the Lissajous-curve for a=b=c=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaadkgacqGH9aqpcaWGJbGaeyypa0JaaGymaaaa@3C6E@ and n=3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaaiodaaaa@38A2@ . Lissajou-curves are not always closed.

     

  • The spiral

     i

    left mouse: rotateright mouse: context menu

    spiral
    t(tcost,tsint,t),t[0,20] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaacIcacaWG0bGaeyyXICTaci4yaiaac+gacaGGZbGaamiDaiaacYcacaWG0bGaeyyXICTaci4CaiaacMgacaGGUbGaamiDaiaacYcacaWG0bGaaiykaiaacYcacaaMf8UaamiDaiabgIGiolaacUfacaaIWaGaaiilaiaaikdacaaIWaGaaiyxaaaa@53CE@

    Display by JavaView

    and the closed curve of Viviani

     i

    left mouse: rotateright mouse: context menu

    curve of Viviani
    t(1+cost,sint,2sin( t 2 )),t[0,4π] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaacIcacaaIXaGaey4kaSIaci4yaiaac+gacaGGZbGaamiDaiaacYcaciGGZbGaaiyAaiaac6gacaWG0bGaaiilaiaaikdacqGHflY1ciGGZbGaaiyAaiaac6gacaGGOaWaaSaaaeaacaWG0baabaGaaGOmaaaacaGGPaGaaiykaiaacYcacaaMf8UaamiDaiabgIGiolaacUfacaaIWaGaaiilaiaaisdacqaHapaCcaGGDbaaaa@57ED@

    Display by JavaView

    are examples for C MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaeyOhIukaaaaa@3852@ -curves in 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3846@ .

  • The Famous Curves Index is a large collection of curves.

Some basic derivation rules and and a version of the mean value thoerem are valid for differentiable paths.

Proposition:  

  1. For any differential paths v,w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3bGaaiOoaiaacUfacaWGHbGaaiilaiaadkgacaGGDbGaeyOKH4QaeSyhHe6aaWbaaSqabeaacaWGRbaaaaaa@4208@ each linear combination αv+βw:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaamODaiabgUcaRiabek7aIjaadEhacaGG6aGaai4waiaadggacaGGSaGaamOyaiaac2facqGHsgIRcqWIDesOdaahaaWcbeqaaiaadUgaaaaaaa@457A@ is differentiable and

(αv+βw ) =α v +β w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeg7aHjaadAhacqGHRaWkcqaHYoGycaWG3bGabiykayaafaGaeyypa0JaeqySdeMabmODayaafaGaey4kaSIaeqOSdiMabm4Dayaafaaaaa@44A1@
[8.6.4]
  1. If w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@405D@ is a differentiable path then for any two different points r,s[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacYcacaWGZbGaeyicI4Saai4waiaadggacaGGSaGaamOyaiaac2faaaa@3E4C@ there are numbers t ˜ 1 ,, t ˜ k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaiaWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcaceWG0bGbaGaadaWgaaWcbaGaam4Aaaqabaaaaa@3C8B@   between

     i

    i.e.  t ˜ i ]r,s[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaiaWaaSbaaSqaaiaadMgaaeqaaOGaeyicI4SaaiyxaiaadkhacaGGSaGaam4CaiaacUfaaaa@3DFB@ , if r<s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabgYda8iaadohaaaa@38DF@

    and  t ˜ i ]s,r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaiaWaaSbaaSqaaiaadMgaaeqaaOGaeyicI4SaaiyxaiaadohacaGGSaGaamOCaiaacUfaaaa@3DFB@ , if s<r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiabgYda8iaadkhaaaa@38DF@ .

    r and s such that

w(s)=w(r)+(sr)( w 1 ( t ˜ 1 ),, w k ( t ˜ k )) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWGZbGaaiykaiabg2da9iaadEhacaGGOaGaamOCaiaacMcacqGHRaWkcaGGOaGaam4CaiabgkHiTiaadkhacaGGPaGaeyyXICTaaiikaiqadEhagaqbamaaBaaaleaacaaIXaaabeaakiaacIcaceWG0bGbaGaadaWgaaWcbaGaaGymaaqabaGccaGGPaGaaiilaiablAciljaacYcaceWG3bGbauaadaWgaaWcbaGaam4AaaqabaGccaGGOaGabmiDayaaiaWaaSbaaSqaaiaadUgaaeqaaOGaaiykaiaacMcaaaa@53C7@
[8.6.5]
  1. If w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@405D@ is a differentiable path the derivative w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaafaaaaa@36F4@ is integrable and

r s w =w(s)w(r) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaceWG3bGbauaaaSqaaiaadkhaaeaacaWGZbaaniabgUIiYdGccqGH9aqpcaWG3bGaaiikaiaadohacaGGPaGaeyOeI0Iaam4DaiaacIcacaWGYbGaaiykaaaa@43E2@
[8.6.6]

for all r,s[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacYcacaWGZbGaeyicI4Saai4waiaadggacaGGSaGaamOyaiaac2faaaa@3E4C@ .

Proof:  

1.   According to the sum and factor rule all coordinate functions α v i +β w i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiabgUcaRiabek7aIjaadEhadaWgaaWcbaGaamyAaaqabaaaaa@3E43@ are differentiable and

(α v i +β w i ) =α v i '+β w i ' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeg7aHjaadAhadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcqaHYoGycaWG3bWaaSbaaSqaaiaadMgaaeqaaOGabiykayaafaGaeyypa0JaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiaacEcacqGHRaWkcqaHYoGycaWG3bWaaSbaaSqaaiaadMgaaeqaaOGaai4jaaaa@4A6F@

2.   Let w i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGPbaabeaaaaa@3802@ be an arbitrary coordinate function. The mean value theorem [7.9.5] provides a t ˜ i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaiaWaaSbaaSqaaiaadMgaaeqaaaaa@380E@ between r and s such that

w i (s)= w i (r)+(sr) w i '( t ˜ i )= w i (r)+(sr) w i ( t ˜ i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGPbaabeaakiaacIcacaWGZbGaaiykaiabg2da9iaadEhadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamOCaiaacMcacqGHRaWkcaGGOaGaam4CaiabgkHiTiaadkhacaGGPaGaeyyXICTaam4DamaaBaaaleaacaWGPbaabeaakiaacEcacaGGOaGabmiDayaaiaWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiabg2da9iaadEhadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamOCaiaacMcacqGHRaWkcaGGOaGaam4CaiabgkHiTiaadkhacaGGPaGaeyyXICTabm4DayaafaWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiqadshagaacamaaBaaaleaacaWGPbaabeaakiaacMcaaaa@600C@

3.   Every coordinate function w i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGPbaabeaaaaa@3802@ is certainly a primitive for w i '= w i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGPbaabeaakiaacEcacqGH9aqpceWG3bGbauaadaWgaaWcbaGaamyAaaqabaaaaa@3BDF@ .

The integral of a path satisfies the same calaculation rules and has the same properties just as a usual integral.

Proposition:  If v,w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3bGaaiOoaiaacUfacaWGHbGaaiilaiaadkgacaGGDbGaeyOKH4QaeSyhHe6aaWbaaSqabeaacaWGRbaaaaaa@4208@ are integrable paths we have for all r,s,t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacYcacaWGZbGaaiilaiaadshacqGHiiIZcaGGBbGaamyyaiaacYcacaWGIbGaaiyxaaaa@3FF5@ :

  1. r r w = 0 , r s w = s r w , r s w = r t w + t s w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWG3baaleaacaWGYbaabaGaamOCaaqdcqGHRiI8aOGaeyypa0JaaGimaiaacYcacaaMf8+aa8qCaeaacaWG3baaleaacaWGYbaabaGaam4CaaqdcqGHRiI8aOGaeyypa0JaeyOeI0Yaa8qCaeaacaWG3baaleaacaWGZbaabaGaamOCaaqdcqGHRiI8aOGaaiilaiaaywW7daWdXbqaaiaadEhaaSqaaiaadkhaaeaacaWGZbaaniabgUIiYdGccqGH9aqpdaWdXbqaaiaadEhaaSqaaiaadkhaaeaacaWG0baaniabgUIiYdGccqGHRaWkdaWdXbqaaiaadEhaaSqaaiaadshaaeaacaWGZbaaniabgUIiYdaaaa@602F@

[8.6.7]
  1. r s αv+βw =α r s v +β r s w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacqaHXoqycaWG2bGaey4kaSIaeqOSdiMaam4DaaWcbaGaamOCaaqaaiaadohaa0Gaey4kIipakiabg2da9iabeg7aHnaapehabaGaamODaaWcbaGaamOCaaqaaiaadohaa0Gaey4kIipakiabgUcaRiabek7aInaapehabaGaam4DaaWcbaGaamOCaaqaaiaadohaa0Gaey4kIipaaaa@5040@

[8.6.8]
  1. If r and s are different there are numbers t ˜ 1 ,, t ˜ k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaiaWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcaceWG0bGbaGaadaWgaaWcbaGaam4Aaaqabaaaaa@3C8B@ between r and s such that

r s w =(sr)( w 1 ( t ˜ 1 ),, w k ( t ˜ k )) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWG3baaleaacaWGYbaabaGaam4CaaqdcqGHRiI8aOGaeyypa0JaaiikaiaadohacqGHsislcaWGYbGaaiykaiabgwSixlaacIcacaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiqadshagaacamaaBaaaleaacaaIXaaabeaakiaacMcacaGGSaGaeSOjGSKaaiilaiaadEhadaWgaaWcbaGaam4AaaqabaGccaGGOaGabmiDayaaiaWaaSbaaSqaaiaadUgaaeqaaOGaaiykaiaacMcaaaa@5192@
[8.6.9]

Proof:  We prove the respective identity coordinatewise and succeed in

1.    with [8.2.2] - [8.2.4].

2.    with [8.2.5]/[8.2.7].

3.    with [8.2.8].

The estimate [8.2.11] is transferable to continuous paths, an essential step in our theory. The notation and the proof as well use the common dot product

 i

The dot product

x·y i=1 k x i y i ,x,y k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabl+y6NjaadMhacqGH9aqpdaaeWbqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHflY1caWG5bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaam4AaaqdcqGHris5aOGaaGPaVlaacYcacaaMf8UaamiEaiaacYcacaWG5bGaeyicI4SaeSyhHe6aaWbaaSqabeaacaWGRbaaaaaa@5245@

in particular provides the length  |x|= x·x = i=1 k x i 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacaGG8bGaeyypa0ZaaOaaaeaacaWG4bGaeS4JPFMaamiEaaWcbeaakiabg2da9maakaaabaWaaabCaeaacaWG4bWaa0baaSqaaiaadMgaaeaacaaIYaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaam4AaaqdcqGHris5aaWcbeaakiaaykW7aaa@49DE@ of a vector x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@3AFA@ . Our proof uses the simple identity

|x | 2 =x·x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacaGG8bWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaamiEaiabl+y6NjaadIhaaaa@3F4C@ ,

and the Cauchy-Schwarz inequality

|x·y||x||y| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqWIpM+zcaWG5bGaaiiFaiabgsMiJkaacYhacaWG4bGaaiiFaiabgwSixlaacYhacaWG5bGaaiiFaaaa@4651@

Its proof and further properties of the dot product, as e.g. the triangle inequality

|x+y||x|+|y| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHRaWkcaWG5bGaaiiFaiabgsMiJkaacYhacaWG4bGaaiiFaiabgUcaRiaacYhacaWG5bGaaiiFaaaa@435B@

are to be found in part 9.13.

in k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGRbaaaaaa@3879@ .

Proposition:  For any continuous path w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@405D@ the following inequality holds:

| a b w | a b |w| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaapehabaGaam4DaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipakiaacYhacqGHKjYOdaWdXbqaaiaacYhacaWG3bGaaiiFaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipaaaa@460F@
[8.6.10]

Proof:  First we note that the paths w and |w| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadEhacaGG8baaaa@38E8@ are continuous and thus integrable. With the abbreviation

c a b w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2da9maapehabaGaam4DaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipaaaa@3D0C@ ,  thus  c i = a b w i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBaaaleaacaWGPbaabeaakiabg2da9maapehabaGaam4DamaaBaaaleaacaWGPbaabeaaaeaacaWGHbaabaGaamOyaaqdcqGHRiI8aaaa@3F3F@

the following calculation is easier to follow. For |c|=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadogacaGG8bGaeyypa0JaaGimaaaa@3A94@ there is nothing to prove because the right side of [8.6.10] is always positive. So we may assume |c|0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadogacaGG8bGaeyiyIKRaaGimaaaa@3B55@ . According to the Cauchy-Schwarz inequality and due to the integral monotony [8.2.10] we may estimate as follows:

|c|| a b w |=|c | 2 =c·c= i=1 k c i a b w i = a b i=1 k c i w i = a b c·w a b |c·w| a b |c||w| =|c| a b |w| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9EC6@

We are now going to calculate the length of a continuous path w. The actual key concept in calculating areas then was to approximate the region in question by a sequence of elementary regions (union of rectangles) with a known area. Analogously we will now try to approximate the path w by elementary paths, namely the union of line segments.

We start with some technical preparations: A finite sequence Z=( t 0 ,, t n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwaiabg2da9iaacIcacaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiablAciljaacYcacaWG0bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaaaa@3FB7@ in [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ , n1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaaigdaaaa@3960@ , such that

a= t 0 < t 1 << t n1 < t n =b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaadshadaWgaaWcbaGaaGimaaqabaGccqGH8aapcaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaeyipaWJaeSOjGSKaeyipaWJaamiDamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGH8aapcaWG0bWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamOyaaaa@48B6@

is called a partition of the interval [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ with the number max{ t 1 t 0 ,, t n t n1 } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacggacaGG4bGaai4EaiaadshadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiablAciljaacYcacaWG0bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamiDamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccaGG9baaaa@48DB@ being its fineness.

For a given path w any partition Z features  n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgUcaRiaaigdaaaa@387C@ points w( t 0 ),,w( t n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiykaiaacYcacqWIMaYscaGGSaGaam4DaiaacIcacaWG0bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaaaa@4123@ of w and thus n consecutive line segments. We use them to create the traverse   p Z MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBaaaleaacaWGAbaabeaaaaa@37EC@ :

p Z (t)=w( t i1 )+ t t i1 t i t i1 (w( t i )w( t i1 )) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBaaaleaacaWGAbaabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadEhacaGGOaGaamiDamaaBaaaleaacaWGPbGaeyOeI0IaaGymaaqabaGccaGGPaGaey4kaSYaaSaaaeaacaWG0bGaeyOeI0IaamiDamaaBaaaleaacaWGPbGaeyOeI0IaaGymaaqabaaakeaacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaamiDamaaBaaaleaacaWGPbGaeyOeI0IaaGymaaqabaaaaOGaaiikaiaadEhacaGGOaGaamiDamaaBaaaleaacaWGPbaabeaakiaacMcacqGHsislcaWG3bGaaiikaiaadshadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiykaiaacMcaaaa@5BA6@ ,  if t[ t i1 , t i ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgIGiolaacUfacaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacYcacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaaiyxaaaa@40BB@
[8.6.11]

Each traverse p Z MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBaaaleaacaWGAbaabeaaaaa@37EC@ is a path with initial point w(a)=w( t 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWGHbGaaiykaiabg2da9iaadEhacaGGOaGaamiDamaaBaaaleaacaaIWaaabeaakiaacMcaaaa@3E6B@ and final point w(b)=w( t n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWGIbGaaiykaiabg2da9iaadEhacaGGOaGaamiDamaaBaaaleaacaWGUbaabeaakiaacMcaaaa@3EA5@ . Traverses approximate a continuous path w in the following way:

Proposition:  Let w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@405D@ be a continuous path. Then there is δ>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaeyOpa4JaaGimaaaa@3953@ for each ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ such that

|w(t) p Z (t)|<ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadEhacaGGOaGaamiDaiaacMcacqGHsislcaWGWbWaaSbaaSqaaiaadQfaaeqaaOGaaiikaiaadshacaGGPaGaaiiFaiabgYda8iabew7aLbaa@432E@   for all t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@3CA6@ .
[8.6.12]

for any partition Z with a fineness less than δ.

Proof:  Take an arbitrary ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ . As each coordinate function w j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGQbaabeaaaaa@3803@ is continuous function on the closed interval [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ it is actually uniformly continuous (see [6.5.5]). Thus there is a δ j >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaadQgaaeqaaOGaeyOpa4JaaGimaaaa@3A78@ for ε 2 k >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaH1oqzaeaacaaIYaWaaOaaaeaacaWGRbaaleqaaaaakiabg6da+iaaicdaaaa@3B36@ such that

s,t[a,b]|st|< δ j | w j (s) w j (t) | 2 < ε 2 4k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacYcacaWG0bGaeyicI4Saai4waiaadggacaGGSaGaamOyaiaac2facaaMf8Uaey4jIKTaaGzbVlaacYhacaWGZbGaeyOeI0IaamiDaiaacYhacqGH8aapcqaH0oazdaWgaaWcbaGaamOAaaqabaGccaaMf8UaeyO0H4TaaGzbVlaacYhacaWG3bWaaSbaaSqaaiaadQgaaeqaaOGaaiikaiaadohacaGGPaGaeyOeI0Iaam4DamaaBaaaleaacaWGQbaabeaakiaacIcacaWG0bGaaiykaiaacYhadaahaaWcbeqaaiaaikdaaaGccqGH8aapdaWcaaqaaiabew7aLnaaCaaaleqabaGaaGOmaaaaaOqaaiaaisdacaWGRbaaaaaa@635E@

Setting δmin{ δ 1 ,, δ k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaeyypa0JaciyBaiaacMgacaGGUbGaai4Eaiabes7aKnaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaeqiTdq2aaSbaaSqaaiaadUgaaeqaaOGaaiyFaaaa@454C@ we therefor have for any s,t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacYcacaWG0bGaeyicI4Saai4waiaadggacaGGSaGaamOyaiaac2faaaa@3E4E@ with |st|<δ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadohacqGHsislcaWG0bGaaiiFaiabgYda8iabes7aKbaa@3D73@ :

|w(s)w(t)|= j=1 k | w j (s) w j (t) | 2 < k ε 2 4k = ε 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadEhacaGGOaGaam4CaiaacMcacqGHsislcaWG3bGaaiikaiaadshacaGGPaGaaiiFaiabg2da9maakaaabaWaaabCaeaacaGG8bGaam4DamaaBaaaleaacaWGQbaabeaakiaacIcacaWGZbGaaiykaiabgkHiTiaadEhadaWgaaWcbaGaamOAaaqabaGccaGGOaGaamiDaiaacMcacaGG8bWaaWbaaSqabeaacaaIYaaaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaam4AaaqdcqGHris5aaWcbeaakiabgYda8maakaaabaGaam4AamaalaaabaGaeqyTdu2aaWbaaSqabeaacaaIYaaaaaGcbaGaaGinaiaadUgaaaaaleqaaOGaeyypa0ZaaSaaaeaacqaH1oqzaeaacaaIYaaaaaaa@5D1B@ [1]

Now let Z=( t 0 ,, t n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwaiabg2da9iaacIcacaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiablAciljaacYcacaWG0bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaaaa@3FB7@ be an arbitrary partition with a fineness not exceeding δ. If t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@3CA6@ , say t[ t i1 , t i ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgIGiolaacUfacaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacYcacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaaiyxaaaa@40BB@ , we thus know:

|t t i1 |<δ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadshacqGHsislcaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacYhacqGH8aapcqaH0oazaaa@4040@  and  | t i t i1 |<δ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadshadaWgaaWcbaGaamyAaaqabaGccqGHsislcaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacYhacqGH8aapcqaH0oazaaa@4164@ .

Using [8.6.11], the triangle inequality and the estimate [1] we now see that

|w(t) p Z (t)| |w(t)w( t i1 )|+|w( t i1 ) p Z (t)| =|w(t)w( t i1 )|+ | t t i1 t i t i1 | 1 |w( t i )w( t i1 )| < ε 2 + ε 2 =ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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aiabgwSixlaacYhacaWG3bGaaiikaiaadshadaWgaaWcbaGaamyAaaqabaGccaGGPaGaeyOeI0Iaam4DaiaacIcacaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacMcacaGG8baabaaabaGaeyipaWZaaSaaaeaacqaH1oqzaeaacaaIYaaaaiabgUcaRmaalaaabaGaeqyTdugabaGaaGOmaaaacqGH9aqpcqaH1oqzaaaaaa@9757@

The length L( p Z ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWGWbWaaSbaaSqaaiaadQfaaeqaaOGaaiykaaaa@3A20@ of a traverse is easily calculated by adding up the lengths of the line segments involved. Thus we set

L( p Z )= i=1 n |w( t i )w( t i1 )| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWGWbWaaSbaaSqaaiaadQfaaeqaaOGaaiykaiabg2da9maaqahabaGaaiiFaiaadEhacaGGOaGaamiDamaaBaaaleaacaWGPbaabeaakiaacMcacqGHsislcaWG3bGaaiikaiaadshadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiykaiaacYhaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa@4E83@
[8.6.13]

As the line segment represents the shortest distance between w( t i1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacMcaaaa@3C06@ and w( t i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacIcacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@3A5E@ we expect the length of w to be an upper bound for all L( p Z ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWGWbWaaSbaaSqaaiaadQfaaeqaaOGaaiykaaaa@3A20@ . Considering [8.6.12], a sound candidate for the prospective path length would be the supremum

 i

We remember the completeness axiom :
In MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375C@ each non-empty bounded subset M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlabl2riHcaa@3A2A@ has a least upper bound, its supremum (termed supM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacwhacaGGWbGaamytaaaa@39A4@ ).

An upper bound s of M equals supM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacwhacaGGWbGaamytaaaa@39A4@ if and only if sε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiabgkHiTiabew7aLbaa@3978@ is no upper bound of M for each ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ .

of {L( p Z )|Zpartition of[a,b]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadYeacaGGOaGaamiCamaaBaaaleaacaWGAbaabeaakiaacMcacaGG8bGaamOwaiaabMgacaqGZbGaaeiDaiaabccacaqGAbGaaeyzaiaabkhacaqGSbGaaeyzaiaabEgacaqG1bGaaeOBaiaabEgacaqGGaGaaeODaiaab+gacaqGUbGaai4waiaadggacaGGSaGaamOyaiaac2facaGG9baaaa@5185@ .

The depicted example on the right shows a part of one of RenĂ© Descartes'  Folium of Descartes, that is the path  t( 12t 1+ t 3 , 12 t 2 1+ t 3 ),  t[0.4,10] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaacIcadaWcaaqaaiaaigdacaaIYaGaamiDaaqaaiaaigdacqGHRaWkcaWG0bWaaWbaaSqabeaacaaIZaaaaaaakiaacYcadaWcaaqaaiaaigdacaaIYaGaamiDamaaCaaaleqabaGaaGOmaaaaaOqaaiaaigdacqGHRaWkcaWG0bWaaWbaaSqabeaacaaIZaaaaaaakiaacMcacaaMc8UaaiilaiaaywW7caWG0bGaeyicI4Saai4waiabgkHiTiaaicdacaGGUaGaaGinaiaacYcacaaIXaGaaGimaiaac2faaaa@54F0@ . For a three dimensional example we replenish the spiral

 i

left mouse: rotateright mouse: context menu

spiral
t(tcost,tsint,t),t[0,20] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaacIcacaWG0bGaeyyXICTaci4yaiaac+gacaGGZbGaamiDaiaacYcacaWG0bGaeyyXICTaci4CaiaacMgacaGGUbGaamiDaiaacYcacaWG0bGaaiykaiaacYcacaaMf8UaamiDaiabgIGiolaacUfacaaIWaGaaiilaiaaikdacaaIWaGaaiyxaaaa@53CE@

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from a former example.

Definition: A continuous path w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@405D@ is a path of finite length (or a rectifiable path) if the set {L( p Z )|Zpartition of[a,b]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadYeacaGGOaGaamiCamaaBaaaleaacaWGAbaabeaakiaacMcacaGG8bGaamOwaiaabMgacaqGZbGaaeiDaiaabccacaqGAbGaaeyzaiaabkhacaqGSbGaaeyzaiaabEgacaqG1bGaaeOBaiaabEgacaqGGaGaaeODaiaab+gacaqGUbGaai4waiaadggacaGGSaGaamOyaiaac2facaGG9baaaa@5185@ is bounded. In this case the number

L(w)sup{L( p Z )|Zpartition of[a,b]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWG3bGaaiykaiabg2da9iGacohacaGG1bGaaiiCaiaacUhacaWGmbGaaiikaiaadchadaWgaaWcbaGaamOwaaqabaGccaGGPaGaaiiFaiaadQfacaqGPbGaae4CaiaabshacaqGGaGaaeOwaiaabwgacaqGYbGaaeiBaiaabwgacaqGNbGaaeyDaiaab6gacaqGNbGaaeiiaiaabAhacaqGVbGaaeOBaiaacUfacaWGHbGaaiilaiaadkgacaGGDbGaaiyFaaaa@5897@
[8.6.14]

is called the length of w.

We will find out that smooth paths are rectifiable and, in addition, integrating the length of their derivative w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaafaaaaa@36F4@ will provide an upper bound for all L( p Z ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWGWbWaaSbaaSqaaiaadQfaaeqaaOGaaiykaaaa@3A20@ . Smoothness however is only a sufficient and not a necessary condition for a finite length, as the non-smooth path  w:t(t,|t|),   t[1,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaWG0bGaeSOPHeMaaiikaiaadshacaGGSaGaaiiFaiaadshacaGG8bGaaiykaiaacYcacaaMe8UaaGjbVlaadshacqGHiiIZcaGGBbGaeyOeI0IaaGymaiaacYcacqGHsislcaaIXaGaaiyxaaaa@4C5A@

 i

If Z=( t 0 ,, t n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwaiabg2da9iaacIcacaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiablAciljaacYcacaWG0bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaaaa@3FB7@ is a partition of [1,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgkHiTiaaigdacaGGSaGaaGymaiaac2faaaa@3ABF@ we find a suitable k such that t k 0< t k+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBaaaleaacaWGRbaabeaakiabgsMiJkaaicdacqGH8aapcaWG0bWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaaaaa@3F30@ . That means:

(| t i || t i1 |) 2 ={ ( t i t i1 ) 2 ,  if  i>k+1 ( t i + t i1 ) 2 ,  if  ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6B01@

and so we have (| t i || t i1 |) 2 = ( t i t i1 ) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacYhacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaaiiFaiabgkHiTiaacYhacaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacYhacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaiikaiaadshadaWgaaWcbaGaamyAaaqabaGccqGHsislcaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaa@4D1E@ and consequently

|w( t i )w( t i1 )|= ( t i t i1 ) 2 + (| t i || t i1 |) 2 = 2 ( t i t i1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadEhacaGGOaGaamiDamaaBaaaleaacaWGPbaabeaakiaacMcacqGHsislcaWG3bGaaiikaiaadshadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiykaiaacYhacqGH9aqpdaGcaaqaaiaacIcacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaamiDamaaBaaaleaacaWGPbGaeyOeI0IaaGymaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaiikaiaacYhacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaaiiFaiabgkHiTiaacYhacaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacYhacaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaGccqGH9aqpdaGcaaqaaiaaikdaaSqabaGccaGGOaGaamiDamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadshadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiykaaaa@65A2@

for all ik+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgcMi5kaadUgacqGHRaWkcaaIXaaaaa@3B2E@ . Taking into account that

|w( t k+1 )w(0)|= t k+1 2 +| t k+1 | 2 = 2 | t k+1 |= 2 t k+1 |w(0)w( t k )|= t k 2 +| t k | 2 = 2 | t k |= 2 t k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7EC6@

we now may estimate L( p Z ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWGWbWaaSbaaSqaaiaadQfaaeqaaOGaaiykaaaa@3A20@ as follows:

L( p Z ) = i=1 n |w( t i )w( t i1 )| i=1 k |w( t i )w( t i1 )| +|w( t k+1 )w(0)|+|w(0)w( t k )| + i=k+2 n |w( t i )w( t i1 )| = 2 ( i=1 k t i t i1 + t k+1 t k + i=k+2 n t i t i1 ) = 2 ( t n t 0 )=2 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C3DB@
  proves.

Proposition:  If w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@405D@ is a smooth path then for any partition Z of [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ the estimate

L( p Z ) a b | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWGWbWaaSbaaSqaaiaadQfaaeqaaOGaaiykaiabgsMiJoaapehabaGaaiiFaiqadEhagaqbaiaacYhaaSqaaiaadggaaeaacaWGIbaaniabgUIiYdaaaa@4313@
[8.6.15]

holds and w is thus rectifiable.

Proof:  If Z=( t 0 ,, t n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwaiabg2da9iaacIcacaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiablAciljaacYcacaWG0bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaaaa@3FB7@ is an arbitrary partition we see from the properties [8.6.6], [8.6.7] and the estimate [8.6.10] that

L( p Z )= i=1 n |w( t i )w( t i1 )| = i=1 n | t i1 t i w | i=1 n t i1 t i | w | = a b | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWGWbWaaSbaaSqaaiaadQfaaeqaaOGaaiykaiabg2da9maaqahabaGaaiiFaiaadEhacaGGOaGaamiDamaaBaaaleaacaWGPbaabeaakiaacMcacqGHsislcaWG3bGaaiikaiaadshadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiykaiaacYhaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaeyypa0ZaaabCaeaacaGG8bWaa8qCaeaaceWG3bGbauaaaSqaaiaadshadaWgaaadbaGaamyAaiabgkHiTiaaigdaaeqaaaWcbaGaamiDamaaBaaameaacaWGPbaabeaaa0Gaey4kIipakiaacYhaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaeyizIm6aaabCaeaadaWdXbqaaiaacYhaceWG3bGbauaaaSqaaiaadshadaWgaaadbaGaamyAaiabgkHiTiaaigdaaeqaaaWcbaGaamiDamaaBaaameaacaWGPbaabeaaa0Gaey4kIipakiaacYhaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaeyypa0Zaa8qCaeaacaGG8bGabm4DayaafaaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaaiiFaaaa@7C1E@ .

Surprisingly, [8.6.15] could be replaced by a stronger result: The integral of | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadEhagaqbaiaacYhaaaa@38F4@ is more than a common upper bound of {L( p Z )|Zpartition of[a,b]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadYeacaGGOaGaamiCamaaBaaaleaacaWGAbaabeaakiaacMcacaGG8bGaamOwaiaabMgacaqGZbGaaeiDaiaabccacaqGAbGaaeyzaiaabkhacaqGSbGaaeyzaiaabEgacaqG1bGaaeOBaiaabEgacaqGGaGaaeODaiaab+gacaqGUbGaai4waiaadggacaGGSaGaamOyaiaac2facaGG9baaaa@5185@ , it is the least one and thus the length of w.

Proposition:  For the length of a smooth path w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@405D@ we have:

L(w)= a b | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWG3bGaaiykaiabg2da9maapehabaGaaiiFaiqadEhagaqbaiaacYhaaSqaaiaadggaaeaacaWGIbaaniabgUIiYdaaaa@4156@
[8.6.16]

Proof:  Due to [8.6.15] the integral a b | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGG8bGabm4DayaafaGaaiiFaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipaaaa@3D2A@ is an upper bound of {L( p Z )|Z  partition of  [a,b]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadYeacaGGOaGaamiCamaaBaaaleaacaWGAbaabeaakiaacMcacaGG8bGaamOwaiaabMgacaqGZbGaaeiDaiaabccacaqGAbGaaeyzaiaabkhacaqGSbGaaeyzaiaabEgacaqG1bGaaeOBaiaabEgacaqGGaGaaeODaiaab+gacaqGUbGaai4waiaadggacaGGSaGaamOyaiaac2facaGG9baaaa@5185@ . It is thus sufficient to find a partition Z for each ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ such that

a b | w | εL( p Z ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGG8bGabm4DayaafaGaaiiFaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipakiabgkHiTiabew7aLjabgsMiJkaadYeacaGGOaGaamiCamaaBaaaleaacaWGAbaabeaakiaacMcaaaa@45B1@ .

As the coordinate functions w j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4DayaafaWaaSbaaSqaaiaadQgaaeqaaaaa@380F@ are continuously differentiable, each ( w j ) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadEhagaqbamaaBaaaleaacaWGQbaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaa@3A5B@ is continuous and on the closed interval [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ even uniformly continuous. Thus there is a δ j >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaadQgaaeqaaOGaeyOpa4JaaGimaaaa@3A78@ for ε ¯ ε 2 k (ba) 2 >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaebacqGH9aqpdaWcaaqaaiabew7aLnaaCaaaleqabaGaaGOmaaaaaOqaaiaadUgacaGGOaGaamOyaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaaIYaaaaaaakiabg6da+iaaicdaaaa@4313@   such that

|ts|< δ j ( w j (t)) 2 ( w j (s)) 2 | ( w j (t)) 2 ( w j (s)) 2 |< ε ¯ ( w j (t)) 2 < ( w j (s)) 2 + ε ¯ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7E4F@ [2]

for all s,t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacYcacaWG0bGaeyicI4Saai4waiaadggacaGGSaGaamOyaiaac2faaaa@3E4E@ . Now take a partition Z=( t 0 ,, t n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwaiabg2da9iaacIcacaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiablAciljaacYcacaWG0bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaaaa@3FB7@ with a fineness less than δmin{ δ 1 ,, δ k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaeyypa0JaciyBaiaacMgacaGGUbGaai4Eaiabes7aKnaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaeqiTdq2aaSbaaSqaaiaadUgaaeqaaOGaaiyFaaaa@454C@ . According to [8.2.8] and [8.6.5] resp. we find numbers

  • y ˜ i ] t i1 , t i [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaiaWaaSbaaSqaaiaadMgaaeqaaOGaeyicI4SaaiyxaiaadshadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiilaiaadshadaWgaaWcbaGaamyAaaqabaGccaGGBbaaaa@41F3@  such that  a b | w | = i=1 n t i1 t i | w | = i=1 n ( t i t i1 )| w ( y ˜ i )| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6762@ [3]

  • x ˜ 1,i ,, x ˜ k,i ] t i1 , t i [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaiaWaaSbaaSqaaiaaigdacaGGSaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiqadIhagaacamaaBaaaleaacaWGRbGaaiilaiaadMgaaeqaaOGaeyicI4SaaiyxaiaadshadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiilaiaadshadaWgaaWcbaGaamyAaaqabaGccaGGBbaaaa@49AF@  such that 

    L( p Z )= i=1 n |w( t i )w( t i1 )| = i=1 n ( t i t i1 )| w 1 ( x ˜ 1,i ),, w k ( x ˜ k,i )| = i=1 n ( t i t i1 ) j=1 k ( w j ( x ˜ j,i )) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9158@ [4]

As  y ˜ i , x ˜ 1,i ,, x ˜ k,i ] t i1 , t i [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaiaWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiqadIhagaacamaaBaaaleaacaaIXaGaaiilaiaadMgaaeqaaOGaaiilaiablAciljaacYcaceWG4bGbaGaadaWgaaWcbaGaam4AaiaacYcacaWGPbaabeaakiabgIGiolaac2facaWG0bWaaSbaaSqaaiaadMgacqGHsislcaaIXaaabeaakiaacYcacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaai4waaaa@4C90@ we have for all i:

| y ˜ i x ˜ j,i |<δ δ j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadMhagaacamaaBaaaleaacaWGPbaabeaakiabgkHiTiqadIhagaacamaaBaaaleaacaWGQbGaaiilaiaadMgaaeqaaOGaaiiFaiabgYda8iabes7aKjabgsMiJkabes7aKnaaBaaaleaacaWGQbaabeaaaaa@45F7@ .

With [2] we thus know that ( w j ( y ˜ i )) 2 < ( w j ( x ˜ j,i )) 2 + ε ¯ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadEhagaqbamaaBaaaleaacaWGQbaabeaakiaacIcaceWG5bGbaGaadaWgaaWcbaGaamyAaaqabaGccaGGPaGaaiykamaaCaaaleqabaGaaGOmaaaakiabgYda8iaacIcaceWG3bGbauaadaWgaaWcbaGaamOAaaqabaGccaGGOaGabmiEayaaiaWaaSbaaSqaaiaadQgacaGGSaGaamyAaaqabaGccaGGPaGaaiykamaaCaaaleqabaGaaGOmaaaakiabgUcaRiqbew7aLzaaraaaaa@4B35@ and according to [3] and [4] we may estimate the integral of | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadEhagaqbaiaacYhaaaa@38F1@ as follows (note that x+y x + y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWG4bGaey4kaSIaamyEaaWcbeaakiabgsMiJoaakaaabaGaamiEaaWcbeaakiabgUcaRmaakaaabaGaamyEaaWcbeaaaaa@3DC0@ for any positive x, y):

a b | w | = i=1 n ( t i t i1 ) j=1 k ( w j ( y ˜ i )) 2 < i=1 n ( t i t i1 ) j=1 k ( ( w j ( x ˜ j,i )) 2 + ε ¯ ) i=1 n ( t i t i1 ) j=1 k ( w j ( x ˜ j,i )) 2 + i=1 n ( t i t i1 ) k ε ¯ =L( p Z )+(ba) k ε ¯ =L( p Z )+ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C2B2@

This proves the required estimate a b | w | εL( p Z ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGG8bGabm4DayaafaGaaiiFaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipakiabgkHiTiabew7aLjabgsMiJkaadYeacaGGOaGaamiCamaaBaaaleaacaWGAbaabeaakiaacMcaaaa@45B1@ . Therefor a b | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGG8bGabm4DayaafaGaaiiFaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipaaaa@3D2A@ is the least upper bound of {L( p Z )|Z  partition of  [a,b]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadYeacaGGOaGaamiCamaaBaaaleaacaWGAbaabeaakiaacMcacaGG8bGaamOwaiaabMgacaqGZbGaaeiDaiaabccacaqGAbGaaeyzaiaabkhacaqGSbGaaeyzaiaabEgacaqG1bGaaeOBaiaabEgacaqGGaGaaeODaiaab+gacaqGUbGaai4waiaadggacaGGSaGaamOyaiaac2facaGG9baaaa@5185@ , i.e. its supremum.

In a former comment paths were considered as means of describing the movement of a particle. With this perception [8.6.16] just says that the length of the path traced by the particle is the integral of its magnitude of speed | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadEhagaqbaiaacYhaaaa@38F4@ .

Although [8.6.16] could replace computing a supremum (usually a difficult task) with calculating an integral, the length of a path is rarely simple to get as the integrand is always the length of a vector, which actually is the root of a sum. The following examples will demonstrate this.

Example:  

  • The curve belonging to the smooth path

    w:ta+t(ba)=( a 1 +t( b 1 a 1 ),, a k +t( b k a k )),t[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaWG0bGaeSOPHeMaamyyaiabgUcaRiaadshacaGGOaGaamOyaiabgkHiTiaadggacaGGPaGaeyypa0JaaiikaiaadggadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWG0bGaaiikaiaadkgadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaacYcacqWIMaYscaGGSaGaamyyamaaBaaaleaacaWGRbaabeaakiabgUcaRiaadshacaGGOaGaamOyamaaBaaaleaacaWGRbaabeaakiabgkHiTiaadggadaWgaaWcbaGaam4AaaqabaGccaGGPaGaaiykaiaacYcacaaMf8UaamiDaiabgIGiolaacUfacaaIWaGaaiilaiaaigdacaGGDbaaaa@629A@

    is the line segment joining the points a=( a 1 , a k ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaacIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaadggadaWgaaWcbaGaam4AaaqabaGccaGGPaaaaa@3EE6@ and b=( b 1 , b k ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2da9iaacIcacaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaadkgadaWgaaWcbaGaam4AaaqabaGccaGGPaaaaa@3EE9@ in k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGRbaaaaaa@3879@ . With the constant derivative w =( b 1 a 1 ,, b k a k )=ba MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4DayaafaGaeyypa0JaaiikaiaadkgadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWGIbWaaSbaaSqaaiaadUgaaeqaaOGaeyOeI0IaamyyamaaBaaaleaacaWGRbaabeaakiaacMcacqGH9aqpcaWGIbGaeyOeI0Iaamyyaaaa@4937@ the length of w straight away calculates to

    L(w) = 0 1 |ba| =|ba| 0 1 1 =|ba| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaiaadYeacaGGOaGaam4DaiaacMcaaeaacqGH9aqpdaWdXbqaaiaacYhacaWGIbGaeyOeI0IaamyyaiaacYhaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdaakeaaaeaacqGH9aqpcaGG8bGaamOyaiabgkHiTiaadggacaGG8bWaa8qCaeaacaaIXaaaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaGcbaaabaGaeyypa0JaaiiFaiaadkgacqGHsislcaWGHbGaaiiFaaaaaaa@52EF@

    which, as expected, is the distance between a and b.


     
  • For an arbitrary a>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg6da+iaaicdaaaa@3894@ we compute the length of

    w:t(tcost,tsint,t),t[0,a] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaWG0bGaeSOPHeMaaiikaiaadshacqGHflY1ciGGJbGaai4BaiaacohacaWG0bGaaiilaiaadshacqGHflY1ciGGZbGaaiyAaiaac6gacaWG0bGaaiilaiaadshacaGGPaGaaiilaiaaywW7caWG0bGaeyicI4Saai4waiaaicdacaGGSaGaamyyaiaac2faaaa@54F8@

    thus measuring the spiral from our initial example:

    From w =(cosXsin,sin+Xcos,1) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4DayaafaGaeyypa0JaaiikaiGacogacaGGVbGaai4CaiabgkHiTiaadIfacqGHflY1ciGGZbGaaiyAaiaac6gacaGGSaGaci4CaiaacMgacaGGUbGaey4kaSIaamiwaiabgwSixlGacogacaGGVbGaai4CaiaacYcacaaIXaGaaiykaaaa@4EE1@ we get (note the Pythagorean theorem : sin 2 + cos 2 =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGymaaaa@4020@ )

    | w | = (cosXsin) 2 + (sin+Xcos) 2 + 1 2 = cos 2 2Xcossin+ X 2 sin 2 + sin 2 +2Xsincos+ X 2 cos 2 +1 = X 2 +2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaiaacYhaceWG3bGbauaacaGG8baabaGaeyypa0ZaaOaaaeaacaGGOaGaci4yaiaac+gacaGGZbGaeyOeI0IaamiwaiabgwSixlGacohacaGGPbGaaiOBaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGaci4CaiaacMgacaGGUbGaey4kaSIaamiwaiabgwSixlGacogacaGGVbGaai4CaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaWaaWbaaSqabeaacaaIYaaaaaqabaaakeaaaeaacqGH9aqpdaGcaaqaaiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWGybGaeyyXICTaci4yaiaac+gacaGGZbGaeyyXICTaci4CaiaacMgacaGGUbGaey4kaSIaamiwamaaCaaaleqabaGaaGOmaaaakiabgwSixlGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabgUcaRiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWGybGaeyyXICTaci4CaiaacMgacaGGUbGaeyyXICTaci4yaiaac+gacaGGZbGaey4kaSIaamiwamaaCaaaleqabaGaaGOmaaaakiabgwSixlGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdaaSqabaaakeaaaeaacqGH9aqpdaGcaaqaaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaaaleqaaaaaaaa@906B@

    To calculate the integral L(w)= 0 a X 2 +2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWG3bGaaiykaiabg2da9maapehabaWaaOaaaeaacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaaWcbeaaaeaacaaIWaaabaGaamyyaaqdcqGHRiI8aaaa@419F@ we need the hyperbolic functions sinh und cosh

     i

    Hyperbolic sine (sinus hyperbolicus) and hyperbolic cosine (cosinus hyperbolicus) are two functions based on the exponential function exp (see [5.9.18]). For x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHcaa@39DD@ we set
     

    sinhx exp(x)exp(x) 2 coshx exp(x)+exp(x) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiGacohacaGGPbGaaiOBaiaacIgacaWG4bGaeyypa0ZaaSaaaeaaciGGLbGaaiiEaiaacchacaGGOaGaamiEaiaacMcacqGHsislciGGLbGaaiiEaiaacchacaGGOaGaeyOeI0IaamiEaiaacMcaaeaacaaIYaaaaaqaaiGacogacaGGVbGaai4CaiaacIgacaWG4bGaeyypa0ZaaSaaaeaaciGGLbGaaiiEaiaacchacaGGOaGaamiEaiaacMcacqGHRaWkciGGLbGaaiiEaiaacchacaGGOaGaeyOeI0IaamiEaiaacMcaaeaacaaIYaaaaaaaaaa@5B86@ [0]
     
    sinh and cosh are arbitrary often differentiable and, as exp =exp MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyzaiaacIhacaGGWbGaai4jaiabg2da9iGacwgacaGG4bGaaiiCaaaa@3D53@ (see [7.5.8]), their derivatives are easily calculated to
     
    sinh =cosh MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiiAaiaacEcacqGH9aqpciGGJbGaai4BaiaacohacaGGObaaaa@3F20@  and  cosh =sinh MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAaiaacEcacqGH9aqpciGGZbGaaiyAaiaac6gacaGGObaaaa@3F20@ .
     
    The following properties are also straight forward from the definition [0]:
    • cosh+sinh=exp MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAaiabgUcaRiGacohacaGGPbGaaiOBaiaacIgacqGH9aqpciGGLbGaaiiEaiaacchaaaa@4232@

    • cosh 2 sinh 2 =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAamaaCaaaleqabaGaaGOmaaaakiabgkHiTiGacohacaGGPbGaaiOBaiaacIgadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIXaaaaa@4203@

    • cosh(x+y)=coshxcoshy+sinhxsinhy MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAaiaacIcacaWG4bGaey4kaSIaamyEaiaacMcacqGH9aqpciGGJbGaai4BaiaacohacaGGObGaamiEaiabgwSixlGacogacaGGVbGaai4CaiaacIgacaWG5bGaey4kaSIaci4CaiaacMgacaGGUbGaaiiAaiaadIhacqGHflY1ciGGZbGaaiyAaiaac6gacaGGObGaamyEaaaa@5759@

    • sinh(x+y)=sinhxcoshy+coshxsinhy MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiiAaiaacIcacaWG4bGaey4kaSIaamyEaiaacMcacqGH9aqpciGGZbGaaiyAaiaac6gacaGGObGaamiEaiabgwSixlGacogacaGGVbGaai4CaiaacIgacaWG5bGaey4kaSIaci4yaiaac+gacaGGZbGaaiiAaiaadIhacqGHflY1ciGGZbGaaiyAaiaac6gacaGGObGaamyEaaaa@575E@

    • cosh(x)=coshx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAaiaacIcacqGHsislcaWG4bGaaiykaiabg2da9iGacogacaGGVbGaai4CaiaacIgacaWG4baaaa@42B0@

    • sinh(x)=sinhx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiiAaiaacIcacqGHsislcaWG4bGaaiykaiabg2da9iabgkHiTiGacohacaGGPbGaaiOBaiaacIgacaWG4baaaa@43A7@

    sinh is injective (according to [7.9.6] as sinh (x)=coshx>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiiAaiaacEcacaGGOaGaamiEaiaacMcacqGH9aqpciGGJbGaai4BaiaacohacaGGObGaamiEaiabg6da+iaaicdaaaa@4435@ for all x) and surjective as well (a consequence of the intermediate value theorem [6.6.2], because sinh is continuous and lim x± sinh(x)=± MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHXcqScqGHEisPaeqaaOGaci4CaiaacMgacaGGUbGaaiiAaiaacIcacaWG4bGaaiykaiabg2da9iabgglaXkabg6HiLcaa@49C7@  ). Thus sinh is bijective and its inverse function

    arcsinh sinh 1 : MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaai4CaiaacMgacaGGUbGaaiiAaiabg2da9iGacohacaGGPbGaaiOBaiaacIgadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGG6aGaeSyhHeQaeyOKH4QaeSyhHekaaa@48A6@

    is called arcussinus hyberbolicus. In a similar way arcuscosinus hyperbolicus is introduced, the inverse function of cosh| 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAaiaacYhacqWIDesOdaahaaWcbeqaaiabgwMiZkaaicdaaaaaaa@3EC8@ :
     

    arccosh (cosh| 0 ) 1 : 1 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaai4yaiaac+gacaGGZbGaaiiAaiabg2da9iaacIcaciGGJbGaai4BaiaacohacaGGObGaaiiFaiabl2riHoaaCaaaleqabaGaeyyzImRaaGimaaaakiaacMcadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGG6aGaeSyhHe6aaWbaaSqabeaacqGHLjYScaaIXaaaaOGaeyOKH4QaeSyhHe6aaWbaaSqabeaacqGHLjYScaaIWaaaaaaa@5481@

    . As 1 2 (sinhcosh+X) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaci4CaiaacMgacaGGUbGaaiiAaiabgwSixlGacogacaGGVbGaai4CaiaacIgacqGHRaWkcaWGybGaaiykaaaa@4458@ is a primitive of cosh 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAamaaCaaaleqabaGaaGOmaaaaaaa@3A94@

     i

    We use the fundamental theorem [8.2.13] and employ integration by parts (see [8.3.1]) to compute for any x the integral

    0 x cosh 2 =sinhcosh | 0 x 0 x sinh 2 =sinh(x)cosh(x) 0 x cosh 2 + 0 x 1 2 0 x cosh 2 =sinh(x)cosh(x)+x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmWaaaqaaaqaamaapehabaGaci4yaiaac+gacaGGZbGaaiiAamaaCaaaleqabaGaaGOmaaaaaeaacaaIWaaabaGaamiEaaqdcqGHRiI8aaGcbaGaeyypa0Jaci4CaiaacMgacaGGUbGaaiiAaiabgwSixlGacogacaGGVbGaai4CaiaacIgacaGG8bWaa0baaSqaaiaaicdaaeaacaWG4baaaOGaeyOeI0Yaa8qCaeaaciGGZbGaaiyAaiaac6gacaGGObWaaWbaaSqabeaacaaIYaaaaaqaaiaaicdaaeaacaWG4baaniabgUIiYdaakeaaaeaaaeaacqGH9aqpciGGZbGaaiyAaiaac6gacaGGObGaaiikaiaadIhacaGGPaGaeyyXICTaci4yaiaac+gacaGGZbGaaiiAaiaacIcacaWG4bGaaiykaiabgkHiTmaapehabaGaci4yaiaac+gacaGGZbGaaiiAamaaCaaaleqabaGaaGOmaaaaaeaacaaIWaaabaGaamiEaaqdcqGHRiI8aOGaey4kaSYaa8qCaeaacaaIXaaaleaacaaIWaaabaGaamiEaaqdcqGHRiI8aaGcbaGaeyO0H4TaaGzbVdqaaiaaikdadaWdXbqaaiGacogacaGGVbGaai4CaiaacIgadaahaaWcbeqaaiaaikdaaaaabaGaaGimaaqaaiaadIhaa0Gaey4kIipaaOqaaiabg2da9iGacohacaGGPbGaaiOBaiaacIgacaGGOaGaamiEaiaacMcacqGHflY1ciGGJbGaai4BaiaacohacaGGObGaaiikaiaadIhacaGGPaGaey4kaSIaamiEaaaaaaa@9409@

    1 2 (sinhcosh+X) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaci4CaiaacMgacaGGUbGaaiiAaiabgwSixlGacogacaGGVbGaai4CaiaacIgacqGHRaWkcaWGybGaaiykaaaa@4458@ is thus a primitive of cosh 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAamaaCaaaleqabaGaaGOmaaaaaaa@3A94@ .

     the integral now could be solved with the substitution formula (see [8.3.5]). We substitute g= 2 sinh MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9maakaaabaGaaGOmaaWcbeaakiGacohacaGGPbGaaiOBaiaacIgaaaa@3C83@ and thus get

    L(w) = arcsinh0 arcsinh a 2 2 sinh 2 +2 2 cosh =2 0 arcsinh a 2 sinh 2 +1 cosh =2 0 arcsinh a 2 cosh 2 cosh =2 0 arcsinh a 2 cosh 2 =sinhcosh+X | 0 arcsinh a 2 = a 2 cosh(arcsinh a 2 )+arcsinh a 2 = a 2 1+ a 2 2 +arcsinh a 2 = a 2 2+ a 2 +arcsinh a 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@F352@

     
  • Surprisingly we can't find a single term that calculates the length of the ellipse given by

    w:t(acost,bsint),t[0,2π] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaWG0bGaeSOPHeMaaiikaiaadggacqGHflY1ciGGJbGaai4BaiaacohacaWG0bGaaiilaiaadkgacqGHflY1ciGGZbGaaiyAaiaac6gacaWG0bGaaiykaiaacYcacaaMf8UaamiDaiabgIGiolaacUfacaaIWaGaaiilaiaaikdacqaHapaCcaGGDbaaaa@54BD@

    The integral

    L(w)= 0 2π | w | = 0 2π |(asin,bcos)| = 0 2π a 2 sin 2 + b 2 cos 2 =a 0 2π 1 cos 2 + b 2 a 2 cos 2 =a 0 2π 1c cos 2  , wobei  c a 2 b 2 a 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A6D9@

    is a so called elliptical integral that could only be solved in the trivial cases c=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2da9iaaigdaaaa@3895@ and c=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2da9iaaicdaaaa@3894@ . In the latter however, i.e. a=b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaadkgaaaa@38BF@ , the ellipse is circle with radius ra=b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iaadggacqGH9aqpcaWGIbaaaa@3ABC@ . Its circumference thus is

    L(w)=r 0 2π 1 =2πr MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWG3bGaaiykaiabg2da9iaadkhadaWdXbqaaiaaigdaaSqaaiaaicdaaeaacaaIYaGaeqiWdahaniabgUIiYdGccqGH9aqpcaaIYaGaeqiWdaNaamOCaaaa@45E6@ .
     

Exercise:

  • For w:t( 3 2 t, t 3 ),   t[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaWG0bGaeSOPHeMaaiikamaaleaaleaacaaIZaaabaGaaGOmaaaakiaadshacaGGSaWaaOaaaeaacaWG0bWaaWbaaSqabeaacaaIZaaaaaqabaGccaGGPaGaaiilaiaaysW7caaMe8UaamiDaiabgIGiolaacUfacaaIWaGaaiilaiaaigdacaGGDbaaaa@4B22@   we have  | | w |= ? |( 3 2 , 3 2 X )| = 9 4 + 9 4 X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadEhagaqbaiaacYhacqGH9aqpcaGG8bGaaiikamaalaaabaGaaG4maaqaaiaaikdaaaGaaiilamaalaaabaGaaG4maaqaaiaaikdaaaWaaOaaaeaacaWGybaaleqaaOGaaiykaiaacYhacqGH9aqpdaGcaaqaamaaleaaleaacaaI5aaabaGaaGinaaaakiabgUcaRmaaleaaleaacaaI5aaabaGaaGinaaaakiaadIfaaSqabaaaaa@4845@   and thus
     

    | L(w)= ? 3 2 0 1 1+X = substituting       g=X1 3 2 1 2 X = X 3 | 1 2 = 8 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@62B3@

     
  • For the parabola segment  w:t(t, t 2 ),   t[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaWG0bGaeSOPHeMaaiikaiaadshacaGGSaGaamiDamaaCaaaleqabaGaaGOmaaaakiaacMcacaGGSaGaaGzbVlaadshacqGHiiIZcaGGBbGaaGimaiaacYcacaaIXaGaaiyxaaaa@47E6@   we have | | w |= ? |(1,2X)|= 1+4 X 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadEhagaqbaiaacYhacqGH9aqpcaGG8bGaaiikaiaaigdacaGGSaGaaGOmaiaadIfacaGGPaGaaiiFaiabg2da9maakaaabaGaaGymaiabgUcaRiaaisdacaWGybWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@458E@ . Thus
     

    | L(w)= ? 0 1 1+4 X 2 = substituting g= 1 2 sinh 1 2 0 arcsinh(2) 1+ sinh 2 =cosh cosh = 1 2 0 arcsinh(2) cosh 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8CD8@

    and as | ? we knowthat 1 2 (sinhcosh+X) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaci4CaiaacMgacaGGUbGaaiiAaiabgwSixlGacogacaGGVbGaai4CaiaacIgacqGHRaWkcaWGybGaaiykaaaa@4458@ is a primitive of cosh 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiiAamaaCaaaleqabaGaaGOmaaaaaaa@3A94@ we eventually get
     

    | L(w)= ? 1 4 (sinh 1+ sinh 2 +X) | 0 arcsinh(2) = 1 4 (2 5 +arcsinh(2)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6706@

The length of smooth path w:tw(t),   t[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaWG0bGaeSOPHeMaam4DaiaacIcacaWG0bGaaiykaiaacYcacaaMe8UaamiDaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@469D@ would be easy to calculate if the derivative was a vector of constant length. For | w |=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadEhagaqbaiaacYhacqGH9aqpcaaIXaaaaa@3AB5@ we even have

L(w|[a,x])= a x 1 =xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWG3bGaaiiFaiaacUfacaWGHbGaaiilaiaadIhacaGGDbGaaiykaiabg2da9maapehabaGaaGymaaWcbaGaamyyaaqaaiaadIhaa0Gaey4kIipakiabg2da9iaadIhacqGHsislcaWGHbaaaa@4852@

for all x[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@3CAA@ , i.e. the section of the path is as long as the section of the respective interval. In that case we say that w is parameterized by the arc length. Regular paths are always parameterizable in this way:

Proposition:  If w:[a,b] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@405D@ is a regular path then there is a D 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGymaaaaaaa@379D@ -bijection  ϕ:[0,L(w)][a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOMaaiOoaiaacUfacaaIWaGaaiilaiaadYeacaGGOaGaam4DaiaacMcacaGGDbGaeyOKH4Qaai4waiaadggacaGGSaGaamOyaiaac2faaaa@44F0@ such that

  1. wϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiablIHiVjabew9aQbaa@39EE@   is parameterized by the arc length.

  2. wϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiablIHiVjabew9aQbaa@39EE@   and w generate the same curve.

  3. L(wϕ)=L(w) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWG3bGaeSigI8Maeqy1dOMaaiykaiabg2da9iaadYeacaGGOaGaam4DaiaacMcaaaa@4044@ .

[8.6.17]

Proof:  Due to the fundamental theorem [8.2.13] the function  s:[a,b][0,L(w)] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacQdacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiabgkziUkaacUfacaaIWaGaaiilaiaadYeacaGGOaGaam4DaiaacMcacaGGDbaaaa@441C@ given by

s(x) a x | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacIcacaWG4bGaaiykaiabg2da9maapehabaGaaiiFaiqadEhagaqbaiaacYhaaSqaaiaadggaaeaacaWG4baaniabgUIiYdaaaa@4194@

is a prinitive of | w | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadEhagaqbaiaacYhaaaa@38F4@ . As w is regular we have: s(x)>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacIcacaWG4bGaaiykaiabg6da+iaaicdaaaa@3AFC@ for all x]a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facaWGHbGaaiilaiaadkgacaGGDbaaaa@3CAC@ . From [7.9.6] we thus know that s is injective and that ϕ s 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOMaeyypa0Jaam4CamaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@3B8B@ is differentiable (see [7.5.4]) at each xs([a,b]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadohacaGGOaGaai4waiaadggacaGGSaGaamOyaiaac2facaGGPaaaaa@3EFB@ with

ϕ (x)= 1 s (ϕ(x)) = 1 | w |(ϕ(x)) = 1 | w | ϕ(x)0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqy1dOMbauaacaGGOaGaamiEaiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaaceWGZbGbauaacaGGOaGaeqy1dOMaaiikaiaadIhacaGGPaGaaiykaaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaGG8bGabm4DayaafaGaaiiFaiaacIcacqaHvpGAcaGGOaGaamiEaiaacMcacaGGPaaaaiabg2da9maalaaabaGaaGymaaqaaiaacYhaceWG3bGbauaacaGG8baaaiablIHiVjabew9aQjaacIcacaWG4bGaaiykaiabgcMi5kaaicdaaaa@5974@

Further, the intermediate value theorem [6.6.2] guarantees that s([a,b])=[0,L(w)] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacIcacaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiaacMcacqGH9aqpcaGGBbGaaGimaiaacYcacaWGmbGaaiikaiaadEhacaGGPaGaaiyxaaaa@43D0@ because s(a)=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacIcacaWGHbGaaiykaiabg2da9iaaicdaaaa@3AE3@ and s(b)=L(w) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacIcacaWGIbGaaiykaiabg2da9iaadYeacaGGOaGaam4DaiaacMcaaaa@3D50@ . Now due to the chain rule [7.7.8] the path wϕ:[0,L(w)] k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiablIHiVjabew9aQjaacQdacaGGBbGaaGimaiaacYcacaWGmbGaaiikaiaadEhacaGGPaGaaiyxaiabgkziUkabl2riHoaaCaaaleqabaGaam4Aaaaaaaa@4576@ is regular and

(wϕ ) =(( w 1 ϕ) ϕ ,,( w k ϕ) ϕ ) =(( w 1 ϕ) 1 | w | ϕ,,( w k ϕ) 1 | w | ϕ) = w | w | ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8423@

Finally we show:

1.    |(wϕ ) |=| w | w | ϕ|= | w | | w | ϕ=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaacIcacaWG3bGaeSigI8Maeqy1dOMabiykayaafaGaaiiFaiabg2da9iaacYhadaWcaaqaaiqadEhagaqbaaqaaiaacYhaceWG3bGbauaacaGG8baaaiablIHiVjabew9aQjaacYhacqGH9aqpdaWcaaqaaiaacYhaceWG3bGbauaacaGG8baabaGaaiiFaiqadEhagaqbaiaacYhaaaGaeSigI8Maeqy1dOMaeyypa0JaaGymaaaa@536C@

2.   As ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@37B8@ is bijective we see that:

{wϕ(t)|t[0,L(w)]}={w(ϕ(t))|t[0,L(w)]}={w(t)|t[a,b]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadEhacqWIyiYBcqaHvpGAcaGGOaGaamiDaiaacMcacaGG8bGaamiDaiabgIGiolaacUfacaaIWaGaaiilaiaadYeacaGGOaGaam4DaiaacMcacaGGDbGaaiyFaiabg2da9iaacUhacaWG3bGaaiikaiabew9aQjaacIcacaWG0bGaaiykaiaacMcacaGG8bGaamiDaiabgIGiolaacUfacaaIWaGaaiilaiaadYeacaGGOaGaam4DaiaacMcacaGGDbGaaiyFaiabg2da9iaacUhacaWG3bGaaiikaiaadshacaGGPaGaaiiFaiaadshacqGHiiIZcaGGBbGaamyyaiaacYcacaWGIbGaaiyxaiaac2haaaa@6961@

3.    L(wϕ)= 0 L(w) 1 =L(w) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacaWG3bGaeSigI8Maeqy1dOMaaiykaiabg2da9maapehabaGaaGymaaWcbaGaaGimaaqaaiaadYeacaGGOaGaam4DaiaacMcaa0Gaey4kIipakiabg2da9iaadYeacaGGOaGaam4DaiaacMcaaaa@4858@


8.5. 8.7.