Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

We can define the path integral of a continuous function $G: \Bbb{R}^N \to \Bbb{R}$ on every path $\gamma:[0,1] \to \Bbb{R}^N$ for which the following makes sense $$ \int_\gamma G ds = \int_0^1 G(\gamma(t))|\gamma'(t)|dt. $$

We know that path integrals are independent of the parametrization for $C^1$ paths, by the change of variables formula.

I was wondering if the following is true:

Supose that we have a piecewise $C^1$ path $\gamma:[0,1] \to \Bbb{R}^N$ with $|\gamma'|>0$, without self intersections, and a Lipschitz continuous path $\beta :[0,1]\to \Bbb{R}^N$ which maps $[0,1]$ onto $\gamma([0,1])$ bijectively (i.e. $\beta$ travels along the same path as $\gamma$ but, perhaps with less regularity).

Is it true then that $$ \int_\beta G ds=\int_\gamma G ds \ ?$$

share|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.