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
add comment

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.