Suppose we have a smooth and bounded domain $D\subset\mathbb{R}^2$, and a continuous function $f:\mathbb{R}^2\rightarrow\mathbb{R}$. Let $\phi:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be some diffeomorphism. Then by the transformation rule for integrals we have

$$\int\limits_{D}f(x) \,dx=\int\limits_{\phi^{-1}(D)}f(\phi(x)) \cdot \vert \text{Det}(d\phi(x)) \vert\, dx.$$

I am wondering, if there is a similar formula for line integrals, i.e. a formula of the type:

$$\int\limits_{\partial D} f(s)\, ds=\int\limits_{\phi^{-1}(\partial D) }f(\phi(s)) \cdot (...) \,ds$$

with some appropriate expression in place of $''(...)''$?

Of course, if $\phi$ is an isometry, then we can replace (...) by $1$. But I am interested in more general functions like for example $\phi: (0,\infty)\times\mathbb{R}\ni(x,y) \mapsto(x^2,y)\in (0,\infty)\times \mathbb{R}$. Is there anything possible in that direction?

Best wishes


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