Change of variables in $n-1$ dimensional integral

Suppose $E \subset \mathbb R^n$ is a set with smooth boundary and let $\phi :E \to \mathbb R^n$ be a $C^\infty$ map. Then by change of variables $$\int_{\phi(E)} f = \int_{E} (f \circ \phi) |\det d \phi|$$ where $d\phi$ is the matrix of partial derivatives of $\phi$.

How can I express the integral on the boundary? The formula $$\int_{ \phi(\partial E)} f = \int_{\partial E} (f \circ \phi) |\det d \phi|$$ clearly does not work (for example taking $\phi$ a dilation by $k$, one would obtain that the integral changes like $k^n$, while it should change like $k^{n-1}$). What should I put instead of $|\det d \phi|$?

• Just asking: Do you know about manifolds? – Andrew D. Hwang Jun 7 '17 at 16:12
• You need further properties for $\phi$ even for the volume statement. – zhw. Jun 7 '17 at 21:10

I think that what you want is the area formula for Lipschitz functions. You can find it in the book of Evans and Gariepy "Measure theory and fine properties of functions". Essentially, if you consider a local chart $\varphi:V\to \mathbb{R}^n$ for $\partial E$, where $V\subset \mathbb{R}^{n-1}$, then (locally) $$\int_{\partial E}g(x)\,dS=\int_Vg(\varphi(y))[[D\varphi(y)]]\,dy,$$ where $[[D\varphi(y)]]=\sqrt{\det((D\varphi(y))^tD\varphi(y))}$. On the other hand, $\phi\circ\varphi:V\to \mathbb{R}^n$ is a local chart for $\phi(\partial E)$ so you get $$\int_{\phi(\partial E)}h(x)\,dS=\int_Vh(\phi(\varphi(y)))[[D(\phi\circ\varphi)(y)]]\,dy.$$ So now you want to make the two right-hand sides equal. I don't know how messy are the computations using the chain rule for $[[D(\phi\circ\varphi)(y)]]$.
• Indeed, looking for an $h$ which satisfies $$\int_{\phi(\partial E)} f = \int_{\partial E} (f \circ \phi) h,$$ which on a local chart $\psi_i : U_i \to \partial E$ becomes $$\int_{U_i} f \circ \phi \circ \psi_i [[d (\phi \circ \psi_i)]] = \int_{U_i} f \circ \phi \circ \psi_i [[d \psi_i]] h,$$ one can just take $$h = \dfrac{[[d (\phi \circ \psi_i)]]}{[[d \psi_i]]}.$$ Using a partition of the unit I believe one can patch up a global $h$ if $\partial E$ is orientable. Does this $h$ has a specific name/symbol not chart dependent? I can't seem to find a reference. – User11111 Jun 7 '17 at 23:48