Change of variable formula when integrand involves partial derivatives

Question

A well known change of variable theorem of integral calculus says the following.

Let $X$ and $Y$ be two open sets in $\mathbb{R}^2$ and $\phi: X \rightarrow Y$ be a diffeomorphism. Then \begin{eqnarray}\tag{1} \int\limits_{Y} f(y_1,y_2) dy_1 dy_2=\int\limits_{X} f(\phi(x_1,x_2)) \left|J_{\phi}(x_1,x_2)\right| dx_1 dx_2, \end{eqnarray} where $J_{\phi}$ denotes the Jacobian matrix of $\phi.$

What is the corresponding change of variable formula if the integrand involves derivatives. More precisely, for $f \in C^1$ what is the change of variable formula if the integrand is \begin{eqnarray} \int\limits_{Y} \left(\partial_1f(y_1,y_2)+\partial_2f(y_1,y_2)\right) dy_1 dy_2=\int\limits_{X} \cdot\cdot\cdot\cdot dx_1 dx_2. \end{eqnarray} How to prove the change of variable formula for such integrand using the change of formula (1) ?

Answer 1

One element of confusion is coming from a somewhat imprecise notation. Writing $\partial_1 f$ is asking for trouble when you invoke a change of variables. If instead you write $\frac{\partial f}{\partial y_1}$ or $\partial_{y_1} f$ this removes (in principle) any ambiguity. For example if $f(y_1,y_2)=y_1^2 y_2$ then (omitting domains) $$\int \partial_{y_1} f (y_1,y_2) dy_1\, dy_2 =\int 2 y_1 y_2 \, dy_1\, dy_2 $$ when changing variables gives $$ \int \partial_{y_1}f\circ \phi(x_1,x_2) \; J(\phi)\; dx_1 dx_2 = \int 2\phi_1(x_1,x_2)\phi_2(x_1,x_2)\; J(\phi)\;dx_1 dx_2$$

Answer 2

I think that you just has a confusion with the notation, by example we will have that

$$ \int_A \partial _1 f(x,y)d(x,y)=\int_{\phi ^{-1}(A)}(\partial _1f\circ \phi)(x,y)|\det [\partial \phi(x,y)]|d(x,y) \tag1 $$

for some diffeomorphism $\phi$. Here $\partial_1 f$ is a function as any other (and above I used the symbol $\partial$ alone without subscript to denote the Fréchet derivative of the function $\phi$). Note that $\partial _1 f\circ \phi $ and $\partial _1(f\circ \phi )$ are two different things, the first one is the composition of the functions $\partial _1 f$ and $\phi $, but the second is the derivative respect to the first argument of the function $f\circ \phi $. Maybe something like that was your source of confusion.

Also observe that I used the notation $(h\circ l)(x)$ instead of $h(l(x))$, the first is preferably in almost all situations by many reasons, one of them is the enhanced clarity in it meaning because there is a separation between the notation for functions and it arguments.

Also note that the notation $\partial_1$ is more clear and simple than something like $\partial_x$: just observe that in (1) I didn't needed to denote in the second integral the arguments using different letters, however if I would used $\partial_x$ instead then I would need to change the name of the argument $x$ to something else because otherwise will be confusing, but it will be confusing anyway because then there is no argument $x$ to know what is the partial derivative that you took!