Suppose we have two coordinate systems: $\displaystyle (x_1, x_2, x_3)$ (representing the usual Cartesian coordinates, for example) and $\displaystyle (X_1, X_2, X_3)$ (representing cylindrical polars, say). I think it's true that:

$\displaystyle \int f(x_1, x_2, x_3) dx_1 = \int \sum_{i=1}^3 f(x_1(X_1,X_2,X_3), x_2(X_1,X_2,X_3), x_3(X_1,X_2,X_3)) \frac{\partial x_1}{\partial X_i} dX_i $,

but I can't rigorously prove this. (It feels right based on the chain rule but I can't write a formal proof, and haven't found anything online.) Can anyone help please?

On a similar note, for functions $\displaystyle f_1, f_2, f_3$ is it true that

$\displaystyle \int \sum_i \sum_j f_i(x_1, x_2, x_3) \frac{\partial f_i}{\partial x_j} dx_j = \int \sum_i \sum_j f_i(x_1(X_1,X_2,X_3), x_2(X_1,X_2,X_3), x_3(X_1,X_2,X_3)) \frac{\partial f_i}{\partial X_j} dX_j $ ?

I'm less sure that this second formula holds in general.