# Thread: Change of coordinate system when integrating

1. ## Change of coordinate system when integrating

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

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

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 $f_1, f_2, f_3$ is it true that

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

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

2. ## Re: Change of coordinate system when integrating

Both are correct,