# Change of variable in multiple integrals

## Main Question or Discussion Point

Hello. Does someone has studied the Change of Variables Theorem for multiple integrals in Apostol's Mathematical Analysis? (First Edition:not Lebesgue but Riemann).

I hope that some of you has the same edition, because if not, it will be sort of dificult to make a legible copy of the equations. It is Theorem 10.30, pg 271.

1.- See pg. 272, after the first 2 paragraphs. ¿Why does Apostol uses "t" to denote a variable vector in set A as well as a variable vector in set B? Is it a typo in my edition?

2.- I have more or less managed to follow the proof up to its last part, in page 274. Here is my problem. In his equation (11), Apostol has a one-dimensional Riemann integral with this product as the integrand function:

F(theta(u)) (Jacobian of function theta in vector(u)) (11)

He says: now we make the one dimensional change of variable
u_n = phi_n (t) in the inner integral and replace the dummy variables u_1, ..., u_n-1 by t_1, ..., t_n-1 and (11) becomes:

(I only copy the integrand function)

F(g(t)) (Jacobian of theta in "t")(Jacobian of phi in "t") dt_n. (*)

Then he equals this integrand function with this one:

F(g(t)) (Jacobian of function g in "t"). (**)

Two questions here:

2.1. How does he goes from (11) to (*)? I know the multiplication theorem for Jacobians (T. 7.2, pg. 140 in the same book) but I can not see how this theorem would justify Apostol's step. It does not match.

2.2. Maybe there is a typo in (*) and what he meant was:

F(g(t)) (Jacobian of theta in "phi(t)")(Jacobian of phi in "t") ???