- #1
Castilla
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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") ?
Please send some aid.
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") ?
Please send some aid.