Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Change of variable in multiple integrals

  1. Mar 30, 2009 #1
    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.
  2. jcsd
  3. Mar 30, 2009 #2
    I am attaching Apostol T. 10.30.

    Attached Files:

  4. Mar 30, 2009 #3
    Or at least you can suggest some website where I can learn completely about jacobians?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook