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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?
     
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