Jacobian determinant in multiple integration

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Discussion Overview

The discussion centers around the mathematical proof of the theorem related to the Jacobian determinant and its application in computing multiple integrals through transformations. Participants explore the educational resources that provide this proof, including textbooks and courses.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about the type of math course that includes a proof of the Jacobian theorem for multiple integrals.
  • Another participant suggests "Calculus on manifolds" by Spivak as a source that contains a proof.
  • Multiple participants mention Mary Boas as a potential source for the proof, although one notes that their version does not include it.
  • One participant expresses uncertainty about the inclusion of the proof in Boas' text, stating they might be wrong.
  • A participant shares their own proof developed in a vector analysis course, indicating a personal contribution to the discussion.

Areas of Agreement / Disagreement

There is no consensus on the availability of the proof in Mary Boas' textbook, as participants express differing experiences with their versions. The discussion remains unresolved regarding the definitive sources for the proof.

Contextual Notes

Participants reference different editions of textbooks, which may lead to variations in content, including the presence or absence of the proof in question.

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In what kind of math course would one learn the proof of the theorem that introduces the Jacobian to computing multiple integrals under various transformations?

My calculus textbook has this theorem, and uses it to derive the triple integral formulas for cylindrical and spherical coordinates, and the double integral formulas for polar coordinates. It also uses it to show how many integration problems can be solved much more easily by applying transformations and using the Jacobian.

But it does not give a proof. In what kind of textbook would I find a proof of this theorem?
Thanks.

BiP
 
Physics news on Phys.org
"Calculus on manifolds" by Spivak contains a proof of the theorem.
 
Mary Boas gives one too.
 
rollingstein said:
Mary Boas gives one too.

Not in my version of Boas.
 
micromass said:
Not in my version of Boas.

I might be wrong.
 
Here's my own proof I did in a vector analysis course I took. Cheers.
 

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