Jacobian determinant in multiple integration

In summary, the conversation discusses the theorem that introduces the Jacobian to computing multiple integrals under various transformations. The theorem is used in calculus textbooks to derive formulas for cylindrical and spherical coordinates, as well as to solve integration problems more easily. However, the textbooks do not provide a proof of the theorem. The books "Calculus on Manifolds" by Spivak and "Mary Boas" are mentioned as containing a proof of the theorem, but the latter may not be included in all versions. The person also mentions having their own proof from a vector analysis course.
  • #1
Bipolarity
776
2
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
 
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  • #2
"Calculus on manifolds" by Spivak contains a proof of the theorem.
 
  • #3
Mary Boas gives one too.
 
  • #4
rollingstein said:
Mary Boas gives one too.

Not in my version of Boas.
 
  • #5
micromass said:
Not in my version of Boas.

I might be wrong.
 
  • #6
Here's my own proof I did in a vector analysis course I took. Cheers.
 

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FAQ: Jacobian determinant in multiple integration

What is the Jacobian determinant in multiple integration?

The Jacobian determinant is a mathematical concept that is used in multiple integration to measure the change in volume when transforming from one coordinate system to another. It is represented by the capital letter J and is calculated by taking the partial derivatives of the transformation equations.

Why is the Jacobian determinant important in multiple integration?

The Jacobian determinant is important because it helps us to transform integrals from one coordinate system to another, making it possible to solve integration problems in non-Cartesian coordinates. It also allows us to calculate the volume of objects in multidimensional space.

How is the Jacobian determinant used in calculating integrals?

The Jacobian determinant is used in calculating integrals by first transforming the integral from one coordinate system to another using the Jacobian. This allows us to change the variables of integration and solve the integral in the new coordinate system. The Jacobian also provides a scaling factor that accounts for the change in volume when transforming between coordinate systems.

Can the Jacobian determinant be negative?

Yes, the Jacobian determinant can be negative. This occurs when the transformation between coordinate systems results in a reversal of the orientation of the coordinate axes. In this case, the Jacobian will have a negative value, which still represents the change in volume but with a negative sign.

How is the Jacobian determinant related to the change of variables formula?

The Jacobian determinant is closely related to the change of variables formula, which is used to transform integrals from one coordinate system to another. The Jacobian is a crucial component of this formula, as it represents the change in volume when transforming between coordinate systems. Without it, the change of variables formula would not be accurate.

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