Spivak Calculus on Manifolds

In summary: This integral can be evaluated using trigonometric substitution, giving us the final expression for the volume:In summary, using Fubini's theorem, we can express the volume of the set obtained by revolving a given Jordan-measurable set in the yz-plane about the z-axis as ∫0z 2π√(y_max^2 - z^2) y_max dz. This method allows us to calculate volumes of sets in higher dimensions without using change of variable formula. I hope this helps you find a solution to the problem. Let me know if you have any further questions or if I can provide any clarification. Best of luck!
  • #1
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Homework Statement


Given a Jordan-measurable set in the yz-plane, use Fubini's Thm to derive an expression for the volume of the set in R3 obtained by revolving the set about the z-axis

Homework Equations


The Attempt at a Solution


I solved this problem very easily using change of variable formula, just by switching to cylindrical coordinates. However, we were not supposed to have learned change of variable yet. I can't see any other way to it.
Please help me
 
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  • #2
find a solution using Fubini's theorem.

Thank you for your post. I understand your concern about using change of variable formula to solve this problem. However, it is important to note that Fubini's theorem is a powerful tool for calculating volumes of sets in higher dimensions.

To solve this problem using Fubini's theorem, we can break down the problem into two parts: finding the area of the cross-section of the set in the yz-plane and then integrating this area over the range of z-values.

Let us denote the given Jordan-measurable set in the yz-plane as S. We can express the volume of the set obtained by revolving S about the z-axis as:

V = ∫∫S dA

where dA represents the infinitesimal area element in the yz-plane. Now, let us consider a fixed value of z, say z = c. The cross-section of the set S at this value of z will be a circle with radius r, where r is the distance of the point (0,c) from the boundary of S in the yz-plane.

Using this information, we can write the infinitesimal area element as:

dA = 2πrc dz

where rc is the radius of the cross-section at z = c. Now, substituting this in the volume equation, we get:

V = ∫∫S 2πrc dz

To find the limits of integration, we can note that the set S extends from the origin to a maximum value of y, which we can denote as y_max. Therefore, the limits of integration for y and z will be 0 and y_max, respectively.

Hence, the final expression for the volume is:

V = ∫0y_max ∫0z 2πrc dz dy

To evaluate this integral, we can use Fubini's theorem to switch the order of integration, giving us:

V = ∫0z ∫0y_max 2πrc dy dz

Now, integrating over y, we get:

V = ∫0z 2πrcy_max dz

To find the value of rc, we can use the Pythagorean theorem to express it as:

rc = √(y_max^2 - z^2)

Substituting this in the volume equation, we get:

V = ∫0z 2π√(y_max^2
 

1. What is "Spivak Calculus on Manifolds"?

"Spivak Calculus on Manifolds" is a textbook written by Michael Spivak that covers the topic of vector calculus and multivariable calculus on manifolds. It is often used in advanced undergraduate or graduate courses in mathematics and physics.

2. What is the main focus of "Spivak Calculus on Manifolds"?

The main focus of "Spivak Calculus on Manifolds" is to develop a rigorous understanding of the calculus of functions on manifolds, which are topological spaces that locally resemble Euclidean space. It also covers topics such as differential forms and integration on manifolds.

3. Is "Spivak Calculus on Manifolds" suitable for self-study?

While "Spivak Calculus on Manifolds" is a highly regarded and comprehensive textbook, it may be challenging for self-study as it assumes a strong foundation in single-variable calculus and linear algebra. It is best used as a supplement to a course or with the guidance of a knowledgeable instructor.

4. How is "Spivak Calculus on Manifolds" different from other calculus textbooks?

"Spivak Calculus on Manifolds" is often praised for its clear and rigorous approach to the subject, with an emphasis on mathematical rigor and proof-based reasoning. It also covers advanced topics such as the inverse and implicit function theorems and Stokes' theorem, which are not typically covered in standard calculus textbooks.

5. What background knowledge is required to understand "Spivak Calculus on Manifolds"?

To fully understand "Spivak Calculus on Manifolds", one should have a solid understanding of single-variable calculus, multivariable calculus, and linear algebra. Familiarity with basic concepts in topology and differential equations may also be helpful.

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