Help finding more info on some theorems [Vector Integrals]

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In summary, Boas Mathematical Methods in the Physical Sciences, Chapter 6 section 11 problem 17 contains a list of 7 theorems called "Vector Integral Theorems". These theorems are derived from the Divergence and Stoke's theorems and are commonly used in various fields of physics, such as electromagnetism and fluid dynamics. They can also be found under the names Gauß' divergence theorem and Stokes theorem. Further information and examples of these theorems can be found in various sources, including the Physics Forums website.
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mishima
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Hi, in Boas Mathematical Methods in the Physical Sciences, Chapter 6 section 11 problem 17 has a list of 7 theorems it calls "Vector Integral Theorems". For example,

$$\int \vec \nabla \times \vec V \ d\tau = \oint \vec n \times \vec V \ d\sigma$$

I understand their derivations from the Divergence and Stoke's theorems, but I was looking for a reference where I could find more information on how these theorems are applied in practical situations of science. Or perhaps they have a more formal name in mathematics I could use to search for more information on my own. Thanks.
 
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Here's a small list of theorems about differentiation and integration with their mathematical names:
https://www.physicsforums.com/insights/pantheon-derivatives-part-v/#toggle-id-2
and also a long list of sources.

Stokes theorem and its (many) variations occur all over the place of physics, from electromagnetism to fluid dynamics. It's one of the main tools. And here is an example (problem #4) (+ solution, post #19) of Gauß' divergence theorem: https://www.physicsforums.com/threads/intermediate-math-challenge-august-2018.952511/
 
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1. What are vector integrals?

Vector integrals are mathematical expressions that involve the integration of vector quantities. They are used in many areas of science, including physics, engineering, and mathematics, to solve problems involving vector quantities, such as force, velocity, and displacement.

2. How are vector integrals different from regular integrals?

Vector integrals are different from regular integrals in that they involve the integration of vector quantities, rather than scalar quantities. This means that the result of a vector integral is a vector, rather than a single value. Additionally, vector integrals require the use of vector calculus, which involves the manipulation of vectors and vector operations.

3. What are some applications of vector integrals?

Vector integrals have many applications in science and engineering. They are used to calculate work, energy, and momentum in physics, as well as to solve problems involving motion and forces. In engineering, vector integrals are used to analyze stresses and strains in structures, and to solve problems related to fluid flow and electromagnetism.

4. How do I solve a vector integral?

Solving a vector integral involves applying the principles of vector calculus, such as the gradient, divergence, and curl operators. It also requires knowledge of vector algebra and trigonometry. The specific method for solving a vector integral will depend on the problem at hand and the specific techniques that are appropriate for that problem.

5. Are there any resources available to help me understand vector integrals?

Yes, there are many resources available to help you understand vector integrals. These include textbooks, online tutorials, and videos that explain the concepts and provide step-by-step examples. You can also consult with a math or physics tutor for personalized help and guidance with specific problems.

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