Path integral/Stokes's and Green's theorem

In summary, Path integral, Stokes's theorem, and Green's theorem are mathematical concepts used in vector calculus. They differ in their applications and types of integrals. They are related through the fundamental theorem of calculus and have physical applications in fields such as electromagnetism and fluid mechanics. A solid understanding of vector calculus, multivariable calculus, and linear algebra is necessary to comprehend these concepts, and they can also be applied in other areas of science such as engineering, computer science, and image processing.
  • #1
PhysicsGente
89
3
I meant Line integral.

Homework Statement



I want to find the path integral of a vector function F over a closed path in Euclidean space with z = 0.

Homework Equations


The Attempt at a Solution



I was wondering if it is allowed to first use Stoke's theorem and then Green's theorem. I would end up getting div(curl(F)) which I believe equals to zero. But I am not sure if I'm allowed to use Green's theorem in this case.
 
Last edited:
Physics news on Phys.org
  • #2
Can you post the original problem and your work so far?
 
  • #3
I have figured it out. Thank you!
 

1. What is the difference between Path integral, Stokes's theorem, and Green's theorem?

Path integral, Stokes's theorem, and Green's theorem are all mathematical concepts used in the field of vector calculus. The main difference between them is their applications and the types of integrals they involve. Path integral is used to calculate the work done by a vector field along a specific path. Stokes's theorem relates the surface integral of a vector field over a surface to the line integral of the same vector field along the boundary of that surface. Green's theorem relates the double integral of a two-dimensional vector field over a region to a line integral along the boundary of that region.

2. How are Path integral, Stokes's theorem, and Green's theorem related?

Path integral, Stokes's theorem, and Green's theorem are all related to each other through the fundamental theorem of calculus. This theorem states that the integral of a function over a path can be calculated by finding the derivative of the function over the path. In other words, Stokes's theorem and Green's theorem can be seen as generalizations of the fundamental theorem of calculus to higher dimensions.

3. What is the physical significance of Path integral, Stokes's theorem, and Green's theorem?

Path integral, Stokes's theorem, and Green's theorem have various physical applications, especially in the study of electromagnetism and fluid mechanics. For example, Stokes's theorem is used in fluid dynamics to calculate the circulation of a fluid around a closed path. Green's theorem is used in electromagnetism to calculate the electric field or potential at a point in space using the charge density distribution in that region.

4. What is the mathematical background required to understand Path integral, Stokes's theorem, and Green's theorem?

A solid understanding of vector calculus, multivariable calculus, and linear algebra is necessary to understand Path integral, Stokes's theorem, and Green's theorem. These concepts involve vector fields, surface and line integrals, and partial derivatives, which are all fundamental concepts in these mathematical fields.

5. Can Path integral, Stokes's theorem, and Green's theorem be applied in other areas of science?

Yes, Path integral, Stokes's theorem, and Green's theorem have applications in various areas of science, including physics, engineering, and computer science. For example, Green's theorem is used in image processing to calculate the average brightness of pixels in an image. Stokes's theorem is used in fluid dynamics to study the rotation of a fluid around a closed loop. These theorems have also found applications in the study of chaos theory and quantum mechanics.

Similar threads

Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
279
Applying Stoke's Theorem: A Hint
    • Calculus and Beyond Homework Help
Replies
26
Views
2K
Singularities when applying Stokes' theorem
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Finding the Wrong Answer with Stokes' Theorem
    • Calculus and Beyond Homework Help
Replies
4
Views
818
Complex Integration Along Given Path
    • Calculus and Beyond Homework Help
Replies
3
Views
869
Determining between direct evaluation or vector theorems
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
Verify Green's Theorem in the given problem
    • Calculus and Beyond Homework Help
Replies
10
Views
447
Using Green's Theorem for a quadrilateral
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Calculating the Line Integral of F over C: Stokes' Theorem and Symmetry
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
Stokes' Theorem, how to apply for this surface?
    • Calculus and Beyond Homework Help
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top