Stokes' theorem over a circular path

In summary, Stokes' theorem can be used to check the function \vec v =ay\hat x + bx\hat y for a circular path of radius R centered at the origin of the xy plane. To compute the area element and line integral for this path, you can use polar or cartesian coordinates and remember to choose the appropriate surface bounded by the circle. The curl of the function can be used to determine the answer for the line integral. A parametrization of x=Rcos t, y=Rsin t, 0<=t<=2pi is suitable for the line integral calculation. Remember to also draw a picture for better visualization.
  • #1
Reshma
749
6
I need complete assistance on this :-)

Check the Stokes' theorem using the function [tex]\vec v =ay\hat x + bx\hat y[/tex]
(a and b are constants) for the circular path of radius R, centered at the origin of the xy plane.

As usual Stokes' theorem suggests:
[tex]\int_s {(\nabla\times \vec v).d\vec a = \oint_p\vec v.d\vec r [/tex]

How do you compute:
1. the area element [tex]d\vec a[/tex]
2. the line integral
For the circular path in this case.

Hints will do!
 
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  • #2
You can do everything in polar plane coordinates,or in rectangular cartesian.It's your choice.

Make it.

Daniel.
 
  • #3
You don't need the expression for the area element. Just compute the curl and you'll immediately see what the answer should be (draw a picture as well).
Remember that you're at liberty to choose the surface that is bounded by the circle.

For the line integral the parametrization x=Rcos t, y=Rsin t, 0<=t<=2pi will do.
 

Related to Stokes' theorem over a circular path

What is Stokes' theorem over a circular path?

Stokes' theorem over a circular path is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field around the boundary of that surface. It is named after the mathematician George Gabriel Stokes.

How is Stokes' theorem over a circular path used in science?

Stokes' theorem over a circular path is used in many fields of science, such as physics, engineering, and fluid dynamics. It allows for the calculation of flux, circulation, and work done by a vector field, which are important concepts in these fields.

What are the conditions for applying Stokes' theorem over a circular path?

The conditions for applying Stokes' theorem over a circular path are that the surface must be closed, the vector field must be differentiable, and the boundary of the surface must be a simple, closed curve. Additionally, the surface and curve must be oriented consistently.

What is the difference between Stokes' theorem over a circular path and Green's theorem?

Stokes' theorem over a circular path is a generalization of Green's theorem, which applies to two-dimensional vector fields. Stokes' theorem applies to three-dimensional vector fields and can be seen as a higher-dimensional version of Green's theorem.

Can Stokes' theorem over a circular path be applied to any shape or path?

No, Stokes' theorem over a circular path can only be applied to closed surfaces and their corresponding boundaries. It cannot be applied to open surfaces or paths that are not simple, closed curves.

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