Stokes' theorem over a circular path

[Reshma]

I need complete assistance on this :-)

Check the Stokes' theorem using the function $$\vec v =ay\hat x + bx\hat y$$
(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:
$$\int_s {(\nabla\times \vec v).d\vec a = \oint_p\vec v.d\vec r$$

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

Hints will do!

 

[dextercioby]

You can do everything in polar plane coordinates,or in rectangular cartesian.It's your choice.

Make it.

Daniel.

 

[Galileo]

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.