Check Stokes's theorem

Problem:

Check Stokes's theorem for the vector function ## v=(x-y)^3 \hat{x} + (x-y)^3 \hat{y} ## using the area and perimeter shown below.

Where do I start?

 

Thanks Dick. I computed the surface integral & I got 101/6. Is this correct?

 

SteamKing, the region is the fourth quadrant. Positive y is pointing vertically upwards. This is just the way the problem was given.

According to Stokes' Theorem, ## \int\limits_S \, (\nabla \times \vec{v}) \cdot d\vec{a} = \oint\limits_P \vec{v} \cdot d\vec{\ell} ##.

For the RHS I did the following:

##y=2x-2 \Rightarrow dy=2dx ##

## \vec{v} \cdot d\vec{\ell}=(x-3)^3 \hat{x} + (x-y)^3 \hat{y} ##

Is this an acceptable answer? Does it make sense that the answer is negative?

## \int \vec{v} \cdot d\vec{\ell} = \int\limits_0^1 (x-3)^3dx + \int\limits_0^1 (x-(2x-2))^32dx = -20 ##.

 

Anyone?

 

I skipped the other 2 integrals because I believe they come out to be 0.

 

## \vec{v} \cdot d\vec{\ell}=(x-3)^3 \hat{x} + (x-y)^3 \hat{y} ## should have been## \vec{v} \cdot d\vec{\ell}=(x-3)^3 dx + (x-y)^3 dy ##

 

Certainly! I'll type it out right now. Thanks for the help by the way!

 

So for the LHS of Stokes's theorem, I get:

## \nabla \times \vec{v} = (\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y})\hat{z} = 6(x-y)^2 \hat{z}## (since ## \vec{v} ## does not have a z-component)

## (\nabla \times \vec{v}) \cdot d\vec{a} = [6(x-y)^2 \hat{z}] \cdot [dx \ dy \ \hat{z}] = 6(x-y)^2dx \ dy##

## \int (\nabla \times \vec{v}) \cdot d\vec{a} = \int \int 6(x-y)^2dx \ dy = 6 \int \int (x-(2x-2))^2dx \ dy = 6 \int \int (2-x)^2dx \ dy = 6 \int \int (x^2-4x+4) dx \ dy##.

So putting in the limits & carrying out the integration, is just left for the LHS. Right?

 

So with the limits, it should be this ## 6 \int_0^{-2} \int_0^{\frac{2+y}{2}} (x^2-4x+4) dx \ dy##, correct?

 

So that means I must integrate over y first and then x?

 

The limits would then be y: 0 -> 2x-2 and x: 1 -> 0?

 

I integrated wrt to y & then x & got an answer of 7. I wasn't sure if the x limits should be from 0->1 or 1->0. I picked 1->0 because that follows the counter-clockwise path, is that correct?