Verifying Stokes' Theorem help

[joe kutil]

Homework Statement

Verify Stokes' theorem
∫_c F • t ds = ∫∫_s n ∇ × F dS
in each of the following cases:

(a) F=i z² + j y²
C, the square of side 1 lying in the x,z-plane and directed as shown
S, the five squares S₁, S₂, S₃, S₄, S₅ as shown in the figure.

(b) F = iy + jz + kx
C, the three quarter circle arcs C₁, C₂, and C₃ directed as shown in the figure.
S, the octant of the sphere x² + y₂ + z₂ = 1 enclosed by the three arcs.

(c) F = iy - jx + kz
C, the circle of radius R lying in the xy-plae, centered at (0,0,0) and directed as shown in the figure.
S, the curved upper surfaces of the cylinder of radius R and height h.

Homework Equations

∫_c F • t ds = ∫∫_s n ∇ × F dS Stokes' theorem

The Attempt at a Solution

I have been staring at this problem for two weeks and every time I think i get it, i find out I'm wrong. Please help!

 

[kuruman]

You need to calculate both sides of the equation and verify that you get the same thing.
Do you understand line integrals?
Do you know how to take the curl of a vector?
More importantly, the left and right sides of the equation tell you to execute a series of operations. Can you figure out what these are?
Start with the line integral on the left side. What do the symbols stand for?

 

[joe kutil]

i understand how to take the curl, i just don't understand the concept of taking one over the surfaces

 

[kuruman]

i understand how to take the curl, i just don't understand the concept of taking one over the surfaces

Do you understand the meaning of all the symbols in ∫∫_s n ∇ × F dS in terms of a picture?
Specifically
1. What does n represent?
2. What does S represent?
3. What is the relation between n and S?
Also note that the correct expression for the integrand is $\hat{n} \cdot \vec{\nabla}\times \vec{F}dS.$ The "dot" between n and ∇ × F is important and means more than "times".

If you don't know the answers to the three questions, please find out.

 

[joe kutil]

Thank you I figured it out

 

[kuruman]

Thank you I figured it out

Great! Are you good to go with the line integral too?