Green's Theorem & Line Integrals

[kingwinner]

Homework Statement

http://www.geocities.com/asdfasdf23135/advcal26.JPG
Note:
path-connected means arcwise-connected
C^0 means continuous
⊿f=Laplacian=(f)xx+(f)yy
df/dn = grad f ． n

2. Relevant material
Green's theorem, line integrals, ...

The Attempt at a Solution

The only part I can finish is part (ii):
(ii) follows from the vector version of Green's theorem

by letting F=grad f

How can I do parts (i) and (iii)?
And provided that I have proved (i), (ii), and (iii), how would these results aid me in proving the actual problem?

Any help or hints would be greatly appreciated!https://www.physicsforums.com/images/smilies/smile.gif

 

[kingwinner]

I seriously need help with parts (i) and (iii)...

 

[kingwinner]

Well, I've solved part (i) as well:

Then use squeeze theorem...


And now I am stuck only on part (iii), please help!

 

[kingwinner]

Provided that I have proved (i), (ii), and (iii), how would these results aid me in proving the actual problem? (i.e. to prove that f is constant on U)