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Surface integral, grad, and stokes theorem

  • Thread starter trelek2
  • Start date
Hi
I'm practicing for my exam but I totally suck at the vector fields stuff.
I have three questions:


1.
Compute the surface integral

[tex]\int_{}^{} F \cdot dS[/tex]

F vector is=(x,y,z)
dS is the area differential
Calculate the integral over a hemispherical cap defined by [tex]x ^{2}+y ^{2}+z ^{2}=b ^{2}[/tex] for z>0

2.
If C is a closed curve in the x-y plane use Stokes theorem to show that area enclosed by C is given by:
[tex]S= \frac{1}{2} \oint_{}^{C}(xdy-ydx)[/tex]
I don't see how :(
And then use it to obtain the area S enclose by ellipse [tex]\frac{x ^{2} }{a ^{2} }+ \frac{y ^{2} }{b ^{2} } =1[/tex]. Use the folowing parametrisation: vector r=[tex]acos \phi _{x}+bsin \phi _{y}[/tex] I also can't get PI*a*b....

3. (probably the easiest one)
Calculate grad of scalar field
(a dot r)/r^2
Where a is a constant vector, r is simply the vector (x,y,z). Supposed to use the chain rule...

Thanks in advance for any help. Please also include the answers to 1 and 3 as my book has very few examples and i want to be sure that what i'm doing is correct:)
 

gabbagabbahey

Homework Helper
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Hi trelek2, we are not here to do your homework (or homework-like problems) for you.

Why not make an attempt at each problem and post your work so we can see where you are getting stuck?:smile:

As a hint for problem 1, the integration is easiest in spherical coordinates.
 
I already did 1 and 3, however I'm worried they might be wrong. I got 2Pi*b^2 in the first one. I did use spherical polar coordinates and computed dS by taking the corss product of dr/dphi and dr/dtheta. Then just integration.
And in 3 i get vector a divided by length r... I guess its just one step, I don't know if i'm using the chain rule correctly.
As for 2 I have no clue what to do. I guess curl sth?
 

gabbagabbahey

Homework Helper
Gold Member
5,001
6
I already did 1 and 3, however I'm worried they might be wrong. I got 2Pi*b^2 in the first one. I did use spherical polar coordinates and computed dS by taking the corss product of dr/dphi and dr/dtheta. Then just integration.
Your method looks good, but you somehow got the wrong answer.....you'd b etter post you calculations so I can see where you went wrong...

And in 3 i get vector a divided by length r... I guess its just one step, I don't know if i'm using the chain rule correctly.
You should be getting [tex]\frac{\vec{a}-2(\vec{a}\cdot\hat{r})\hat{r}}{r^2}[/tex], so you must be applying the chain rule incorrectly....again, post your calculations and I'll be able to see where you are going wrong.

As for 2 I have no clue what to do. I guess curl sth?
As a hint, [itex]\vec{\nabla}\times(-y\hat{x}+x\hat{y})[/itex]=____?
 
alright, great thanks. 2 works fine. The answer to 3 seems obvious now that I see it. So for ex:
grad r^n will be for example nr^(n-2)*r-vector, right?
If your saying the method is right maybe I did get the right answer in 1. As a matter of fact I got 2Pi*b^3, but copied into here incorrectly. If thats still incorrect I totally give up and I'll have to copy all the calculations in here. Thanks again
 

gabbagabbahey

Homework Helper
Gold Member
5,001
6
alright, great thanks. 2 works fine. The answer to 3 seems obvious now that I see it. So for ex:
grad r^n will be for example nr^(n-2)*r-vector, right?
Yep.:smile:

If your saying the method is right maybe I did get the right answer in 1. As a matter of fact I got 2Pi*b^3, but copied into here incorrectly. If thats still incorrect I totally give up and I'll have to copy all the calculations in here. Thanks again
Your answer is still incorrect; better post your calculation...

EDIT: Your answer is correct: when I first read the problem statement, I had thought that the surface was the entire sphere, re-reading it I see that it is only the z>0 hemisphere.
 
Last edited:

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