- #1

trelek2

- 88

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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:)