- #1
trelek2
- 88
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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:)
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:)