Surface integral questions

May 2010
7
0
I need to find the surface integrals for these various questions. Some I don't know where to begin, others I don't know how to continue. Help would be appreciated



Not sure about this one



I get: \(\displaystyle \int \int F \cdot n dS\)
\(\displaystyle F = (x,y,2z)\)
\(\displaystyle n = \frac {2x}{\sqrt {4x^2 + 4y^2 + 1}} + \frac {2y}{\sqrt {4x^2 + 4y^2 + 1}} + \frac {1}{\sqrt {4x^2 + 4y^2 + 1}}\)
\(\displaystyle \int \int F \cdot n dS = \int \int 2 dA \)

But not sure what the limits are to proceed



I'm guessing similar manner to question above, but do I get z on its own i.e. \(\displaystyle z = 3-3x-\frac {3}{2}y\) then proceed. But once again, not sure what the limits would be



Using divF = 3 then \(\displaystyle \int^1_0 \int^1_0 \int^1_0 3 dzdxdy = 3 \) Is that right?



Need to use divergence again, but not sure how to proceed.



I could do the same as below, but not sure how to proceed



I got the curl to be i + j + k and \(\displaystyle \sqrt { ( \frac {\partial f}{\partial x})^2 + ( \frac {\partial f}{\partial y})^2 + 1 } = \sqrt {3}\)
Not sure how to proceed
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
I didn't read all your problems - way too many in one post! But I can tell you the first one is easily solved by the divergence theorem. Find \(\displaystyle \mbox{div } F\) and integrate \(\displaystyle \mbox{div }F\ dV\) inside the cube.
 
Jul 2007
894
298
New Orleans
I need to find the surface integrals for these various questions. Some I don't know where to begin, others I don't know how to continue. Help would be appreciated



Not sure about this one



I get: \(\displaystyle \int \int F \cdot n dS\)
\(\displaystyle F = (x,y,2z)\)
\(\displaystyle n = \frac {2x}{\sqrt {4x^2 + 4y^2 + 1}} + \frac {2y}{\sqrt {4x^2 + 4y^2 + 1}} + \frac {1}{\sqrt {4x^2 + 4y^2 + 1}}\)
\(\displaystyle \int \int F \cdot n dS = \int \int 2 dA \)

But not sure what the limits are to proceed



I'm guessing similar manner to question above, but do I get z on its own i.e. \(\displaystyle z = 3-3x-\frac {3}{2}y\) then proceed. But once again, not sure what the limits would be



Using divF = 3 then \(\displaystyle \int^1_0 \int^1_0 \int^1_0 3 dzdxdy = 3 \) Is that right?



Need to use divergence again, but not sure how to proceed.



I could do the same as below, but not sure how to proceed



I got the curl to be i + j + k and \(\displaystyle \sqrt { ( \frac {\partial f}{\partial x})^2 + ( \frac {\partial f}{\partial y})^2 + 1 } = \sqrt {3}\)
Not sure how to proceed

for the second problem change it into polar and use the limits

\(\displaystyle \int^{1}_{0}\int^{2\pi}_{0}d\theta dr\)

for the 3rd one the limits are

\(\displaystyle \int^{\frac{3}{2}}_{0} \int^{-2x+3}_{0} dydx\)

Correct on 4

For question 5 find the divergence and then use

\(\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} drd\theta dz\)
 
Last edited:
May 2010
7
0
For question 5 find the divergence and then use

\(\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} drd\theta dz\)
Thanks. For the above would an r need to be added? I remember something like that when i did coordinates, i.e. should it be \(\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} r drd\theta dz\)

Also, any ideas for the final 2? I've gone through my book twice and it doesn't have a proper example
 
Jul 2007
894
298
New Orleans
Thanks. For the above would an r need to be added? I remember something like that when i did coordinates, i.e. should it be \(\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} r drd\theta dz\)

Also, any ideas for the final 2? I've gone through my book twice and it doesn't have a proper example
Yes for the 6th one remember what Stokes theorem says. It does not matter what surface you use as long as C is the same. In this case it be alot easier to use the surface of z=1 with the domain being projected to the xy plane and you get the integral

\(\displaystyle \int^{1}_{0} \int^{2\pi}_{0} d\theta dr\)

Another way to do 6 is let

\(\displaystyle r(t)= \cos{t}i +\sin{t}j + k\)

\(\displaystyle f(r(t)) = \cos^2{t}i + \sin^2{t}j +k\)

\(\displaystyle r'(t) = -\sin{t}i +\cos{t}j + 0k\)

Now just evaluate the line integral
 
Jul 2007
894
298
New Orleans
Thanks. For the above would an r need to be added? I remember something like that when i did coordinates, i.e. should it be \(\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} r drd\theta dz\)

Also, any ideas for the final 2? I've gone through my book twice and it doesn't have a proper example

And for the last one I got a different curl than you did.

Mine was

\(\displaystyle -i +j -k\)

so

\(\displaystyle \int \int [-Mf_x - Nf_y +P]dxdy\)

f_x = -1

f_y = -1

and your integrals would be

\(\displaystyle \int^{1}_{0} \int^{1-x}_{0} dydx\)