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  1. #1 thoroughly Stuck! 
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    alright, this bit of homework has really got to me, so I will post the original question here, along with the work I've done. Please help me out guys:

    Calculate the surface integral Where and S is the entire sphere with outward orientation

    SO, the tip I got from lectures is that we are basically only considering F on S. So:

    I redifined S in terms of spherical coordinates such that because the radius is a constant, I don't need to consider it (as independent) when I make the parameterized surface, since it is always 3. I can rewrite the integrals now as:

    and, because I switched to spherical coordinates, I also need a jacobian of which, in this case, is

    Now, I take the derivative of this separetly with respect to each variable, and then cross them (still the tips from the lectures), which would, effectively, be

    ==

    okay, that all said, here is where it's my own brain, and I feel like I've missed something obvious, but here goes anyway.

    for my varibles,

    the fun part:

    = and I have no idea where to go from here. I only combined like terms and pulled out a ... any helpful hints appreciated.


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  3. #2 Re: thoroughly Stuck! 
    . DrRocket's Avatar
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    Try using Stokes Theorem (or maybe you have heard it called the Divergence Theorem for this case).


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  4. #3  
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    Life is good. thank you Dr.

    becomes
    which, when we substitute back in our original problems we get:
    which becomes:
    and, as the last 2 terms are opposites, they cancel, and all I have is
    which, when evaluated over spherical coordinates, becomes
    if I'm not mistaken. Which, will evaluate to:
    which is, more elegantly, , yes?
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  5. #4  
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    Quote Originally Posted by Arcane_Mathamatition
    Life is good. thank you Dr.

    becomes
    which, when we substitute back in our original problems we get:
    which becomes:
    and, as the last 2 terms are opposites, they cancel, and all I have is
    which, when evaluated over spherical coordinates, becomes
    if I'm not mistaken. Which, will evaluate to:
    which is, more elegantly, , yes?
    yep

    It is even easier when you know the formula for the volume of a sphere -- -- and just substitute the radius of 3 and multiply by the constant that you are integrating which is 3 again. You can do the whole problem without evaluating a single integral directly.
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  6. #5  
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    ah, good to know, but I have a sneaking suspicion we wont have a sphere to integrate over on our final. good tip though, thanks!
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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