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Math Help - Vector Calculus

  1. #1
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    Vector Calculus

    2) Let S be the surface of the region bounded by the coordinate planes and the planes x + 2z = 4 and y = 2. Use the Divergence Theorem to the flux of F(x,y,z) = (2xz)i + (xyz)j + (yz)k through S

    3) Let S be the first octant portion of the plane x+y+z =1. Verify Stokes' Theorem for the vector field F=y^2i + z^2j + x^2k

    If you could show your work on how to do these much would be appreciated. Thank you
    Last edited by CaptainBlack; July 27th 2008 at 12:39 AM. Reason: delete double posted question
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  2. #2
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    Quote Originally Posted by davidson89 View Post
    1) Let S be the first-octant portion of the paraboloid z= x^2 + y^2 that is cut off by the plane z=4. If F(x,y,z)=(x^2+z)i + (zy^2)j + (x^2 + y^2 + z)k, find the flux of F through S

    2) Let S be the surface of the region bounded by the coordinate planes and the planes x + 2z = 4 and y = 2. Use the Divergence Theorem to the flux of F(x,y,z) = (2xz)i + (xyz)j + (yz)k through S

    3) Let S be the first octant portion of the plane x+y+z =1. Verify Stokes' Theorem for the vector field F=y^2i + z^2j + x^2k

    If you could show your work on how to do these much would be appreciated. Thank you
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  3. #3
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    Quote Originally Posted by davidson89 View Post
    [snip]
    2) Let S be the surface of the region bounded by the coordinate planes and the planes x + 2z = 4 and y = 2. Use the Divergence Theorem to the flux of F(x,y,z) = (2xz)i + (xyz)j + (yz)k through S
    [snip]
    You have to become more familiar with the basic theorems and their application, as well as setting up and calculating double integrals.

    Do you know what the divergence theorem says?
    Can you take the divergence of F?
    Have you drawn a diagram of the closed surface (without doing this you have no hope of getting the integral terminals for the triple (volume) integral)?
    Can you set up the triple (volume) integral?
    Can you solve the triple (volume) integral?

    The integral terminals for the triple (volume) integral, using the order dx dy dz, are:
    z = 0 to z = (4 - x)/2
    x = 0 to x = 4
    y = 0 to y = 2.

    (The region in the xy-plane is a rectangle).
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  4. #4
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    Quote Originally Posted by davidson89 View Post
    [snip]

    3) Let S be the first octant portion of the plane x+y+z =1. Verify Stokes' Theorem for the vector field F=y^2i + z^2j + x^2k

    If you could show your work on how to do these much would be appreciated. Thank you
    Where are you stuck? Doing the line integral? Or getting the flux of curl F through S?

    Have you drawn a diagram? You can get the edges of S as follows:

    In the xy-plane, z = 0 and the edge is x + y = 1.

    In the xz-plane, y = 0 and the edge is x + z = 1.

    In the yz-plane, x = 0 and the edge is z + y = 1.

    I think you need to put a lot more time into understanding the basics .... line integrals, double integrals, surface integrals etc.

    Do you have textbook or class notes with examples to follow? Do you have access to a library that has textbooks that you can browse?
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