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Math Help - Vector field and flow

  1. #1
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    Vector field and flow

    Vector field F(x,y,z) = xi + 0j + zk and the cylinder obtained by rotation the straight x=1, y=t, z=0 with -1 \leq t \leq 1 around the OY axis. Determine the flow of the field F in the direction normal outside of the cylinder. (the normal vector points to out)
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  2. #2
    MHF Contributor Calculus26's Avatar
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    If you know the divergence theorem it is simply 2*V = 4pi

    Even though this is not a closed surface the flux through the right and left faces is 0 (j component of F is 0) you can close up the cylinder and use the divergence theorem

    If Not

    The equation of the cylinder is then x^2 +z^2 = 1 -1< y <1

    You'll have to break this up into the upper half and lower half over the rectangle -1 < x < 1 and -1 < y <1

    For the upper half N = x/z i + k For the lower half N = -x/z i - k

    You'll get 2pi in both cases
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  3. #3
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    Quote Originally Posted by Calculus26 View Post
    If you know the divergence theorem it is simply 2*V = 4pi

    Even though this is not a closed surface the flux through the right and left faces is 0 (j component of F is 0) you can close up the cylinder and use the divergence theorem

    If Not

    The equation of the cylinder is then x^2 +z^2 = 1 -1< y <1

    You'll have to break this up into the upper half and lower half over the rectangle -1 < x < 1 and -1 < y <1

    For the upper half N = x/z i + k For the lower half N = x/z i - k

    You'll get 2pi in both cases
    divergence theorem is:
    \frac{d(x)}{dx} + \frac{d(0)}{dy} + \frac{d(z)}{dz} = 2
    2 \int_0^{2 \pi} ds = 2 \pi it is ?

    Sory
    I do not understand
    " For the upper half N = x/z i + k For the lower half N = x/z i - k "

    I do not know well integral surface.
    What are the steps to calculate?

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  4. #4
    MHF Contributor Calculus26's Avatar
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    Divergence Theorem : Flux is triple integral of divergence

    Here div = 2 volume is pi (1)^2 *2 = 4pi

    Not sure where the integral comes from but even with this integral is 2*2pi is still 4pi

    If you're not going to use the divergence theorem

    then you're just going to have to look up calculating N for a surface

    where z = f(x,y)---do the work
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  5. #5
    MHF Contributor Calculus26's Avatar
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    If you're using the formula for a flux integral over a parameterized surface

    I see where comes from but this does not involve the divergence---
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  6. #6
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    Quote Originally Posted by Calculus26 View Post
    If you're using the formula for a flux integral over a parameterized surface

    I see where comes from but this does not involve the divergence---
    How can I calculate the normal ?
    Is there a trick to parameterization ?

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  7. #7
    MHF Contributor Calculus26's Avatar
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    I'll tell you what --we've seen 3 different ways of showing the flux is 4pi.

    Submit your work and expanation of your work and then I'll comment on what you are doing right or not right
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