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Math Help - Divergent and rotational

  1. #1
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    Divergent and rotational

    Calculate the div and rot:

    Correct?














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  2. #2
    MHF Contributor Calculus26's Avatar
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    The divergence is correct

    Not sure what you have in mind for rotF

    but it is 5z^2 i +3xz^2 j +4 y^4x k

    The middle row of the determinant are the operators d/dx d/dy and d/dz
    and the last row are the x , y , and z components of F not the parial derivatives
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  3. #3
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    In the middle row is the partial derivatives (div) ?
    In the last line components i, j and k of F ?

    I solved this way not correct ?


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  4. #4
    MHF Contributor Calculus26's Avatar
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    No the middle row is just the operators d/dx etc

    Write F =f i + g j + h k

    then the last row is f - g - h

    Then rotF = (dh/dy-dg/dz)i - (dh/dx-df/dz) j + (dg/dx-df/dy)k
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  5. #5
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    Thank you very much. Look this is correct

    F(x,y,z) = ln(x)i + e^{xyz}j + arctg(\frac{x}{z})k

    div = x^{-1} + xze^{xyz} + \frac{x}{(x^2+y^2)}

    rot = -xye^{xyz}i + \frac{z}{x^2+z^2}j + yze^{xyz}k
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  6. #6
    MHF Contributor Calculus26's Avatar
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    Yes except the last term in your calculation of divF should be

    -x/(x^2+z^2)
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  7. #7
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    Thanks

    How do I calculate : \bigtriangledown . (F x G)

    F(x,y,z) = 2xi + j + 4yk
    G(x,y,z) = xi + yj - zk
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  8. #8
    MHF Contributor Calculus26's Avatar
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    Compute FxG then compute the divergence of the result
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  9. #9
    MHF Contributor Calculus26's Avatar
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    you can also use del*(FxG) = G*(del x F)-F*(del x G)
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  10. #10
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    The symbol \bigtriangledown means divergent ?

    If I have: \bigtriangledown . ( \bigtriangledown x F) and have F how can I calculate ?

    How can I verify that the vector projection r have the property:
    1) \bigtriangledown ||r|| = \frac{r}{||r||}


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  11. #11
    MHF Contributor Calculus26's Avatar
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    is an operator and means d/dx i +d/dy j +d/dz k where d/dx d/dy d/dz ar the partial derivative operators

    f is the gradient where f is a scalar valued function

    X F is the curl or rotation where F is a vector field

    *F is the divergence where F is a vector field * means dot product

    From the previous identity * ( x F) = 0

    r= xi +yj +zk

    ||r|| = (x^2 +y^2 +z^2)^(1/2)

    Compute ||r|| and the result follows directly
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  12. #12
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    Thanks, you're helping me a lot

    check my learning:

    1) \bigtriangledown . (FxG)
    F(x,y,z) = yzi + xzj + xyk
    G(x,y,z) = xyj + xyzk

    My solution:

    FxG = (x^2yz^2 - x^2y^2)i + (-xy^2z^2)j + (xy^2z)k
    div = (2xyz^2 - 2xy^2)i + (-2xyz^2)j + (xy^2)k

    2) \bigtriangledown . (\bigtriangledown x F)
    F(x,y,z) = sen(x)i + cos(x-y)j + zk

    My solution:

    rotF = 0i - 0j + (-sen(x-y))k
    div(rotF) = 0i + 0j + 0k

    3) Check that: \bigtriangledown ||r|| = \frac{r}{||r||}
    r = xi + yj + zk

    My solution:

    div||r|| = \frac{x}{\sqrt{x^2+y^2+z^2}} + \frac{y}{\sqrt{x^2+y^2+z^2}} + \frac{z}{\sqrt{x^2+y^2+z^2}}

    \bigtriangledown ||r|| = \frac{r}{||r||} is correct

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  13. #13
    MHF Contributor Calculus26's Avatar
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    Looks good you have learned well.

    Note

    is always 0 regardless of what F is. so you don't have to calculate but it was a good exercise to verify this.
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  14. #14
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    thanks. good Note
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