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Math Help - conservative and independent of path - line integral

  1. #1
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    Question conservative and independent of path - line integral

    question.

    Find a potential function for the vector field

    F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k


    I don't know how to do this question. Please help me . Thank you very much.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by kittycat View Post
    question.

    Find a potential function for the vector field

    F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k


    I don't know how to do this question. Please help me . Thank you very much.
    F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k

    note that F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k = \bigtriangledown \Phi(x,y,z)

    so:
    2xy^3z^4 = \frac{\partial \Phi}{\partial x}

    3x^2 y^2 z^4 = \frac{\partial \Phi}{\partial y}

    4x^2 y^3 z^3 = \frac{\partial \Phi}{\partial z}

    \implies \Phi (x,y,z) = x^2y^3z^4 + g(y,z)

    from here,
    \frac{\partial \Phi}{\partial y} = 3x^2y^2z^4 + \frac{\partial g}{\partial y} and \frac{\partial \Phi}{\partial y} = 4x^2y^3z^3 + \frac{\partial g}{\partial z}

    \implies \frac{\partial g}{\partial y} = 0 and \frac{\partial g}{\partial z} = 0 (why?)

    \implies g(y,z) = c_1 and g(y,z) = c_2 (or g(y,z) = C)

    \implies \Phi (x,y,z) = x^2y^3z^4 + C
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  3. #3
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    Check to see if it is conservative.
    \begin{array}{l}<br />
 J_z  = 12x^2 y^2 z^3  = K_y  \\ <br />
 K_x  = 8xy^3 z^3  = I_z  \\ <br />
 J_x  = 6xy^2 z^4  = I_y  \\ <br />
 \end{array}
    It is.

    Then find the primitive function.
    f = x^2 y^3 z^4 .
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  4. #4
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    Question



    hi kalagota,
    Why + g(y,z) ??? Could you please explain this point to me?

    Thank you very much.
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  5. #5
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    Quote Originally Posted by kittycat View Post

    Why + g(y,z) ??? Could you please explain this point to me?
    The g(y,z) would act as a constant function when differentiating with respect to x.
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  6. #6
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    1) Kalagota use \nabla instead of the monster you are using.

    2) You can see if curl is a zero vector to know if it has a scalar potentional.
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  7. #7
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    Quote Originally Posted by ThePerfectHacker View Post
    You can see if curl is a zero vector to know if it has a scalar potentional.
    If you note, that is exactly what I did in my solution.
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