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Math Help - [SOLVED] General potential of a field

  1. #1
    MHF Contributor arbolis's Avatar
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    [SOLVED] General potential of a field

    Let \bold K(x,y,z)=(4xyz,2x^2z+2yz^2,2x^2y+2y^2z+4z^3) defined in \mathbb{R}^3.
    Show that the field is irrotational and calculate the general potential of this field.
    I've calculated its curl (0), so the field is irrotational.
    Now I must find, I believe, a vector field whose curl is \bold K. I called it \bold A=(a_1,a_2,a_3) but I've too much unknowns. I don't know how to solve the last part. Can someone help me?

    In details I wrote \frac{\partial a_3}{\partial y}-\frac{\partial a_2}{\partial z}=4xyz and 2 other relations of the same kind. At last I ended up with at least 9 unknowns and only 3 equations.
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  2. #2
    Super Member Random Variable's Avatar
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    Since K is irrotational, it's the gradient of a function.
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  3. #3
    Member Black's Avatar
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    I believe you need to find \varphi such that

    \mathbf{K} = -\nabla \varphi , so you have

    \varphi_x = -4xyz

    \varphi_y = -2x^2z-2yz^2

    \varphi_z = -2x^2y-2y^2z-4z^3.
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  4. #4
    MHF Contributor arbolis's Avatar
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    Ah thanks to both. I get it. I forgot the theorem or consequence of a theorem that if a field is irrotational, then it's the gradient of some function.
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  5. #5
    MHF Contributor arbolis's Avatar
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    Clueless

    I thought I had it, but I realize I made a mistake.
    Now I got it, but I think I got it by chance. Would someone explain me how to reach \varphi from \varphi _x, \varphi _y and \varphi _z?
    I realized that \int \varphi _z dz =\varphi. Which gives \varphi =-2x^2yz -y^2z^2-z^4.

    I was thinking about integrating \varphi _x with respect to dx and sum up a function g(y,z) then integrating \varphi _y with respect to y and sum up a function g_1(x,z) and integrating \varphi _z with respect to z and sum up a function g_2 (x,y). Lastly, summing all these functions and try to determine g, g_1 and g_2.
    I just don't understand how I got the right result with what I've done...
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  6. #6
    Super Member Random Variable's Avatar
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    That's what I would do.

     \varphi = -2x^{2}yz + g_{1}(y,z) = -2x^{2}yz-y^{2}z^{2} + g_{2}(x,z) = 2x^{2}yz - y^{2}z^{2}-z^{4} + g_{3}(x,y)

    so  g_{1}(y,z) = -y^{2}z^{2}-z^{4}, g_{2}(x,z) = -z^{4}, \text{and} \ g_{3}(x,y) =0
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  7. #7
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Random Variable View Post
    That's what I would do.

     \varphi = -2x^{2}yz + g_{1}(y,z) = -2x^{2}yz-y^{2}z^{2} + g_{2}(x,z) = 2x^{2}yz - y^{2}z^{2}-z^{4} + g_{3}(x,y)

    so  g_{1}(y,z) = -y^{2}z^{2}-z^{4}, g_{2}(x,z) = -z^{4}, \text{and} \ g_{3}(x,y) =0
    Makes perfectly sense.
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