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    corresponding potential function

    The vector field F(x, y, z) = 2xi+zj+yk is conservative. Find a corresponding potential function. That is, find a function f(x, y, z) such that F(x, y, z) = (upside down triangle)f(x, y, z).
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    as I understand, we need to solve the system

    \frac{\partial f}{\partial x}=2x
    \frac{\partial f}{\partial y}=z
    \frac{\partial f}{\partial z}=y

    Integrating first equation we obtain f(x,y,z)=x^2+g(y,z). Plugging to the second and integrating, we get g(y,z)=yz+h(z), and then plugging that to the third equation and integrating, we get h(z)=const. Hence f(x,y,z)=x^2+yz+c for any constant c
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by wik_chick88 View Post
    The vector field F(x, y, z) = 2xi+zj+yk is conservative. Find a corresponding potential function. That is, find a function f(x, y, z) such that F(x, y, z) = (upside down triangle)f(x, y, z).
    have you done differential equations before? this is done in much the same way as the method of exact equations. essentially, we are looking for a function f(x,y,z) so that \bold{F}(x,y,z) = f_x \bold{i} + f_y \bold{j} + f_z \bold{k}.

    so we have,
    f_x = 2x .............(1)
    f_y = z ...............(2)
    f_z = y ...............(3)

    integrating (1) with respect to x, we get

    f(x,y,z) = x^2 + g(y,z) ...........we think of our constant as a function of y and z, for (hopefully) obvious reasons

    now, differentiate this with respect to y

    \Rightarrow f_y = g_y(y,z)

    comparing this to (2), we see that g_y(y,z) = z

    so that g(y,z) = yz + h(z), by integrating g_y(y,z) with respect to y, leaving a constant in terms of a funciton of z.

    so now, go back to f, we see that,

    f(x,y,z) = x^2 + yz + h(z)

    differentiate this with respect to z and compare to (3), we see that:

    f_z = y + h_z(z) = y \implies h_z(z) = 0 \implies h(z) = C, a constant, so finally,

    f(x,y,z) = x^2 + yz + C

    it is trivial to verify that \nabla f = \bold{F}
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