CactuarEnigma
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Alright, so the field is \mathbf{F} = (z^2 + 2xy,x^2,2xz)
it's a gradient only when f_x = z^2 + 2xy, f_y = x^2 and f_z = 2xz
integrate the first equation with respect to x to get f(x,y,z) = \int z^2 +2xy\,dx = xz^2 + x^2y + g(y,z)
now, f_z(x,y,z) = g_z(y,x) which is 2xz
integrate that with respect to z, g(y,z) = \int 2xz\,dz = xz^2 + h(y)
plug that into our previous expression f(x,y,z) = xz^2 + x^2y + g(y,z) = 2xz^2 + x^2y + h(y)
derive that with respect to y for f_y(x,y,z) = x^2 + h'(y) = x^2
so h'(y) = 0 and h(y) = c and we can set c = 0 and now the potential function is f(x,y,z) = 2xz^2 + x^2y which is wrong. It should be f(x,y,z) = xz^2 + x^2y. Help me please. Sorry if my LaTeX is wonky.
it's a gradient only when f_x = z^2 + 2xy, f_y = x^2 and f_z = 2xz
integrate the first equation with respect to x to get f(x,y,z) = \int z^2 +2xy\,dx = xz^2 + x^2y + g(y,z)
now, f_z(x,y,z) = g_z(y,x) which is 2xz
integrate that with respect to z, g(y,z) = \int 2xz\,dz = xz^2 + h(y)
plug that into our previous expression f(x,y,z) = xz^2 + x^2y + g(y,z) = 2xz^2 + x^2y + h(y)
derive that with respect to y for f_y(x,y,z) = x^2 + h'(y) = x^2
so h'(y) = 0 and h(y) = c and we can set c = 0 and now the potential function is f(x,y,z) = 2xz^2 + x^2y which is wrong. It should be f(x,y,z) = xz^2 + x^2y. Help me please. Sorry if my LaTeX is wonky.
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