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.
$\displaystyle F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k$
note that $\displaystyle F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k = \bigtriangledown \Phi(x,y,z)$
so:
$\displaystyle 2xy^3z^4 = \frac{\partial \Phi}{\partial x}$
$\displaystyle 3x^2 y^2 z^4 = \frac{\partial \Phi}{\partial y}$
$\displaystyle 4x^2 y^3 z^3 = \frac{\partial \Phi}{\partial z}$
$\displaystyle \implies \Phi (x,y,z) = x^2y^3z^4 + g(y,z)$
from here,
$\displaystyle \frac{\partial \Phi}{\partial y} = 3x^2y^2z^4 + \frac{\partial g}{\partial y}$ and $\displaystyle \frac{\partial \Phi}{\partial y} = 4x^2y^3z^3 + \frac{\partial g}{\partial z}$
$\displaystyle \implies \frac{\partial g}{\partial y} = 0$ and $\displaystyle \frac{\partial g}{\partial z} = 0$ (why?)
$\displaystyle \implies g(y,z) = c_1$ and $\displaystyle g(y,z) = c_2$ (or $\displaystyle g(y,z) = C)$
$\displaystyle \implies \Phi (x,y,z) = x^2y^3z^4 + C$