$\displaystyle F(x,y,z)=(4xe^z )i +(cosy)j+[(2x^2+cos^2z)e^z]k$
what is the general theory for finding the potential of this.
i know only how to find on 2d functions
The potential function W, if exists, satisfies the conditions...
$\displaystyle \frac{\partial{W}}{\partial{x}} = - 4 x e^{z}$ (1)
$\displaystyle \frac{\partial{W}}{\partial{y}} = - \cos y$ (2)
$\displaystyle \frac{\partial{W}}{\partial{z}} = - (2 x^{2}+ \cos^{2} z) e^{z}$ (3)
The (1), (2) and (3) can be integrated independently. The (1) and (2) have no problem, for (3) You take into account that ...
$\displaystyle \int e^{z}\ \cos^{2} z\ dz = \frac{e^{z}}{5}\ (\cos^{2} z + 2 \sin z\ \cos z + 2) + c$ (4)
... so that is...
$\displaystyle W(x,y,z)= - \sin y - \frac{e^{z}}{5}\ (10 x^{2} + \cos^{2} z + 2 \sin z\ \cos z + 2) + c$ (5)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$