# Thread: finding potential function of F(x,y,z)

1. ## finding potential function of F(x,y,z)

$\displaystyle F(x,y,z)=(4xe^z )i +(cosy)j+[(2x^2+cos^2z)e^z]k$

what is the general theory for finding the potential of this.

i know only how to find on 2d functions

2. ## Re: finding potential function of F(x,y,z)

Originally Posted by transgalactic
$\displaystyle F(x,y,z)=(4xe^z )i +(cosy)j+[(2x^2+cos^2z)e^z]k$

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

$\displaystyle \chi$ $\displaystyle \sigma$