1. ## Potential function

Find a potential function

$\displaystyle F(x,y,z) = (yz)i + (xz+z)j + (xy+y-1)k$

My solution:

$\displaystyle rotF = [(x+1)-(x+1)]i - [y-y]j + [z-z]k$
$\displaystyle rotF = (0,0,0)$ is conservative

$\displaystyle \frac{ \partial \phi}{ \partial x} = yz$
$\displaystyle \phi (x,y) = yz + f(y)$
$\displaystyle z + f'(y) = \frac{ \partial \phi}{ \partial y}$
$\displaystyle f'(y) = xz ==> f(y) = xyz$

$\displaystyle \phi (x,y,z) = yz + xyz + f(z)$
$\displaystyle y + xy + f'(z) = \frac{ \partial \phi}{ \partial z}$
$\displaystyle y + xy + f'(z) = xy +y -1$
$\displaystyle f'(z) = -1 ==> f(z) = -1 + k$

Potential function:
$\displaystyle \phi (x,y,z) = yz + xyz -1 + k$

Correct ?

2. No her I'm using f for the potential fn and d/dx etc are partial derivatives

in your first integration f = xyz + h(y,z) not yz

df/dy = xz + dh/dy = xz +z

h = yz + g(z)

f= xyz + yz +g(z)

df/dz = xy+y +g ' (z) = xy + y -1

g(z) = -z

f= xyz + y z - z

You can always check by taking grad(f)

3. Ok. Thank you