# Potential function for conservative vector field

1. Apr 30, 2008

### kasse

[SOLVED] Potential function for conservative vector field

1. The problem statement, all variables and given/known data

Find a potential function for the conservative vector field F = <x + y, x - z, z - y>

2. The attempt at a solution

OK, we know that

(1) fx = x + y
(2) fy = x - z og
(3) fz = z - y

We can then integrate (1) with respect to x, and we get

(4) f = (1/2)x2 + xy + C(y,z)

We then differentiate (4) wrt y and get:

(5) fy = x + C'(y,z)

Comparing (2) and (5):

(6) C'(y,z) = -z

Integrate this wrt z:

(7) C(y,z) = -(1/2)z2 + C(y)

We then have the following expression for the potential function so far:

(8) f = (1/2)x2 + xy + -(1/2)z2 + C(y)

Then I'm stuck. Is my method correct so far?

2. Apr 30, 2008

### Dick

In step (6) when you write C'(y,z)=(-z) the prime means derivative wrt y. So I wouldn't integrate wrt z. You're on the right track. Just pay more attention to what variable the derivatives are taken wrt.

3. Apr 30, 2008

### dynamicsolo

I was with you up to here. Why did you compare (2) and (5) and then integrate with respect to z? It seems to me that if you'd integrated with respect to y first, you'd have gotten

C(y,z) = -yz + C(z) ,

from which point you should be all right...

I guess I prefer integrating each partial differential equation separately and then intercomparing all three. As long as the equations are too complicated, things usually fall into place pretty well.