# Help with this one

1. Feb 3, 2004

### jlmac2001

Problem: Verify that F= i - zj - jk is conservative, and find the scalar potential V(x,y,z) such that F=-gradV.

How do you tell if this is conservative?

will the scalar potential be: F=-gradV=-(-k-j)= (k+j)

2. Feb 3, 2004

### HallsofIvy

Staff Emeritus
Uh- did you notice that (k+ j) is not even a scalar? Or that "grad" is only defined for scalar functions so that "-grad V" is not defined?

You are going the wrong way: you need to find a function V(x,y,z) such that -gradF= V. That is, we seek a function V(x,y,z) such that FVx= -1, Vy= -z and Vz= OOPs, I have absolutely no idea what you mean by "-jk". I am going to assume that you meant "-yk" and mistyped: Fz= -y.

Okay, if there exist such a function then Vxy=
(-1)y= 0 and Vyx= (-z)x= 0. Okay that's possible: Vxy= Vyx as expected. Vyz= (-z)z= -1 and Vzy= -yy= -1. Yes! We have Vyz= Vzy.
Finally, Vxz= (-1)z= 0 and Vzx= (-y)x= 0.

Yes, is conservative.

We must have Vx= 1 so V(x,y,z)=x+ g(y,z) (If g depends only on y and z, then it dervative with respect to x is 0).
The Vy= gy(y,z)= -z so g(y,z)= -yz+ f(z) and V(x,y,a)= x- yz+ f(z). Then Vz= -y+ f'(z)= -y (assuming thatit was supposed to by -yk rather than -ik) so f'(z)= 0 and f is a constant, C. V(x,y,z)= x- yz+ C.

3. Feb 3, 2004

### Staff: Mentor

One way is just find the potential function, like Halls did.

Another way is to evalute the curl of the force; if it's zero, the force is conservative and there exists a potential function.