Verifying the Conservativity of F and Finding Scalar Potential for F=-gradV"

[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)

 

[HallsofIvy]

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

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 FV_x= -1, V_y= -z and V_z= OOPs, I have absolutely no idea what you mean by "-jk". I am going to assume that you meant "-yk" and mistyped: F_z= -y.

Okay, if there exist such a function then V_xy=
(-1)_y= 0 and V_yx= (-z)_x= 0. Okay that's possible: V_xy= V_yx as expected. V_yz= (-z)_z= -1 and V_zy= -y_y= -1. Yes! We have V_yz= V_zy.
Finally, V_xz= (-1)_z= 0 and V_zx= (-y)_x= 0.

Yes, is conservative.

We must have V_x= 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 V_y= g_y(y,z)= -z so g(y,z)= -yz+ f(z) and V(x,y,a)= x- yz+ f(z). Then V_z= -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.

 

[Doc Al]

Originally posted by jlmac2001
How do you tell if this is conservative?

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.