1. ## vector force fields

By attempting to find a V such that F = $\displaystyle - \nabla V$ in each case, find which of the following vector force fields are conservative, and obtain an appropriate potential energy function $\displaystyle V$ when it exists.
(i) F $\displaystyle (x, y, z)$ = $\displaystyle 3x^2yz$ i $\displaystyle + (x^3z - cos y)$ j $\displaystyle - x^3y$ k
(ii) F $\displaystyle (x, y, z)$ = $\displaystyle (2xyz^2 - 6xyze^{x^2yz})$ i $\displaystyle + (x^2z^2 - 3x^2ze^{x^2yz})$ j $\displaystyle + (2x^2yz - 3x^2ye^{x^2yz})$ k.

2. 1. to determine which field is conservative Calculuate curlF
If CurlF = 0 then F is conservative

2. To find V
1. Integrate the x component wrt x and add a fn h(y,z) instead of an integration constant
2. Differentiate the result wrt y and set it equal to the y component

Integrate this with respect to find h(y,z) and add on a fn g( z)

3. Differentiate this wrt to z snd set equal to the z- component

Integrat this and add it alll together

I'll illustrate with a simple example

F = yzi + (xz+2y)j +(xy+2z)k

V= xyz +h(y,z)

dV/dy = xz + dh/dy = xz + y

so dh/dy = y so h= y^2 + g(z)

V= xyz +y^2 +g(z)

dV/dz = xy + dg/dz = xy +2z

dg/dz = 2z so g = z^2

V= xyz + y^2 +z^2

3. You want F = -del V (must be a physics problem mathematicians tend to use F = del V)

so must multiply result by a negative.

4. In all honesty sometimes simple inspection does the trick

For ii V = - x^2yz^2 +3e^(x^2yz)