By attempting to find a V such that F = in each case, find which of the following vector force fields are conservative, and obtain an appropriate potential energy function when it exists.
(i) F = i j k
(ii) F = i j k.
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