Conservative Vector Fields
Text A vector field F(p,q,r) = (p(x,y,z),q(x,y,z),r(x,y,z)) is called conservative if there exists a function f(x,y,z) such that F = ∇f.
If f exists, then it is called the potential function of F.
If a threedimensional vector field F(p,q,r) is conservative, then p_{y} = q_{x}, p_{z} = r_{x}, and q_{z} = r_{y}.
Since F is conservative, F = ∇f for some function f and p = f_{x}, q = f_{y}, and r = f_{z}. By the equality of mixed partials,
p_{y} = f_{xy} = f_{yx} = q_{x},
p_{z} = f_{xz} = f_{zx} = r_{x},
q_{z} = f_{yz} = f_{zy} = r_{y}.
If a threedimensional vector field F(p,q,r) is conservative, then its curl is identically zero.
Using the previous part,
\nabla \times F =
\left \begin{array}
\vec{i} & \vec{j} & \vec{k} \\
\frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz} \\
p & q & r
\end{array} \right =
(r_y  q_z, p_z  r_x, q_x  p_y) = (0,0,0).
Demos
Conservative Fields
 
For the most part, this demo works the same way as the demo in 2.5.4. The difference is that in determining a path, you must first choose the "shadow" of the path in the xy plane and then choose the z coordinate of each point.

Exercises 1. For each of the following, use the demo to determine whether or not the vector field F is conservative. If it is conservative, find the potential function of F.
 F = (x, y, z)
 F = (y  z, x + z, x + 2*y)
 F = ( 1, 1, z)
 F = (cos(x), sin(y), arctan(z))
 F = (x/(x^{2}+y^{2}+z^{2}), y/(x^{2}+y^{2}+z^{2}), z/(x^{2}+y^{2}+z^{2}))
 F = ( 0, 0, 9.8)
2. In the 2Variable lab on conservative fields, you have the option of plotting a surface that represents a function whose gradient is the vector field. Why will that not work for 3 variables? What could be used instead?
