HP 50g Graphing Calculator - Manual (Page 476)

Page 15-6
As an example, in an earlier example we attempted to find a potential function
for the vector field F(x,y,z) = (x+y)i + (x-y+z)j + xzk, and got an error message
back from function POTENTIAL. To verify that this is a rotational field (i.e.,
∇×F
≠ 0), we use function CURL on this field:
On the other hand, the vector field F(x,y,z) = xi + yj + zk, is indeed irrotational
as shown below:
Vector potential
Given a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, if there exists a
vector function
Φ(x,y,z) = φ(x,y,z)i+ψ(x,y,z)j+η(x,y,z)k, such that F = curl Φ = ∇
× Φ, then function Φ(x,y,z) is referred to as the vector potential of F(x,y,z).
The calculator provides function VPOTENTIAL, available through the command
catalog (
‚N), to calculate the vector potential, Φ(x,y,z), given the vector
field, F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k. For example, given the vector
field, F(x,y,z) = -(yi+zj+xk), function VPOTENTIAL produces
i.e., Φ(x,y,z) = -x 2 /2j + (-y 2 /2+zx)k.
It should be indicated that there is more than one possible vector potential
functions
Φ for a given vector field F. For example, the following screen shot
shows that the curl of the vector function Φ
1
= [X
2
+Y
2
+Z
2
,XYZ,X+Y+Z] is the
vector F = ∇× Φ
2
= [1-XY,2Z-1,ZY-2Y]. Application of function VPOTENTIAL