HP 49g+, Руководство пользователя, Страница: 473 / 862

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 + xz k , 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 + y j + z k , 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) = -(y i +z j +x k ), function VPOTENTIAL produces
i.e.,
Φ
(x,y,z) = -x
2
/2 j + (-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