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