Curl; Irrotational Fields And Potential Function - HP 48gII User Manual

Graphing calculator.

Curl

The curl of a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, is defined by
a "cross-product" of the del operator with the vector field, i.e.,
curl
F
h
i
y
The curl of vector field can be calculated with function CURL. For example,
for the function F(X,Y,Z) = [XY,X

Irrotational fields and potential function

In an earlier section in this chapter we introduced function POTENTIAL to
calculate the potential function (x,y,z) for a vector field, F(x,y,z) = f(x,y,z)i+
g(x,y,z)j+ h(x,y,z)k, such that F = grad
conditions for the existence of , were:
g/ z = h/ y. These conditions are equivalent to the vector expression
A vector field F(x,y,z), with zero curl, is known as an irrotational field. Thus,
we conclude that a potential function (x,y,z) always exists for an irrotational
field F(x,y,z).
i
j
[ ]
[ ]
F
x
y
f
(
x
,
y
,
z
)
g
(
x
,
y
g
f
h
j
k
z
z
x
2
2
2
+Y
+Z
,YZ], the curl is calculated as follows:
=
. We also indicated that the
f/ y = g/ x, f/ z = h/ x, and
curl F =
F = 0.
k
[ ]
z
,
z
)
h
(
x
,
y
,
z
)
h
g
y
z
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