Irrotational Fields And Potential Function - HP 50g User Manual

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
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:
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).
F F
f
⎛ ⎞
h g
⎜ ⎜ ⎟ ⎟
i
⎝ ⎠
y z
2 2
+Y
curl F =
i
x y
( x , y , z ) g ( x ,
⎛ ⎞
f h
j
⎜ ⎟
⎝ ⎠
z x
2
+Z ,YZ], the curl is calculated as follows:
= . We also indicated that the
f/ y = g/ x, f/ z = h/ x, and g/
F = 0.
j k
z
y , z ) h ( x , y , z
⎛ ⎞
h g
⎜ ⎜ ⎟ ⎟
k
⎝ ⎠
y z
)
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