HP 49g+, Руководство пользователя, Страница: 471 / 862

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function
φ
(x,y,z) does not exist. In such case, function POTENTIAL returns an
error message. For example, the vector field F (x,y,z) = (x+y) i + (x-y+z) j +
xz k , does not have a potential function associated with it, since,
∂
f/
∂
z
≠
∂
h/
∂
x. The calculator response in this case is shown below:
Divergence
The divergence of a vector function, F (x,y,z) = f(x,y,z) i +g(x,y,z) j +h(x,y,z) k ,
is defined by taking a “dot-product” of the del operator with the function, i.e.,
z
h
y
g
x
f
F divF
∂
∂
+
∂
∂
+
∂
∂
= • ∇ =
Function DIV can be used to calculate the divergence of a vector field. For
example, for
F (X,Y,Z) = [XY,X
2
+Y
2
+Z
2
,YZ], the divergence is calculated, in
ALG mode, as follows:
Laplacian
The divergence of the gradient of a scalar function produces an operator
called the Laplacian operator. Thus, the Laplacian of a scalar function
φ
(x,y,z)
is given by
2
2
2
2
2
2
2
x x x ∂
∂
+
∂
∂
+
∂
∂
= ∇ • ∇ = ∇
φ φ φ
φ φ
The partial differential equation
∇ 2 φ
= 0 is known as Laplace’s equation.
Function LAPL can be used to calculate the Laplacian of a scalar function. For
example, to calculate the Laplacian of the function
φ
(X,Y,Z) = (X
2
+Y
2
)cos(Z),
use: