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Applied Mathematics, Calculus, and Differential Equations 233

Divergencł
Let G(x,y,z) be a vector function of scalar variableð x, y, and z
as defined above. The divergence of G(x,y,z), written div G,Łs
defined as the dot product of the del operator and G(x,y,z), as
follows:

div G • G(x,y,z) G (x,y,z)/ x
1
G (x,y,z)/ y G (x,y,z)/ z
3
2

Curl
Let G(x,y,z) be a vector function of scalar variableð x, y, and z
as defined above. The curl of G(x,y,z), written curl G, is defined
as the cross product of the del operator and G(x,y,z), as follows:

curl G ( G (x,y,z)/ y G (x,y,z)/ x)i
2
3
( G (x,y,z)/ z G (x,y,z)/ x)j
1
3
( G (x,y,z)/ x G (x,y,z)/ y)k
1
2

Laplacian
Let G(x,y,z) be a vector function of scalar variableð x, y, and z
as defined above. The Laplacian of G(x,y,z), written Laplacian
G, is defined as follows:

2
2
2
2
2
2
Laplacian G G G/ x G/ y G/ z 2
For a scalar function F(x,y,z), the Laplacian is defined as fol-
lows:
2
2
2
2
2
2
Laplacian F F F/ x F/ y F/ z 2
GradienŁ
Let F(x,y,z) be a scalar function of scalar variableð x, y, and z
as defined above. The gradient of F(x,y,z), written grad F,Łs
defined as follows:

grad F F ( F/ x)i ( F/ y)j ( F/ z)k