Page 491 - Maxwell House

APPENDIX 471

2 2 2
2 (A.23)
+ + + = 0
2 2 2
In some cases, the parameter k may be a function of coordinates.
Curl of Vector Function

The ability of the vector field (, , ) = (, , ) + (, , ) + (, , ) to swirl
0 0 0
is measured by special differential operator denoted by symbol curl F

0 0 0

curl = � � = � − � + � − � + � − � (A.24)
0 0 0

Vector Differential Operator (called Del or Nubla)
The formal differential vector-operator

= 0 + 0 + 0 (A.25)

is used as a shorthand form to simplify many long mathematical expressions. Applying usual
vector operations, we obtain

grad = ∙ = + + (A.26)
0 0 0

∘ grad = ( ∘ ) ∙ = + + (A.27)

The last is the directional derivative or rate of change of a field in the direction of the
vector = + + . The same formal multiplication leads to
0
0
0

div = ∘ = � 0 + 0 + 0 � ∘ � + + � = + + (A.28)
0
0
0

curl = × (A.29)
and so on. Mainly, we can evaluate with the help of the nubla operator differential operators of
second order in Cartesian coordinate system

2
div(grad ) = ∘ ( ∙ ) = ∇
2 2 2 � (A.30)

2
∇ = ∙ = + +
2 2 2
and the vector Laplacian as
∇ = ∇ + ∇ + ∇ = [∇ , ∇ , ∇ ] (A.31)
2
2
2
2
2
2
2

0
0

0

Unfortunately, as we will show later the vector Laplacian in the cylindrical or spherical
coordinate system is a somewhat cumbersome expression. Another example as well is
div(curl ) = ∘ ( × ) ≡ 0, since formally ∥