Page 168 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 159

1. Gradient. The gradient of is defined by
@ @ @ @ @ @

grad ¼r ¼ i þ j þ k ¼ i þ j þ k ð12Þ
@x @y @z @x @y @z
@ @ @
k
¼ i þ j þ
@x @y @z
2. Divergence. The divergence of A is defined by
@ @ @

div A ¼r A ¼ i þ j þ k ðA 1 i þ A 2 j þ A 3 kÞ ð13Þ
@x @y @z
@A 1 @A 2 @A 3
@x @y @z
¼ þ þ
3. Curl. The curl of A is defined by

@ @ @
curl A ¼r A ¼ i þ j þ k ðA 1 i þ A 2 j þ A 3 kÞ ð14Þ
@x @y @z
i j

k
@ @

@

¼
@x @y @z

A 1 A 2 A 3
@ @ @

@ @ @

¼ i @y @z j @x @z þ k @x @y

A 2 A 3 A 1 A 2 A 1 A 2

@A 3 @A 2 @A 1 @A 3 @A 2 @A 1
k
@y @z @z @x @x @y
¼ i þ j þ
Note that in the expansion of the determinant, the operators @=@x; @=@y; @=@z must precede
A 1 ; A 2 ; A 3 . In other words, r is a vector operator, not a vector. When employing it the laws of
vector algebra either do not apply or at the very least must be validated. In particular, r A is a new
vector obtained by the specified partial differentiation on A, while A r is an operator waiting to act
upon a vector or a scalar.
FORMULAS INVOLVING r
If the partial derivatives of A, B, U, and V are assumed to exist, then
1. rðU þ VÞ¼rU þrV or grad ðU þ VÞ¼ grad u þ grad V
2. r ðA þ BÞ¼ r A þr B or div ðA þ BÞþ div A þ div B
3. r ðA þ BÞ¼ r A þr B or curl ðA þ BÞ¼ curl A þ curl B
4. r ðUAÞ¼ðrUÞ A þ Uðr AÞ
5. r ðUAÞ¼ ðrUÞ A þ Uðr AÞ
6. r ðA BÞ¼ B ðr AÞ A ðr BÞ
7. r ðA BÞ¼ ðB rÞA Bðr AÞ ðA rÞB þ Aðr BÞ
8. rðA BÞ¼ðB rÞA þðA rÞB þ B ðr AÞþ A ðr BÞ
2
2
2
@ U @ U @ U
2
9: r ðrUÞ r U þ þ is called the Laplacian of U
@x 2 @y 2 @z 2
@ 2 @ 2 @ 2
2
and r 2 þ 2 þ 2 is called the Laplacian operator:
@x @y @z
10. r ðrUÞ¼ 0. The curl of the gradient of U is zero.