Multivariate Calculus - Bruce E. Shapiro

174 LECTURE 21. VECTOR FIELDS
Solution. The curl is
⎛
0 − ∂ ∂z
∇ × F = ⎜
⎝
∂
∂y
∂
∂z
0 − ∂
∂x
− ∂
∂y
∂
∂x
0
( ∂
=
∂y (1) − ∂ ∂z
=
(0, 0, 2ye y2 − 2ye y2)
= 0
⎞
⎛
⎟ ⎝
⎠
e y2
2xye y2
1
(
2xye y2) , ∂ ∂z
Example 21.7 Find ∇ × F for
Solution. The curl is
⎛
0 − ∂ ∂z
∇ × F = ⎜
⎝
∂
∂y
∂
∂z
0 − ∂
∂x
− ∂
∂y
∂
∂x
0
( ∂
=
∂y (0) − ∂ ∂z (2xy) , ∂ ∂z
= (0, 0, 4y)
⎞
⎠
(
e y2) − ∂
∂x (1), ∂
(
2xye y2) − ∂ (e y2))
∂x
∂y
F = ( x 2 − y 2 , 2xy, 0 )
⎞
⎛
x 2 − y 2 ⎞
⎟ ⎝
⎠ 2xy ⎠
0
(
x 2 − y 2) − ∂
∂x (0), ∂
∂x (2xy) − ∂ (
x 2 − y 2))
∂y
Definition 21.8 The Laplacian of a scalar field f(x, y, z) is given by the product
∇ 2 f = ∇ · ∇f = div gradf = ∂2 f
∂x + ∂2 f
∂y + ∂2 f
∂z
(21.9)
The Laplacian of a scalar field is another scalar field.
Theorem 21.1 Properties of Vector Operators Let f : R 3 ↦→ R be a scalar
field and F : R 3 ↦→ R 3 be a vector field. Then all of the following properties hold:
(a) ∇ · ∇ × F = 0
(b) ∇ × ∇f = 0
(c) ∇ · (fF) = f∇ · F + F · ∇f
(d) ∇ × (fF) = f∇ × F − F × ∇f
(e) ∇ · (F × G) = G · ∇ × F − F · ∇ × G
(f) ∇ · (∇f × ∇g) = 0
Revised December 6, 2006. Math 250, Fall 2006