Multivariate Calculus - Bruce E. Shapiro

LECTURE 21. VECTOR FIELDS 173
Example 21.4 Find the divergence of the vector field
F(x, y, z) = xi + ( y 2 x + z ) j + e x k
Solution.
(
i ∂
∇ · F =
∂x + j ∂ ∂y + k ∂ ∂z
= ∂x
∂x + ∂ (
y 2 x + z ) + ∂ ∂y
∂z ex
= 1 + 2xy
)
· (xi
+ ( y 2 x + z ) j + e x k )
Example 21.5 Show that ∇ · (fa) = a · ∇f
Solution. Suppose that a = (a 1 , a 2 , a 3 ). Then since a 1 , a 2 , and a 3 are all constants,
∇ · (fa) = ∇ · (a 1 f, a 2 f, a 3 f)
( )
∂
=
∂x , ∂
∂y , ∂
·
∂z
= ∂(a 1f)
+ ∂(a 2f)
+ ∂(a 3f)
∂x ∂y ∂z
∂f ∂f ∂
= a 1
∂x + a 2
∂y + a 3
∂z
= (a 1 , a 2 , a 3 ) ·
= a · ∇f
( ∂f
∂x , ∂f
∂y , ∂f
∂z
Definition 21.7 The curl of a vector field F = (F 1 , F 2 , F 3 ) is given by the cross
product
( ∂
curl F = ∇ × F =
∂x , ∂ ∂y , ∂ )
× (F 1 , F 2 , F 3 ) (21.8)
∂z
The curl of a vector field is another vector field.
We can derive a formula for the curl as follows:
⎛
0 − ∂ ⎞
∂ ⎛ ⎞
∂z ∂y
∇ × F = ⎜ ∂
⎝ ∂z
0 − ∂
F 1
⎟ ⎝
∂x⎠
F 2
⎠
F 3
=
− ∂
∂y
Example 21.6 Find ∇ × F for
∂
∂x
0
( ∂F3
∂y − ∂F 2
∂z , ∂F 1
∂z − ∂F 3
∂x , ∂F 2
∂x − ∂F 1
∂y
F =
(
)
e y2 , 2xye y2 , 1
)
)
Math 250, Fall 2006 Revised December 6, 2006.