Math 251 Section 14.5
Curl and Divergence

Definition: For



F ( x, y , z )  P ( x, y , z ) i  Q ( x, y , z ) j  R ( x, y , z ) k
,



i

curl F    F 
x
P
j

y
Q
k
 R Q    R P    Q P  

 



 i 
 k
 j  
z  y z 

x

y
 x z 


R



The magnitude of
curl F ( x0 , y0 , z 0 )
is a measure of the speed of rotation about and axis through

( x0 , y0 , z 0 ) with the axis pointing in the direction of curl F ( x0 , y0 , z 0 ) .

If


curl F ( x0 , y0 , z 0 )  0 , then F
is called irrotational at
( x0 , y 0 , z 0 ) .
From 14.3 and 14.4, we have that for a two dimensional vector field with



F ( x, y )  P ( x, y ) i  Q ( x, y ) j
on a simply connected domain D with P and Q having

continuous partial derivatives in D then,
F
is conservative and independent of path if and only if
 Q P   
curl F  

 k  0 on D.
 x y 

The same is true in three dimensions but requires Stokes’ Theorem of section 14.8. We assume it
without proof now.



Theorem:   F is identically 0 in the simply connected domain, D, if and only if F is
conservative in D, always assuming all partial derivatives are continuous in D. Note: One direction,

  f  0
follows from Clairaut’s Theorem. The other direction follows from Stokes’ theorem.
Examples: 1. Is
F ( x , y , z )  xz i  xy j  yz k


2. Is


x2


conservative?
2 y2
F ( x , y , z )  ( xy  e ) i  ( 2 zy e
2

 x y) j  e
2
y2

k
conservative?
Divergence of a Vector Field
Definition: The divergence of the vector field




F ( x, y , z )  P ( x, y , z ) i  Q ( x, y , z ) j  R ( x, y , z ) k


div (F )    F 

If
is the scalar function
P Q R


.
x y z

div ( F )  0, F
is called incompressible.

If P, Q, and R have continuous 2nd partial derivatives on
from Clairaut’s Theorem.
R 3 then div ( curl ( F ))  0 . This follows