9.7 Divergence and Curl
Vector Fields:
R2  R2
F(x,y)= P(x,y) i + Q(x,y) j
F(x,y,z) = P(x,y,z) i + Q(x,y,z) j + R(x,y,z) k
Example 1: Graph the 2-dim vector field
R3  R3
F(x,y)= -y i + x j
Draw vectors of the same length
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Divergence
P Q R
divF 


x
y
z
Scalar
divF    F
Example 2
2 3
4
5 2
4 6
IF F  ( x y  z )i  (4 x y z ) j  ( y z )k
Find div F
Curl
R Q
P R
Q P
curlF  (

)i  (

) j(

)k
y
z
z
x
x
y
vector
Example 2
2 3
4
5 2
4 6
IF F  ( x y  z )i  (4 x y z ) j  ( y z )k
Find curl F
Curl
i
j k
  
curl F    F 
x y z
P Q R
Remarks:
1)
curl ( grad f )    f  0
2)
div (curl F )    (  F )  0
Cross product of the del
operator and the vector F
WHY Vector Fields
The motion of a wind or fluid can be described by a
vector field.
The concept of a force field plays an important role in
mechanics, electricity, and magnetism.
Physical Interpretations
Curl was introduced by Maxwell
James Clerk Maxwell (1831-1879) Scottish Physicist
[b. Edinburgh, Scotland, June 13, 1831, d. Cambridge, England, November 5,
1879] He published his first scientific paper at age 14, entered the University of
Edinburgh at 16, and graduated from Cambridge University.
Physical Interpretations



Curl is easily understood in connection with the flow of fluids. If a paddle
device, such as shown in fig, is inserted in a flowing fluid, the the curl of
the velocity field F is a measure of the tendency of the fluid to turn the
device about its vertical axis w.
If curl F = 0 then flow of the fluid is said to be irrotational. Which means
that it is free of vortices or whirlpools that would cause the paddle to
rotate.
Note: “irrotational” does not mean that the fluid does not rotate.
Physical Interpretations
P Q R
divF 


x
y
z







The volume of the fluid flowing through an element of surface area per
unit time that is , the flux of the vector field F through the area.
The divergence of a velocity field F near a point p(x,y,z) is the flux per
unit volume.
If div F(p) > 0 then p is said to be a source for F. since there is a net
outward flow of fluid near p
If div F(p) < 0 then p is said to be a sink for F. since there is a net inward
flow of fluid near p
If div F(p) = 0 then there are no sources or sinks near p.
The divergence of a vector field can also be interpreted as a measure of
the rate of change of the density of the fluid at a point.
If div F = 0 the fluid is said to be incompressible