This document discusses CURL and its applications in vector fields. CURL describes the tendency of a fluid to cause rotation and was introduced by physicist James Clerk Maxwell. The CURL of a velocity field F measures how a fluid would turn an inserted paddle device. A flow with zero CURL is considered irrotational without vortices. DIVERGENCE describes the flux of a vector field through a surface, indicating fluid sources and sinks. For incompressible fluids, the DIVERGENCE is zero with no change in density. CURL and DIVERGENCE provide physical insight into fluid motion and flows.

1. CURL and its applications
PREPARED BY:
KUSHAGRA GANERIWAL (130010111009)
GUIDED BY:
PROF. SHAILESH KHANT (EC DEPT.)

2. k
y
P
x
Q
j
x
R
z
P
i
z
Q
y
R
curlF )()()(
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
=
vector
RQP
zyx
kji
FFcurl
∂
∂
∂
∂
∂
∂
=×∇=
Cross product of the del
operator and the vector F
0)( =∇×∇= ffgradcurl
0)()( =×∇⋅∇= FFcurldiv

3. 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.

4. 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.

5. 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.

6. Physical Interpretations
 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
z
R
y
Q
x
P
divF
∂
∂
+
∂
∂
+
∂
∂
=

7. Physical Interpretations
 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
z
R
y
Q
x
P
divF
∂
∂
+
∂
∂
+
∂
∂
=