PHYS 3323 Lesson 2
Div, Grad, Curl
A.
Freshman Calculus
1.
Ordinary Derivatives
We are often interested in knowing how much a function (money in
our bank account, temperature, etc) has changed as we change a
variable (time, altitude, etc) an infinitesimal amount, dx. Calculus is
the math of change and enables us to answer these questions.
The change in a function, df, is found by
 df 
 dx
 dx 
df  
Change in function
Amount the variable changed
Rate of change (slope)
2.
Ordinary Integrals
To find the total change in a function, we simply add up all of the
infinitesimal changes. This is the fundamental theorem of calculus.
b
dF
 dx dx  F(b)  F(a)  ΔF
a
3.
Integration by Parts
Evaluating integrals in closed form can be difficult. A particularly
useful technique for solving integrals is “Integration by Parts”.
b
b
 dg 
 df 
f
dx




 dx
 g dx dx  f g a


a 
a 
b
B.
Multivariable Calculus
In most real world problems, functions (temperature, etc) depend on
more than just one variable. Unfortunately, determining the change in
a multivariable function is somewhat more complicated than for
single variable (ordinary) functions. In particular, the rate of change
(derivative) depends on which variables are changed and how they are
changed. Two new and interesting questions also arise in these
problems:
1.
2.
What is the maximum rate of change for the function?
How must the variables be changed in order to obtain this
maximum rate of change?
To start answering these questions for functions that depend only on
the three spatial dimensions, we begin by determining the change in
the function by allowing only one variable to change at a time. We
then add the change in the function for each variable to find the total
change in the function.
Change in variable x
Total Change
 f 
 f 
 f 
 dx    dy    dz
 x 
 z 
 y 
df  
Rate of change as only x is changed
C.
Del Operator and Gradient
We would now like to develop answers to the questions posed above.
Using our knowledge of the scalar product in Cartesian form, we see
that we can rewrite our equation for the total change in a spatial
function as
 f
 f 
f 
df    î    ĵ    k̂   dx î  dy ĵ  dz k̂ 

 x   y   z   
The last term is the displacement vector in Cartesian form. Thus we
have that the total change in a function is
 


 
df    î    ĵ    k̂  f  d r
 x   y   z  
The first term is an operator with three components like a vector! By
checking its rotation properties, it is possible to demonstrate that it is a
vector!
1.
Del Operator in Cartesian Form - 
  î



 ĵ  k̂
x y
z
The del operator is actually defined by the gradient and will have
more complicated forms in other coordinate systems.
2.
Gradient of a Scalar Function - f
We now have our final result for the total change in a spatial function
as
Del Operator
Total Change

df   f  dr
Gradient of a Function
This equation defines both the gradient of a scalar function and
the del operator. It makes no reference to the coordinate system
being used!!
The gradient of a scalar function is a vector.
1.
2.
The magnitude of the gradient of a scalar function is equal to the
largest rate of change of the scalar function.
The direction of the gradient of the scalar function is in the
direction of the largest rate of change of the function.
These final two observations are a direct consequence of our work on
scalar dot products and answer the two questions we raised about
changes in multivariable functions.
The gradient is used for many applications besides E&M. Computer
programs use the gradient to find the quickest approach to adjusting
parameters when fitting data.
Finally, from our work with the scalar product we see that the gradient
must be perpendicular to constant function surfaces in order for dF to
be zero for all arbitrary displacements. This is the mathematical proof
that electrostatic field lines are perpendicular the equipotential
surfaces that we learned in PHYS2424.
D.

Flux () and Divergence of a Vector -   E
1.
Definition of Flux
In the Mechanical Universe tape that we watched in PHYS2424, we
were told that flux comes from the Greek for “water flow.” We also
developed the following mathematical definition.

dA  dA n̂

E

The flux, E , of the vector E passing through a surface of area A like
the top of the cube shown above is given by
Φ 
E
 
E
  dA
surface
2.
Definition of the Divergence

The divergence of an arbitrary vector E is the scalar product of the del
vector and the arbitrary vector. It follows that the divergence is a
scalar quantity. However, it is important to note that one must be
careful in applying the scalar product when dealing with the del
operator. For instance, the order of the operation is important (i.e.
 
  E  E   ). Thus, the divergence is usually defined by the
following coordinate independent definition.

The divergence of a vector E at some point in space is defined as the
flux per unit volume through an infinitesimal closed surface about that
spatial point.



1
  E  lim
E

d
A
v0 v 
surface
Physically, the divergence tells us if there is a net flow at a particular
point in space. A positive divergence indicates a source of flow (water
faucet for instance). A negative divergence is a flow sink (a drain in
your sink).
We will need the definition of the divergence to develop the very
important Gauss’ Theorem that you memorized in PHYS2424.
3.
Divergence in Cartesian Coordinates
The form of the divergence is simple in Cartesian coordinates.
 E
E y E z
E x 

x
y
z
E.

Circulation and the Curl of a Vector -   E
1.
Definition of Circulation

The circulation of a vector E is defined as the line integral around
some closed path.
 
E
  dl
curve
We previously dealt with circulation in PHYS2424 and in calculating
the work done by a force upon a body over a closed path in
PHYS1224. You may remember that the value for our work integral is
usually path dependent except for conservative forces where the
circulation integral is always zero.
2.
Coordinate Independent Definition of the Curl of a Vector


The curl of a vector E is defined to be the circulation of E per unit
area around an infinitesimal loop. The component in along the
n̂ direction is found by



 
1
n̂    E  lim
E
dl
A0 A 
curve
with the curve being any infinitesimal loop whose surface normal is
n̂ .

The curl is a vector that tells us how much the vector E ”curls around”
a particular point.
3.
Cartesian Component Form of the Curl

The curl of an arbitrary vector E in Cartesian coordinates can be
found using cross product formula
î


  E  Determinan t
x
Ex
F.
ĵ

y
Ey
k̂

z
Ez
Second Order Derivatives in Vector Calculus
The gradient of a scalar function is a vector. Thus, we can take the
divergence or curl of this new vector.
1.
Divergence of the Gradient (Laplacian) of a Scalar Function
The divergence of the gradient occurs so frequently in physics that it
has its own special name and symbol.
 2f    f 
In Cartesian coordinates, the Laplacian has the simple form
 2f  2f  2f
 f 2 2  2
x
y
z
2
In other coordinate systems, the form of the Laplacian is more
complicated.
2.
Curl of the Gradient of a Scalar Function
The curl of the gradient of any scalar function is always zero!!
 f  0
This important result provides a method for developing vector’s that
have zero circulation (i.e. are conservative). Such a vector can be
produced by taking the gradient of a scalar function! In PHYS1224,
the scalar function used to produce the negative of a conservative
force was the potential energy function. In PHYS2424, the gradient of
the electric potential (voltage) was used to produce the negative of the
electric field for electrostatic problems.
3.
Divergence of the Curl of a Vector
The divergence of the curl of any arbitrary vector is always zero!!



   E 0
This important result provides a method for developing vectors with
zero divergence (i.e. no radial flow). Such a vector is produced by
taking the curl of a vector! In PHYS2424, you learned that magnetic
fields have no divergence (Gauss Law of Magnetism). Thus,
 we will
see later that physicists use the magnetic vector potential, A , to insure
that the magnetic field has no divergence.


B A
4.
Curl of the Curl of a Vector
Using our BAC-CAB rule, we obtain the following useful result:

 




    E    E  2 E
G.
Other Potentially Useful Vector Calculus Results
The vector calculus relations in part F should be put to memory as
they are of essential importance in physics and engineering
applications. Other vector calculus relationships can be obtained
either by applying the rules of Calculus or by looking them up in
tables when needed.
f g  fg  gf



  g F  g  F  g  F

 
 
 
 

  F G  G   F- F   G  F   G  G   F

H.
 
   

 


 
  



  g F  g   F  g  F

 
  
  F  G    F  G  F   G

Gauss’s Theorem
The flux of a vector quantity outward through any closed surface, S, is
equal to the integral of the divergence of the function in an enclosed
volume V.
 

E

d
A



E
dv


S
v
This important theorem actually has three different names:
1.
Gauss’s Theorem
2.
Divergence Theorem
3.
Green’s Theorem
It is a consequence of our previous definition of divergence since any
finite size volume can be divided into a large number of infinitesimal
volumes with the internal fluxes canceling as shown in your textbook.
We used this theorem to transform from the integral to differential
form of Maxwell’s Equations in PHYS2424. You should put it to
memory as you will find it to be extremely useful throughout this
course.


I.
Stokes Theorem
The circulation of a vector function around a closed curve C is
equal to the flux of vorticity through any surface S bounded by a
curve C,

 

E

d
l



E

d
A




C
S
This theorem follows from our previous definition of the curl as
shown in the textbook. We also covered this theorem in PHYS2424
and should be put to memory.
J.
Solenoidal and Irrotational
A vector with zero curl is called irrotational.

 E 0
A vector with zero divergence is called solenoidal.

  E 0
K.
Helmholtz Theorem

Let E be differentiable at all points in space with divergence
 

  E  k and curl   E  C . If k and d approach 0 faster than r 2 and



E  0 as r  0 then E   Ψ    A where
k dv'
 

4π
r  r'
all space
Ψ

c dv'
 

4π
r  r'
all space

A
We discussed this theorem conceptually in PHYS2424. The
theorem tells us that if certain boundary conditions are satisfied
we can completely determine a vector as long as we know the
vector’s circulation and divergence. This is why each vector
field in E&M has two equations (1 for circulation and 1 for
divergence). It also provides us a method for finding the vector
provided we know its divergence and curl.