The document is a lecture on vector calculus and operations, primarily aimed at engineering and physics students. It covers fundamental concepts such as fields, functions, scalar and vector fields, as well as time derivatives and differential operations on these fields. Additionally, it discusses mathematical results related to divergence and curl operations on vector fields and provides examples and definitions relevant to scalar and vector calculus.

1. MANMOHAN DASH, PHYSICIST, TEACHER !
Physics for ‘Engineers and Physicists’
“A concise course of important results”
Lecture - 1
Vector Calculus and Operations
Lectures around 9.Nov.2009 + further
content developments this week; 14-18 Aug
2015 !

2. Fields and Functions
What are Fields: A field is a ‘field’ of values of a variable
called a ‘function’, across parameters; such as space, time,
location, distance, speed and momentum or spin, just like
a paddy field is a field of paddy across a region.
What are Functions; A function is an infinite set of values,
for each specification of another value. Eg y is a variable
which is a function of values of x. We write y = f(x) and
say ‘y is a function of x’.
So a field is a collection of values of a function of values of another variable
spread across a chosen parameter. A field is a function. y = f(x) is a field.

3. Fields and Functions

4. Vector and Scalar Fields
There are 2 kinds of fields because there are 2 kinds of
functions; SCALAR and VECTOR.
Scalar as the name suggests “scales”. It scales up, scales down.
It’s a magnitude of anything. Just a magnitude of any
quantity. It can only increase or decrease (or stay constant).
But there are no other properties associated with it.
Eg; time takes to complete a task is a scalar. Its just a number
with no other special attributes.

5. Vector and Scalar Fields
A vector on the other hand is a carrier of a quantity. Hence a
vector carries magnitudes, in as many degrees of freedom, as
possible.
A vector is a quantity defined by a specifed number of scalars,
in different independent directions. These are called as
Degrees of Freedom (DOF) or dimension (Dim).
An usual vector in space and time is 1, 2 or 3-dimensional. But
in general we define n-dimensional vectors where n is any
number from [1 – infinity].

6. Vector and Scalar Fields
A vector field is a vector function and a scalar field is a scalar function. Scalar
Field; T = f(h), temperature as a function of altitude. Vector Field; v = f(t), velocity
as a function of instant of time.
A scalar function is one where the function is defined by only
one value at each parameter-specification. Eg y = f(x).
Physical variables that are scalar are ‘energy’, ‘temperature’
and ‘magnitude of any Physical variable’
A vector function is one where the function is defined by a
vector, that is, n independent scalar values, along n
independent directions, which also satisfy additional rules.

7. Scalar Functions

8. Vector Functions

9. Time derivative of fields
Now that we discussed scalar and vector fields, we can define
their time derivatives.
A vector field has 3 components as stated above
and its time-derivative is given as;
A scalar field looks like this;
A vector field looks like this;
),( trff


),( trAA


So a vector field can be written as; )(ˆ)(ˆ)(ˆ)( tAktAjtAitAA zyx 

Derivative of a vector;
t
A
k
t
A
j
t
A
i
t
tAttA
dt
Ad zyx
t 












ˆˆˆ)()(
lim
0

t
A
k
t
A
j
t
A
i
dt
Ad zyx








 ˆˆˆ


10. Direction of time derivative of fields
Time derivative of a vector field is thus a vector; what angle it
makes with the original vector field? 3 situations arise to
answer this question >>
1. The vector A changes magnitude but not direction; then
resultant time derivative dA/dt is along the original vector A.
2. The vector A changes direction, but not magnitude; then
resultant time derivative dA/dt is perpendicular to the
original vector A.
3. Both the direction and magnitude of A changes, dA/dt
makes arbitrary angles with A.

11. When A does not change direction

12. When A does not change magnitude

13. When A changes in a generic fashion

14. Time derivative of ‘operations on vectors’
We will discuss 4 results of taking time derivatives of different
operations of vectors or vector fields A, B.
1.
2.
3.
4.
dt
Ad
c
dt
Acd


)(
A
dt
df
dt
Ad
f
dt
Afd 


)(
dt
Bd
AB
dt
Ad
dt
BAd




 )(
dt
Bd
AB
dt
Ad
dt
BAd




 )(
c = constant of time
f = function of time
‘•’ = dot product
‘x’ = cross product

15. Differential operations on fields
We will first discuss results of differential operations on scalar
fields. We will discuss the operator grad, nabla or del.
1.
2.
3.
nabladelgrad 

z
ˆ
y
ˆ
x
ˆ








 kji

z
ˆ
y
ˆ
x
ˆ









V
k
V
j
V
iV

The gradient, grad, del or
nabla operator. Here grad
is acting on a scalar
function or field V and
grad is defined from
(x, y, z) = r.
In next slide we will show an important result on gradient of
scalar function >> rdVdV



16. Differential of a scalar field
Lets see how the results we stated in last slide is valid. The
differential of V is dV.
Since
and
So we have
dzkdyjdird ˆˆxˆ 

z
ˆ
y
ˆ
x
ˆ









V
k
V
j
V
iV

It’s a well known mathematical
result that differentials are
given in terms of partials.
if V = V(x, y, z);
dz
V
dy
VV
dV
zy
dx
x 








rdVdV



17. Direction of ‘gradient of scalar field’
The direction of gradient is always along the maximal
change of the scalar field.
Since
The direction of gradient or
above vector E, is normal to
surface of constant V. E is
in the direction along which
most rapid change of V
occurs.
VE 

Or in the given
definition, in direction
opposite to the most
rapid change.
These surfaces are called
as equipotential surface.
The directional derivative of V is its rate of change along n
given by; Vn
dn
dV


ˆ

18. Uniform and non-uniform fields

19. Relation between V and E is comprehensive
V = scalar field <<-->> E = vector field
For every V field there is an E field and vice-a-versa
---------O ---------
V = Potential Energy; E = Corresponding Force
V = Electric Potential; E = Electric Field
V = Gravitational Potential; E = Gravitational Field
V = Simple Harmonic Pot; E = Simple Harmonic Force
V = Molecular Pot. Energy; E = Molecular Force
---------O ---------
etc … …

20. Gradient of ‘operations on scalar field’
The gradient can easily be calculated for addition,
product and ratio of scalar fields in the following way.
i.
ii.
iii.
WVWV 

)( The ‘grad’ is a 3-Dimensional
differentiation.
The usual rules of
differentiation for products
and ratio etc are valid.
In the next slide lets prove the following important
identity for gradient of a scalar;
)()()( WVWVWV 

2
)(
W
WVVW
W
V 



r
V
rrV


 ˆ)(


21. Proof of the following identity
Lets prove
the identity;
)
z
ˆ
y
ˆ
x
ˆ(
rzr
ˆ
yr
ˆ
xr
ˆ
z
ˆ
y
ˆ
x
ˆ



































r
k
r
j
r
i
VrV
k
rV
j
rV
i
V
k
V
j
V
iV

r
ˆ)(:QED
r
ˆ
r
)
zˆyˆxˆ(
r
)(
z
z
,
y
y
,
x
x
)x(;ˆˆxˆ 2/1222






















V
rrV
V
r
r
rV
r
k
r
j
r
i
V
rV
r
r
r
r
r
r
zyrzkyjir



r
V
rrV


 ˆ)(


22. Lets discuss operations on vector field
For vectors basically two operations are possible. One is called
divergence and one is called curl. Lets discuss divergence 1st.
Definitions;
Relations for divergence;
Solenoidal Field; divergence of a vector vanishes everywhere.
zyx
)ˆˆˆ()
z
ˆ
y
ˆ
x
ˆ(
x
x



















zy
zy
AAA
AAdiv
AkAjAikjiAAdiv


)()()(.
)(.
AVAVAVii
BABAi





23. Divergence on vector field
Divergence of a vector field such as that of an electric dipole
offers +ve, –ve and zero divergences at different flux points.

24. Curl of a vector field
Here is how curl of a vector, vector function or vector field is
defined.
Definitions;
Relations for curl;
Irrotational Field; curl of a vector vanishes everywhere.
)-(kˆ)-(jˆ)-(iˆ;
z
,
y
,
x
where
ˆˆˆ
xyxxzxzy
zyx
x
zyx
AAAAAAASo
AAA
kji
AAcurl
yzyz
zy























)()()(.
)(.
AVAVAVii
BABAi





25. Homework
Two short home work questions, if you have carried yourself
this far !
Prove the following;
3r
0r





26. Curl Me !
If vector A is on
x–y plane in the
given fashion, its
curl will span
along z axis. Curl
always follows
right-hand-rule.
Palm curls from
one vector to
another then
right hand thumb
gives resultaing
curl vector.

27. All that we got!
Now that we learned curl, divergence, grad as the
differential operations possible on vector and scalar fields
we have following combinations that are possible, that we
will discuss one by one.
1. Laplacian [On scalar and vectors fields]
2. Curl of grad [of scalar field]
3. Divergence of curl [of vector field]
4. Curl of curl [of vector field]
5. Divergence [of cross product of two vector fields]

28. 1. Laplacian
The divergence of the del operator can act on both scalars
and vectors. Its simply a double differentiation wrt given
parameters, for our discussion space parameters; (x, y, z)=r.
Called as Laplacian it looks like this;
So we have two types of quantities;
A
V
2
2


2
2
2
2
2
2
22
zyx 










29. 2. Curl of grad of scalar field is zero
Curl of grad of a scalar field is zero. This is
because order of partial differentiation does not matter. i.e.
It’s a very important result; if curl of a vector is zero (such
vector is called irrotational) then this vector A can always be
represented as grad of any scalar field. i.e.
Conversely; If a vector field is a grad of a scalar field, its curl
vanishes (its irrotational)
Example;
0 V

yzz
VV




 2
y
2
VA 

0 EVE


30. 3. Div of Curl of a vector field is zero
Divergence of curl of a vector field is zero.
Here is another important result; if a vector field is a curl of
another vector its divergence is zero (its a solenoid field).
Conversely; If the divergence of a vector field is zero (its a
solenoid field) then it can be expressed as curl of a vector.
0 A


31. Curl of Curl and divergence of cross prod.
4. Curl of curl of a vector field;
5. Divergence of cross product of two vector fields;
* If for a scalar field f, its Laplacian is zero, then grad-f is
both solenoidal (divergence = 0) and irrotational (curl = 0) !
AAA
 2
)( 
)()()( BAABBA


0)(,0)(
and
is
02



ff
alirrotation
solenoidalf
f



32. Integrations on vector fields
We can define 3 kinds of integral operation on a vector field;
1. Line or path integral 2. Surface integral 3. Volume
integral !
These definitions come from the kind of differential element
we use for these integrations, a length, an area or a
volume.
Lets discuss the line/path integral of a vector field first !

33. Line/Path Integral of vector fields
Line integral of a vector field A between a, b along a given
path (L);
1. I depends on points a, b and the path between a, b: in
general.
2. If I is independent of the path of integration the vector
field A is called a ‘conservative field’.
3. Line integral of a conservative vector field vanishes along
a closed contour.
4. Example of line integral in physics;
  
b
a
b
a
zyL dzAdyAdAldAI )x( x

  0ldA



QP
ldFWork


34. Line/Path Integral of vector fields

35. Surface Integral of vector fields
Differential surface elements dS are vectors with directions
along the outward normals, n to the surface. An is the
projection of vector A along n and thus the normal
component of vector A. Only the normal component of
vector A contributes to the surface integral.
nAAdSAdSnAI
dSnSdSdAI
n
S
n
S
S
S
S
ˆ,ˆ
ˆ,






Flux of the electric field and work done on a surface are the
example of surface integrals.
 
 SQP
SdESdFW

E,

36. Integral form of curl, div, grad
First lets state the volume integral;
dV is the differential element of Volume.
We have seen the curl, grad, div etc defined as differential
operators. We can also cast them into line, surface and
volume integrals.
We define areas that enclose relevant volumes for surface
integrals and surfaces that are bounded by closed paths
for line intergrals.
i. Gradient ii. Divergence iii. Curl

V
V dVAI

lim
0 V
dS
S
V 






lim
0 V
SdA
A S
V 






n
S
ldA
A C
S
ˆlim
0 







37. Integral and differential theorems
1. Gauss Divergence Theorem;
The ‘volume integral’ of the ‘divergence of vector A’ over a
given volume is equal to the surface integral of the vector
over a closed area enclosing the volume.
 
SV
SdAdVA

1. Stoke’s Theorem;  
CS
ldASdA

)(
The ‘surface integral’ of the ‘curl of vector A’ over a given
surface is equal to the line integral of the vector over a
closed contour/line enclosing the surface.

38. Green’s Theorem for 2 scalar fields
We saw this for divergence;
Lets write A as a gradient of f;
Then we will get;
Subtract -2 from -1 and take volume integral on both sides;
Apply Gauss Div theorem on LHS;
Now we have Green’s Theorem;
)()()( AfAfAf


gA 

2-)(
1-)(
2
2
fgfgfg
gfgfgf




)]()([)(
V
22
  dVfggfdVfggf
V

 
SV
SdfggfdVfggf

)()(
)]()([)(
V
22
  dVfggfSdfggf
S


39. Thank you
We discussed most of what one wishes to learn in vector
calculus at the undergraduate engineering level. Its also
useful for the Physics ‘honors’ and ‘pass’ students.
This was a course I delivered to engineering first years, around
9th November 2009. But I have added contents to make it
more understandable, eg I added all the diagrams and
many explanations only now; 14-18th Aug 2015.
More such lectures will follow soon. Eg electromagnetism and
electromagnetic waves !