SlideShare a Scribd company logo
1 of 39
Download to read offline
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 !
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.
Fields and Functions
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.
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].
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.
Scalar Functions
Vector Functions
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

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.
When A does not change direction
When A does not change magnitude
When A changes in a generic fashion
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
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


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


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
Uniform and non-uniform fields
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 … …
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)(

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)(

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




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.
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




Homework
Two short home work questions, if you have carried yourself
this far !
Prove the following;
3r
0r




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.
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]
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 









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

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

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


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 !
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

Line/Path Integral of vector fields
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,
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 






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.
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

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 !

More Related Content

What's hot

B.tech ii unit-4 material vector differentiation
B.tech ii unit-4 material vector differentiationB.tech ii unit-4 material vector differentiation
B.tech ii unit-4 material vector differentiationRai University
 
Partial differentiation B tech
Partial differentiation B techPartial differentiation B tech
Partial differentiation B techRaj verma
 
Partial differential equations
Partial differential equationsPartial differential equations
Partial differential equationsmuhammadabullah
 
34 polar coordinate and equations
34 polar coordinate and equations34 polar coordinate and equations
34 polar coordinate and equationsmath266
 
Eigenvalue problems .ppt
Eigenvalue problems .pptEigenvalue problems .ppt
Eigenvalue problems .pptSelf-employed
 
Liner algebra-vector space-1 introduction to vector space and subspace
Liner algebra-vector space-1   introduction to vector space and subspace Liner algebra-vector space-1   introduction to vector space and subspace
Liner algebra-vector space-1 introduction to vector space and subspace Manikanta satyala
 
Triple product of vectors
Triple product of vectorsTriple product of vectors
Triple product of vectorsguest581a478
 
Line integral,Strokes and Green Theorem
Line integral,Strokes and Green TheoremLine integral,Strokes and Green Theorem
Line integral,Strokes and Green TheoremHassan Ahmed
 
14.6 triple integrals in cylindrical and spherical coordinates
14.6 triple integrals in cylindrical and spherical coordinates14.6 triple integrals in cylindrical and spherical coordinates
14.6 triple integrals in cylindrical and spherical coordinatesEmiey Shaari
 
Sequences and Series
Sequences and SeriesSequences and Series
Sequences and Seriessujathavvv
 
APPLICATION OF PARTIAL DIFFERENTIATION
APPLICATION OF PARTIAL DIFFERENTIATIONAPPLICATION OF PARTIAL DIFFERENTIATION
APPLICATION OF PARTIAL DIFFERENTIATIONDhrupal Patel
 
Cylindrical and Spherical Coordinates System
Cylindrical and Spherical Coordinates SystemCylindrical and Spherical Coordinates System
Cylindrical and Spherical Coordinates SystemJezreel David
 
Numerical solution of ordinary differential equations GTU CVNM PPT
Numerical solution of ordinary differential equations GTU CVNM PPTNumerical solution of ordinary differential equations GTU CVNM PPT
Numerical solution of ordinary differential equations GTU CVNM PPTPanchal Anand
 
Complex analysis
Complex analysisComplex analysis
Complex analysissujathavvv
 
Linear differential equation with constant coefficient
Linear differential equation with constant coefficientLinear differential equation with constant coefficient
Linear differential equation with constant coefficientSanjay Singh
 
Group Theory and Its Application: Beamer Presentation (PPT)
Group Theory and Its Application:   Beamer Presentation (PPT)Group Theory and Its Application:   Beamer Presentation (PPT)
Group Theory and Its Application: Beamer Presentation (PPT)SIRAJAHMAD36
 
Techniques of Integration ppt.ppt
Techniques of Integration ppt.pptTechniques of Integration ppt.ppt
Techniques of Integration ppt.pptJaysonFabela1
 

What's hot (20)

B.tech ii unit-4 material vector differentiation
B.tech ii unit-4 material vector differentiationB.tech ii unit-4 material vector differentiation
B.tech ii unit-4 material vector differentiation
 
Partial differentiation B tech
Partial differentiation B techPartial differentiation B tech
Partial differentiation B tech
 
Partial differential equations
Partial differential equationsPartial differential equations
Partial differential equations
 
34 polar coordinate and equations
34 polar coordinate and equations34 polar coordinate and equations
34 polar coordinate and equations
 
Calculus of variations
Calculus of variationsCalculus of variations
Calculus of variations
 
Eigenvalue problems .ppt
Eigenvalue problems .pptEigenvalue problems .ppt
Eigenvalue problems .ppt
 
Liner algebra-vector space-1 introduction to vector space and subspace
Liner algebra-vector space-1   introduction to vector space and subspace Liner algebra-vector space-1   introduction to vector space and subspace
Liner algebra-vector space-1 introduction to vector space and subspace
 
Triple product of vectors
Triple product of vectorsTriple product of vectors
Triple product of vectors
 
Line integral,Strokes and Green Theorem
Line integral,Strokes and Green TheoremLine integral,Strokes and Green Theorem
Line integral,Strokes and Green Theorem
 
14.6 triple integrals in cylindrical and spherical coordinates
14.6 triple integrals in cylindrical and spherical coordinates14.6 triple integrals in cylindrical and spherical coordinates
14.6 triple integrals in cylindrical and spherical coordinates
 
Sequences and Series
Sequences and SeriesSequences and Series
Sequences and Series
 
APPLICATION OF PARTIAL DIFFERENTIATION
APPLICATION OF PARTIAL DIFFERENTIATIONAPPLICATION OF PARTIAL DIFFERENTIATION
APPLICATION OF PARTIAL DIFFERENTIATION
 
Cylindrical and Spherical Coordinates System
Cylindrical and Spherical Coordinates SystemCylindrical and Spherical Coordinates System
Cylindrical and Spherical Coordinates System
 
Tensor analysis
Tensor analysisTensor analysis
Tensor analysis
 
Numerical solution of ordinary differential equations GTU CVNM PPT
Numerical solution of ordinary differential equations GTU CVNM PPTNumerical solution of ordinary differential equations GTU CVNM PPT
Numerical solution of ordinary differential equations GTU CVNM PPT
 
Complex analysis
Complex analysisComplex analysis
Complex analysis
 
Linear differential equation with constant coefficient
Linear differential equation with constant coefficientLinear differential equation with constant coefficient
Linear differential equation with constant coefficient
 
Group Theory and Its Application: Beamer Presentation (PPT)
Group Theory and Its Application:   Beamer Presentation (PPT)Group Theory and Its Application:   Beamer Presentation (PPT)
Group Theory and Its Application: Beamer Presentation (PPT)
 
Techniques of Integration ppt.ppt
Techniques of Integration ppt.pptTechniques of Integration ppt.ppt
Techniques of Integration ppt.ppt
 
Chapter 16 2
Chapter 16 2Chapter 16 2
Chapter 16 2
 

Viewers also liked

Uncertainty Principle and Photography. see mdashf.org/2015/06/08/
Uncertainty Principle and Photography. see mdashf.org/2015/06/08/Uncertainty Principle and Photography. see mdashf.org/2015/06/08/
Uncertainty Principle and Photography. see mdashf.org/2015/06/08/Manmohan Dash
 
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan Dash
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan DashConcepts and Problems in Quantum Mechanics, Lecture-II By Manmohan Dash
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan DashManmohan Dash
 
Why prices and stocks get inflated?
Why prices and stocks get inflated?Why prices and stocks get inflated?
Why prices and stocks get inflated?Manmohan Dash
 
Electromagnetic Waves !
Electromagnetic Waves !Electromagnetic Waves !
Electromagnetic Waves !Manmohan Dash
 
Recruiting Lessons from BrightCarbon
Recruiting Lessons from BrightCarbonRecruiting Lessons from BrightCarbon
Recruiting Lessons from BrightCarbonBrightCarbon
 
My PhD Thesis as I presented in my Preliminary Exam.
My PhD Thesis as I presented in my Preliminary Exam. My PhD Thesis as I presented in my Preliminary Exam.
My PhD Thesis as I presented in my Preliminary Exam. Manmohan Dash
 
Being at the fore front of scientfic research !
Being at the fore front of scientfic research !Being at the fore front of scientfic research !
Being at the fore front of scientfic research !Manmohan Dash
 
Heisenberg's Uncertainty Principle !
Heisenberg's Uncertainty Principle !Heisenberg's Uncertainty Principle !
Heisenberg's Uncertainty Principle !Manmohan Dash
 
[Electricity and Magnetism] Electrodynamics
[Electricity and Magnetism] Electrodynamics[Electricity and Magnetism] Electrodynamics
[Electricity and Magnetism] ElectrodynamicsManmohan Dash
 
Concepts and problems in Quantum Mechanics. Lecture-I
Concepts and problems in Quantum Mechanics. Lecture-IConcepts and problems in Quantum Mechanics. Lecture-I
Concepts and problems in Quantum Mechanics. Lecture-IManmohan Dash
 
De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...Manmohan Dash
 
10 Major Mistakes in Physics !
10 Major Mistakes in Physics !10 Major Mistakes in Physics !
10 Major Mistakes in Physics !Manmohan Dash
 
7 Digital Photography Concepts You Have To know
7 Digital Photography Concepts You Have To know7 Digital Photography Concepts You Have To know
7 Digital Photography Concepts You Have To knowYang Ao Wei 楊翱維
 

Viewers also liked (14)

Uncertainty Principle and Photography. see mdashf.org/2015/06/08/
Uncertainty Principle and Photography. see mdashf.org/2015/06/08/Uncertainty Principle and Photography. see mdashf.org/2015/06/08/
Uncertainty Principle and Photography. see mdashf.org/2015/06/08/
 
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan Dash
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan DashConcepts and Problems in Quantum Mechanics, Lecture-II By Manmohan Dash
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan Dash
 
Why prices and stocks get inflated?
Why prices and stocks get inflated?Why prices and stocks get inflated?
Why prices and stocks get inflated?
 
Electromagnetic Waves !
Electromagnetic Waves !Electromagnetic Waves !
Electromagnetic Waves !
 
Cross Section
Cross SectionCross Section
Cross Section
 
Recruiting Lessons from BrightCarbon
Recruiting Lessons from BrightCarbonRecruiting Lessons from BrightCarbon
Recruiting Lessons from BrightCarbon
 
My PhD Thesis as I presented in my Preliminary Exam.
My PhD Thesis as I presented in my Preliminary Exam. My PhD Thesis as I presented in my Preliminary Exam.
My PhD Thesis as I presented in my Preliminary Exam.
 
Being at the fore front of scientfic research !
Being at the fore front of scientfic research !Being at the fore front of scientfic research !
Being at the fore front of scientfic research !
 
Heisenberg's Uncertainty Principle !
Heisenberg's Uncertainty Principle !Heisenberg's Uncertainty Principle !
Heisenberg's Uncertainty Principle !
 
[Electricity and Magnetism] Electrodynamics
[Electricity and Magnetism] Electrodynamics[Electricity and Magnetism] Electrodynamics
[Electricity and Magnetism] Electrodynamics
 
Concepts and problems in Quantum Mechanics. Lecture-I
Concepts and problems in Quantum Mechanics. Lecture-IConcepts and problems in Quantum Mechanics. Lecture-I
Concepts and problems in Quantum Mechanics. Lecture-I
 
De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...
 
10 Major Mistakes in Physics !
10 Major Mistakes in Physics !10 Major Mistakes in Physics !
10 Major Mistakes in Physics !
 
7 Digital Photography Concepts You Have To know
7 Digital Photography Concepts You Have To know7 Digital Photography Concepts You Have To know
7 Digital Photography Concepts You Have To know
 

Similar to Vector Calculus.

Notes notes vector calculus made at home (wecompress.com)
Notes notes vector calculus made at home (wecompress.com)Notes notes vector calculus made at home (wecompress.com)
Notes notes vector calculus made at home (wecompress.com)Ashish Raje
 
EMF.0.11.VectorCalculus-I.pdf
EMF.0.11.VectorCalculus-I.pdfEMF.0.11.VectorCalculus-I.pdf
EMF.0.11.VectorCalculus-I.pdfrsrao8
 
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....loniyakrishn
 
2. Vector Algebra.pptx
2. Vector Algebra.pptx2. Vector Algebra.pptx
2. Vector Algebra.pptxMehrija
 
5. lec5 curl of a vector
5. lec5 curl of a vector5. lec5 curl of a vector
5. lec5 curl of a vectorshabdrang
 
M1 unit v-jntuworld
M1 unit v-jntuworldM1 unit v-jntuworld
M1 unit v-jntuworldmrecedu
 
Electromagnetic theory EMT lecture 1
Electromagnetic theory EMT lecture 1Electromagnetic theory EMT lecture 1
Electromagnetic theory EMT lecture 1Ali Farooq
 
Function of several variables
Function of several variablesFunction of several variables
Function of several variablesKamel Attar
 
M1 unit vii-jntuworld
M1 unit vii-jntuworldM1 unit vii-jntuworld
M1 unit vii-jntuworldmrecedu
 
Derivación e integración de funciones de varias variables
Derivación e integración de funciones de varias variablesDerivación e integración de funciones de varias variables
Derivación e integración de funciones de varias variablesRicardoAzocar3
 
vectorcalculusandlinearalgebra-150518103010-lva1-app6892.pptx
vectorcalculusandlinearalgebra-150518103010-lva1-app6892.pptxvectorcalculusandlinearalgebra-150518103010-lva1-app6892.pptx
vectorcalculusandlinearalgebra-150518103010-lva1-app6892.pptxShalabhMishra10
 
Divergence,curl,gradient
Divergence,curl,gradientDivergence,curl,gradient
Divergence,curl,gradientKunj Patel
 
vectorcalculusandlinearalgebra-150518103010-lva1-app6892 (1).pdf
vectorcalculusandlinearalgebra-150518103010-lva1-app6892 (1).pdfvectorcalculusandlinearalgebra-150518103010-lva1-app6892 (1).pdf
vectorcalculusandlinearalgebra-150518103010-lva1-app6892 (1).pdfShantanuGolande
 
EMF.0.14.VectorCalculus-IV.pdf
EMF.0.14.VectorCalculus-IV.pdfEMF.0.14.VectorCalculus-IV.pdf
EMF.0.14.VectorCalculus-IV.pdfrsrao8
 

Similar to Vector Calculus. (20)

Notes notes vector calculus made at home (wecompress.com)
Notes notes vector calculus made at home (wecompress.com)Notes notes vector calculus made at home (wecompress.com)
Notes notes vector calculus made at home (wecompress.com)
 
EMF.0.11.VectorCalculus-I.pdf
EMF.0.11.VectorCalculus-I.pdfEMF.0.11.VectorCalculus-I.pdf
EMF.0.11.VectorCalculus-I.pdf
 
Emfbook
EmfbookEmfbook
Emfbook
 
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....
 
2. Vector Algebra.pptx
2. Vector Algebra.pptx2. Vector Algebra.pptx
2. Vector Algebra.pptx
 
5. lec5 curl of a vector
5. lec5 curl of a vector5. lec5 curl of a vector
5. lec5 curl of a vector
 
M1 unit v-jntuworld
M1 unit v-jntuworldM1 unit v-jntuworld
M1 unit v-jntuworld
 
Electromagnetic theory EMT lecture 1
Electromagnetic theory EMT lecture 1Electromagnetic theory EMT lecture 1
Electromagnetic theory EMT lecture 1
 
Function of several variables
Function of several variablesFunction of several variables
Function of several variables
 
M1 unit vii-jntuworld
M1 unit vii-jntuworldM1 unit vii-jntuworld
M1 unit vii-jntuworld
 
Derivación e integración de funciones de varias variables
Derivación e integración de funciones de varias variablesDerivación e integración de funciones de varias variables
Derivación e integración de funciones de varias variables
 
vector.pdf
vector.pdfvector.pdf
vector.pdf
 
vectorcalculusandlinearalgebra-150518103010-lva1-app6892.pptx
vectorcalculusandlinearalgebra-150518103010-lva1-app6892.pptxvectorcalculusandlinearalgebra-150518103010-lva1-app6892.pptx
vectorcalculusandlinearalgebra-150518103010-lva1-app6892.pptx
 
Divergence,curl,gradient
Divergence,curl,gradientDivergence,curl,gradient
Divergence,curl,gradient
 
Chapter 16 1
Chapter 16 1Chapter 16 1
Chapter 16 1
 
vectorcalculusandlinearalgebra-150518103010-lva1-app6892 (1).pdf
vectorcalculusandlinearalgebra-150518103010-lva1-app6892 (1).pdfvectorcalculusandlinearalgebra-150518103010-lva1-app6892 (1).pdf
vectorcalculusandlinearalgebra-150518103010-lva1-app6892 (1).pdf
 
Ch06 ssm
Ch06 ssmCh06 ssm
Ch06 ssm
 
Fields Lec 2
Fields Lec 2Fields Lec 2
Fields Lec 2
 
EMF.0.14.VectorCalculus-IV.pdf
EMF.0.14.VectorCalculus-IV.pdfEMF.0.14.VectorCalculus-IV.pdf
EMF.0.14.VectorCalculus-IV.pdf
 
10. functions
10. functions10. functions
10. functions
 

Recently uploaded

Sulphonamides, mechanisms and their uses
Sulphonamides, mechanisms and their usesSulphonamides, mechanisms and their uses
Sulphonamides, mechanisms and their usesVijayaLaxmi84
 
Objectives n learning outcoms - MD 20240404.pptx
Objectives n learning outcoms - MD 20240404.pptxObjectives n learning outcoms - MD 20240404.pptx
Objectives n learning outcoms - MD 20240404.pptxMadhavi Dharankar
 
BÀI TẬP BỔ TRỢ 4 KĨ NĂNG TIẾNG ANH LỚP 8 - CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC ...
BÀI TẬP BỔ TRỢ 4 KĨ NĂNG TIẾNG ANH LỚP 8 - CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC ...BÀI TẬP BỔ TRỢ 4 KĨ NĂNG TIẾNG ANH LỚP 8 - CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC ...
BÀI TẬP BỔ TRỢ 4 KĨ NĂNG TIẾNG ANH LỚP 8 - CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC ...Nguyen Thanh Tu Collection
 
4.4.24 Economic Precarity and Global Economic Forces.pptx
4.4.24 Economic Precarity and Global Economic Forces.pptx4.4.24 Economic Precarity and Global Economic Forces.pptx
4.4.24 Economic Precarity and Global Economic Forces.pptxmary850239
 
6 ways Samsung’s Interactive Display powered by Android changes the classroom
6 ways Samsung’s Interactive Display powered by Android changes the classroom6 ways Samsung’s Interactive Display powered by Android changes the classroom
6 ways Samsung’s Interactive Display powered by Android changes the classroomSamsung Business USA
 
4.9.24 School Desegregation in Boston.pptx
4.9.24 School Desegregation in Boston.pptx4.9.24 School Desegregation in Boston.pptx
4.9.24 School Desegregation in Boston.pptxmary850239
 
Basic cosmetics prepared by my student Mr. Balamurugan, II Maths, 2023-2024
Basic cosmetics prepared by my student Mr. Balamurugan, II Maths, 2023-2024Basic cosmetics prepared by my student Mr. Balamurugan, II Maths, 2023-2024
Basic cosmetics prepared by my student Mr. Balamurugan, II Maths, 2023-2024St.John's College
 
LEVERAGING SYNERGISM INDUSTRY-ACADEMIA PARTNERSHIP FOR IMPLEMENTATION OF NAT...
LEVERAGING SYNERGISM INDUSTRY-ACADEMIA PARTNERSHIP FOR IMPLEMENTATION OF  NAT...LEVERAGING SYNERGISM INDUSTRY-ACADEMIA PARTNERSHIP FOR IMPLEMENTATION OF  NAT...
LEVERAGING SYNERGISM INDUSTRY-ACADEMIA PARTNERSHIP FOR IMPLEMENTATION OF NAT...pragatimahajan3
 
Healthy Minds, Flourishing Lives: A Philosophical Approach to Mental Health a...
Healthy Minds, Flourishing Lives: A Philosophical Approach to Mental Health a...Healthy Minds, Flourishing Lives: A Philosophical Approach to Mental Health a...
Healthy Minds, Flourishing Lives: A Philosophical Approach to Mental Health a...Osopher
 
Sarah Lahm In Media Res Media Component
Sarah Lahm  In Media Res Media ComponentSarah Lahm  In Media Res Media Component
Sarah Lahm In Media Res Media ComponentInMediaRes1
 
CLASSIFICATION OF ANTI - CANCER DRUGS.pptx
CLASSIFICATION OF ANTI - CANCER DRUGS.pptxCLASSIFICATION OF ANTI - CANCER DRUGS.pptx
CLASSIFICATION OF ANTI - CANCER DRUGS.pptxAnupam32727
 
Jason Potel In Media Res Media Component
Jason Potel In Media Res Media ComponentJason Potel In Media Res Media Component
Jason Potel In Media Res Media ComponentInMediaRes1
 
Paul Dobryden In Media Res Media Component
Paul Dobryden In Media Res Media ComponentPaul Dobryden In Media Res Media Component
Paul Dobryden In Media Res Media ComponentInMediaRes1
 
What is Property Fields in Odoo 17 ERP Module
What is Property Fields in Odoo 17 ERP ModuleWhat is Property Fields in Odoo 17 ERP Module
What is Property Fields in Odoo 17 ERP ModuleCeline George
 
Shark introduction Morphology and its behaviour characteristics
Shark introduction Morphology and its behaviour characteristicsShark introduction Morphology and its behaviour characteristics
Shark introduction Morphology and its behaviour characteristicsArubSultan
 
BÀI TẬP BỔ TRỢ TIẾNG ANH 11 THEO ĐƠN VỊ BÀI HỌC - CẢ NĂM - CÓ FILE NGHE (GLOB...
BÀI TẬP BỔ TRỢ TIẾNG ANH 11 THEO ĐƠN VỊ BÀI HỌC - CẢ NĂM - CÓ FILE NGHE (GLOB...BÀI TẬP BỔ TRỢ TIẾNG ANH 11 THEO ĐƠN VỊ BÀI HỌC - CẢ NĂM - CÓ FILE NGHE (GLOB...
BÀI TẬP BỔ TRỢ TIẾNG ANH 11 THEO ĐƠN VỊ BÀI HỌC - CẢ NĂM - CÓ FILE NGHE (GLOB...Nguyen Thanh Tu Collection
 
Self directed Learning - SDL, introduction to SDL
Self directed Learning - SDL, introduction to SDLSelf directed Learning - SDL, introduction to SDL
Self directed Learning - SDL, introduction to SDLspmdoc
 
PART 1 - CHAPTER 1 - CELL THE FUNDAMENTAL UNIT OF LIFE
PART 1 - CHAPTER 1 - CELL THE FUNDAMENTAL UNIT OF LIFEPART 1 - CHAPTER 1 - CELL THE FUNDAMENTAL UNIT OF LIFE
PART 1 - CHAPTER 1 - CELL THE FUNDAMENTAL UNIT OF LIFEMISSRITIMABIOLOGYEXP
 

Recently uploaded (20)

Sulphonamides, mechanisms and their uses
Sulphonamides, mechanisms and their usesSulphonamides, mechanisms and their uses
Sulphonamides, mechanisms and their uses
 
Objectives n learning outcoms - MD 20240404.pptx
Objectives n learning outcoms - MD 20240404.pptxObjectives n learning outcoms - MD 20240404.pptx
Objectives n learning outcoms - MD 20240404.pptx
 
BÀI TẬP BỔ TRỢ 4 KĨ NĂNG TIẾNG ANH LỚP 8 - CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC ...
BÀI TẬP BỔ TRỢ 4 KĨ NĂNG TIẾNG ANH LỚP 8 - CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC ...BÀI TẬP BỔ TRỢ 4 KĨ NĂNG TIẾNG ANH LỚP 8 - CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC ...
BÀI TẬP BỔ TRỢ 4 KĨ NĂNG TIẾNG ANH LỚP 8 - CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC ...
 
4.4.24 Economic Precarity and Global Economic Forces.pptx
4.4.24 Economic Precarity and Global Economic Forces.pptx4.4.24 Economic Precarity and Global Economic Forces.pptx
4.4.24 Economic Precarity and Global Economic Forces.pptx
 
6 ways Samsung’s Interactive Display powered by Android changes the classroom
6 ways Samsung’s Interactive Display powered by Android changes the classroom6 ways Samsung’s Interactive Display powered by Android changes the classroom
6 ways Samsung’s Interactive Display powered by Android changes the classroom
 
4.9.24 School Desegregation in Boston.pptx
4.9.24 School Desegregation in Boston.pptx4.9.24 School Desegregation in Boston.pptx
4.9.24 School Desegregation in Boston.pptx
 
Basic cosmetics prepared by my student Mr. Balamurugan, II Maths, 2023-2024
Basic cosmetics prepared by my student Mr. Balamurugan, II Maths, 2023-2024Basic cosmetics prepared by my student Mr. Balamurugan, II Maths, 2023-2024
Basic cosmetics prepared by my student Mr. Balamurugan, II Maths, 2023-2024
 
LEVERAGING SYNERGISM INDUSTRY-ACADEMIA PARTNERSHIP FOR IMPLEMENTATION OF NAT...
LEVERAGING SYNERGISM INDUSTRY-ACADEMIA PARTNERSHIP FOR IMPLEMENTATION OF  NAT...LEVERAGING SYNERGISM INDUSTRY-ACADEMIA PARTNERSHIP FOR IMPLEMENTATION OF  NAT...
LEVERAGING SYNERGISM INDUSTRY-ACADEMIA PARTNERSHIP FOR IMPLEMENTATION OF NAT...
 
Chi-Square Test Non Parametric Test Categorical Variable
Chi-Square Test Non Parametric Test Categorical VariableChi-Square Test Non Parametric Test Categorical Variable
Chi-Square Test Non Parametric Test Categorical Variable
 
Healthy Minds, Flourishing Lives: A Philosophical Approach to Mental Health a...
Healthy Minds, Flourishing Lives: A Philosophical Approach to Mental Health a...Healthy Minds, Flourishing Lives: A Philosophical Approach to Mental Health a...
Healthy Minds, Flourishing Lives: A Philosophical Approach to Mental Health a...
 
Sarah Lahm In Media Res Media Component
Sarah Lahm  In Media Res Media ComponentSarah Lahm  In Media Res Media Component
Sarah Lahm In Media Res Media Component
 
CLASSIFICATION OF ANTI - CANCER DRUGS.pptx
CLASSIFICATION OF ANTI - CANCER DRUGS.pptxCLASSIFICATION OF ANTI - CANCER DRUGS.pptx
CLASSIFICATION OF ANTI - CANCER DRUGS.pptx
 
Jason Potel In Media Res Media Component
Jason Potel In Media Res Media ComponentJason Potel In Media Res Media Component
Jason Potel In Media Res Media Component
 
Paul Dobryden In Media Res Media Component
Paul Dobryden In Media Res Media ComponentPaul Dobryden In Media Res Media Component
Paul Dobryden In Media Res Media Component
 
What is Property Fields in Odoo 17 ERP Module
What is Property Fields in Odoo 17 ERP ModuleWhat is Property Fields in Odoo 17 ERP Module
What is Property Fields in Odoo 17 ERP Module
 
Shark introduction Morphology and its behaviour characteristics
Shark introduction Morphology and its behaviour characteristicsShark introduction Morphology and its behaviour characteristics
Shark introduction Morphology and its behaviour characteristics
 
BÀI TẬP BỔ TRỢ TIẾNG ANH 11 THEO ĐƠN VỊ BÀI HỌC - CẢ NĂM - CÓ FILE NGHE (GLOB...
BÀI TẬP BỔ TRỢ TIẾNG ANH 11 THEO ĐƠN VỊ BÀI HỌC - CẢ NĂM - CÓ FILE NGHE (GLOB...BÀI TẬP BỔ TRỢ TIẾNG ANH 11 THEO ĐƠN VỊ BÀI HỌC - CẢ NĂM - CÓ FILE NGHE (GLOB...
BÀI TẬP BỔ TRỢ TIẾNG ANH 11 THEO ĐƠN VỊ BÀI HỌC - CẢ NĂM - CÓ FILE NGHE (GLOB...
 
Israel Genealogy Research Assoc. April 2024 Database Release
Israel Genealogy Research Assoc. April 2024 Database ReleaseIsrael Genealogy Research Assoc. April 2024 Database Release
Israel Genealogy Research Assoc. April 2024 Database Release
 
Self directed Learning - SDL, introduction to SDL
Self directed Learning - SDL, introduction to SDLSelf directed Learning - SDL, introduction to SDL
Self directed Learning - SDL, introduction to SDL
 
PART 1 - CHAPTER 1 - CELL THE FUNDAMENTAL UNIT OF LIFE
PART 1 - CHAPTER 1 - CELL THE FUNDAMENTAL UNIT OF LIFEPART 1 - CHAPTER 1 - CELL THE FUNDAMENTAL UNIT OF LIFE
PART 1 - CHAPTER 1 - CELL THE FUNDAMENTAL UNIT OF LIFE
 

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.
  • 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.
  • 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
  • 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 !