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

Ak

t

Aj

t

Ai

t

tAttA

dt

Ad zyx

t

ˆˆˆ)()(lim

0

t

Ak

t

Aj

t

Ai

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

Adc

dt

Acd

)(

Adt

df

dt

Adf

dt

Afd

)(

dt

BdAB

dt

Ad

dt

BAd

)(

dt

BdAB

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

zˆ

yˆ

xˆ

kji

zˆ

yˆ

xˆ

Vk

Vj

ViV

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

zˆ

yˆ

xˆ

Vk

Vj

ViV

It’s a well known mathematical result that differentials are given in terms of partials. if V = V(x, y, z);

dzV

dyVV

dVzy

dxx

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;

Vndn

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

VrrV

ˆ)(

Proof of the following identity

Lets prove the identity;

)z

ˆy

ˆx

ˆ(rzr

ˆyr

ˆxr

ˆ

zˆ

yˆ

xˆ

rk

rj

ri

VrVk

rVj

rVi

Vk

Vj

ViV

r

ˆ)( :QED

r

ˆr

)zˆyˆxˆ(

r)(

z

z,

y

y,

x

x

)x( ;ˆˆxˆ 2/1222

VrrV

Vr

r

rV

rk

rj

ri

VrV

r

r

r

r

r

r

zyrzkyjir

r

VrrV

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

ˆy

ˆx

ˆ(

x

x

zy

zy

AAAAAdiv

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.

)-(k)-(j)-(i ;

z,

y,

x where

ˆ ˆ ˆ

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

222

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

lim0 V

dSS

V

lim0 V

SdA

A S

V

n

S

ldA

A C

Sˆlim

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

dVfggfdVfggfV

SV

SdfggfdVfggf

)()(

)]()([)(V

22

dVfggfSdfggfS

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 !