Vector calculus 1st 2

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Vector Algebra

Mr. HIMANSHU DIWAKARAssistant professor

ECED

Mr. Himanshu Diwakar

JETGI 2Mr. Himanshu Diwakar

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VECTOR CALCULUS

Introduction

Scalars And Vectors

Gradient Of A Scalar

Divergence Of A Vector

Divergence Theorem

Curl Of A Vector

Stokes’s Theorem

Laplacian Of A Scalar


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Introduction• Electromagnetics(EM):-

Interaction between electric charges at rest and

in motion.

• It entails the analysis, synthesis, physical interpretation, and application of electric and magnetic fields.


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Scalars and vectors

• A scalar is a quantity that has only magnitudeEx:- time, mass, distance, temperature, entropy etc.

• A vector is a quantity that has both magnitude and direction• Velocity, force, displacement and electric field intensity.


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Unit vectors


x

z

y

az

ay

ax

Unit vectorsaz ,ay ,az

Similarly a A vector in Cartesian co-ordinate

A=Ax.ax+Ay.ay+Az.az

So Unit vector of A

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Position and distance vector


x

z

y

az

ay

ax

P (3, 4, 5)

O (0, 0, 0)

The distance vector is the displacement from one point to another.

A= 3.ax+ 4.ay+ 5.az

OP distanceOP=

= =7.071

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Vector multiplication

• Scalar (or dot) product = A.B

• Vector (or cross) product = AB

• Scalar triple product :

• Vector triple product :


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Angels


If A and B are vectors then the angle between these vectors can be find

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Differential Length (Cartesian Coordinates )

• Differential elements in length, area, and volume are useful in vector calculus. They are defined in the Cartesian, cylindrical, and spherical coordinate systems.

1. Differential displacement is given by

2. Differential normal area is given by

3. Differential volume is given by


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Cylindrical Coordinates• Notice from Figure that in cylindrical coordinates, differential

elements can be found as follows:1. Differential displacement is given by

2. Differential normal area is given by

3. Differential volume is given by


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Spherical Coordinates

From Figure, we notice that in spherical coordinates,1. The differential displacement is


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2. The differential normal area is


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3. The differential volume is


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Que:- Consider the object shown in Figure and CalculateThe distance BC, The distance CD, The surface area ABCD,

The surface area ABO, The surface area A OFD, The volume ABDCFO


C(0, 5, 0)

B(0, 5, 0)

D(5, 0, 10)

A(5, 0, 0)

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Line, Surface And Volume Integrals

• The familiar concept of integration will now be extended to cases when the integrand involves a vector. By a line we mean the path along a curve in space. We shall use terms such as line, curve, and contour interchangeably.• The line integral is the integral of the tangential component of A

along curve L. Or simplyAnd For closed surface


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Example:- Given that , calculate the circulation of F around the (closed) path shown in Figure

Ans:


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Dell operator

• The del operator, written , is the vector differential operator. In Cartesian coordinates,

• This vector differential operator, otherwise known as the gradient operator, is not a vector in itself, but when it operates on a scalar function, for example, a vector ensues. The operator is useful in defining


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1. The gradient of a scalar V, written, as 2. The divergence of a vector A, written as (• A)3. The curl of a vector A, written as ( X A)4. The Laplacian of a scalar V, written as

Each of these will be denned in detail in the subsequent sections.



So the solution for above equations is

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CLASSIFICATION OF VECTOR FIELDS

• A vector field is uniquely characterized by its divergence and curl. Neither the divergence nor curl of a vector field is sufficient to completely describe the field.

• All vector fields can be classified in terms of their vanishing or non vanishing divergence or curl as follows:


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Typical fields with vanishing and non vanishing divergence or curl.


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•A vector field A is said to be solenoidal (or divergenceless) if .

•A vector field A is said to be irrotational (or potential) if .


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The differential distances are the components of the differential distance vector :

dzdydx ,,

zyx dzdydxd aaaL Ld

However, from differential calculus, the differential temperature:

dzzTdy

yTdx

xTTTdT

12

GRADIENT OF A SCALAR


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But,

z

y

x

ddz

ddyddx

aL

aLaL

So, previous equation can be rewritten as:

Laaa

LaLaLa

dzT

yT

xT

dzTd

yTd

xTdT

zyx

zyx

GRADIENT OF A SCALAR (Cont’d)


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The vector inside square brackets defines the change of temperature corresponding to a vector change in position .This vector is called Gradient of Scalar T.

LddT


For Cartesian coordinate:

zyx zV

yV

xVV aaa


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For Circular cylindrical coordinate:

zzVVVV aaa

1

For Spherical coordinate:

aaa

V

rV

rrVV r sin

11


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EXAMPLE 1

Find gradient of these scalars:

yxeV z cosh2sin

2cos2zU

cossin10 2rW

(a)

(b)

(c)


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SOLUTION TO EXAMPLE 1

(a) Use gradient for Cartesian coordinate:

zz

yz

xz

zyx

yxe

yxeyxe

zV

yV

xVV

a

aa

aaa

cosh2sin

sinh2sincosh2cos2


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SOLUTION TO EXAMPLE 1 (Cont’d)

(b) Use gradient for Circular cylindrical coordinate:

z

z

zzzUUUU

a

aa

aaa

2cos

2sin22cos2

1

2


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(c) Use gradient for Spherical coordinate:

a aa

aaa

sinsin10cos2sin10cossin10

sin11

2

r

rW

rW

rrWW


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DIVERGENCE OF A VECTOR

Illustration of the divergence of a vector field at point P:

Positive Divergence

Negative Divergence

Zero Divergence


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DIVERGENCE OF A VECTOR (Cont’d)

The divergence of A at a given point P is the outward flux per unit volume:

v

dSdiv s

v

AA A lim

0


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What is ?? s

dSA Vector field A at closed surface S


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Where,dSdS

bottomtoprightleftbackfronts

AA

And, v is volume enclosed by surface S



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zA

yA

xA zyx

A


z

AAA z

11A



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A

rA

rAr

rr r sin1sin

sin11 2

2A



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EXAMPLE 11

Find divergence of these vectors:

zx xzyzxP aa 2

zzzQ aaa cossin 2

aaa coscossincos12 r

rW r

(a)

(b)

(c)


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(a) Use divergence for Cartesian coordinate:


xxyz

xzzy

yzxx

zP

yP

xP zyx

2

02

P


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(b) Use divergence for Circular cylindrical coordinate:

cossin2

cos1sin1

11

22

Q

zz

z

zQQ

Q z



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(c) Use divergence for Spherical coordinate:

coscos2

cossin1

cossinsin1cos1

sin1sin

sin11

22

22

W

r

rrrr

Wr

Wr

Wrrr r


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It states that the total outward flux of a vector field A at the closed surface S is the same as volume integral of divergence of A.

VV

dVdS AA

DIVERGENCE THEOREM


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EXAMPLE 12

A vector field exists in the region between two concentric cylindrical surfaces defined by ρ = 1 and ρ = 2, with both cylinders extending between z = 0 and z = 5. Verify the divergence theorem by evaluating:

aD 3

S

dsD

V

DdV

(a)

(b)


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(a) For two concentric cylinder, the left side:

topbottomouterinnerS

d DDDDSD

Where,

10)(

)(

2

0

5

01

4

2

0

5

0 13

z

zinner

dzd

dzdD

aa

aa


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160)(

)(

2

0

5

0 24

2

0

5

02

3

z

zouter

dzd

dzdD

aa

aa

2

1

2

05

3

2

1

2

00

3

0)(

0)(

zztop

zzbottom

ddD

ddD

aa

aa

SOLUTION TO EXAMPLE 12 Cont’d)


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Therefore

150

0016010

SDS

d



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(b) For the right side of Divergence Theorem, evaluate divergence of D

23 41

D

So,

150

4

5

0

2

0

2

14

5

0

2

0

2

1

2

zr

zdzdddVD


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CURL OF A VECTOR

The curl of vector A is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum.


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maxlim0

aA

A A ns

s s

dlCurl

Where,

CURL OF A VECTOR (Cont’d)

dldldacdbcabs

AA


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The curl of the vector field is concerned with rotation of the vector field. Rotation can be used to measure the uniformity of the field, the more non uniform the field, the larger value of curl.


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zyx

zyx

AAAzyx

aaa

A

zxy

yxz

xyz

yA

xA

zA

xA

zA

yA aaaA


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z

z

AAAz

aaa

A 1

z

zz

AA

zAA

zAA

a

aaA

1

1




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ArrAArrr

r

sinsin1

2

aaa

A

a

aaA

r

rr

Ar

rAr

rrAA

rAA

r

)(1

sin11sin

sin1


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EXAMPLE 13

zx xzyzxP aa 2

zzzQ aaa cossin 2

aaa coscossincos12 r

rW r

(a)

(b)

(c)

Find curl of these vectors:


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(a) Use curl for Cartesian coordinate:

zy

zyx

zxy

yxz

xyz

zxzyx

zxzyx

yP

xP

zP

xP

zP

yP

aa

aaa

aaaP

22

22 000


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(b) Use curl for Circular cylindrical coordinate

z

z

zzz

zz

z

z

yQ

xQQ

zQ

zQQ

aa

a

aa

aaaQ

cos3sin1

cos31

00sin

11

3

2

2



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(c) Use curl for Spherical coordinate:

a

aa

a

aaW

22

2

cos)cossin(1

coscos

sin11cossinsincos

sin1

)(1

sin11sin

sin1

rr

rr

rrr

rr

r

Wr

rWr

rrWW

rWW

r

r

r

rr



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a

aa

a

aa

sin1cos2

cossinsin

2cos

sincossin21

cos01sinsin2cossin1

3

2

r

rr

rr

r

rr

r

r

r


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STOKE’S THEOREM

The circulation of a vector field A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L that A and curl of A are continuous on S.

SL

dSdl AA


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STOKE’S THEOREM (Cont’d)


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EXAMPLE 14

By using Stoke’s Theorem, evaluate for

dlA

aaA sincos


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EXAMPLE 14 (Cont’d)


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Stoke’s Theorem,

SL

dSdl AA

where, and

zddd aS Evaluate right side to get left side,

zaA

sin11


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941.4

sin110

0

60

30

5

2

aA zS

dddS


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EXAMPLE 15

Verify Stoke’s theorem for the vector field for given figure by evaluating: aaB sincos

(a) over the semicircular contour.

LB d

(b) over the surface of semicircular contour.

SB d


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(a) To find LB d

321 LLL

dddd LBLBLBLB

Where,

dd

dzddd z

sincos

sincos

aaaaaLB


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So

2021

sincos

2

0

2

0

0

00,0

2

01

LBzzL

ddd

4cos20

sincos

0

0,200

2

22

LBzzL

ddd



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2021

sincos

0

2

2

00,0

0

23

r

zzLddd

LB


Therefore the closed integral,

8242 LB d


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(b) To find SB d

z

z

z

zz

a

aaa

a

aa

aaB

11sin

sinsin100

cossin1

0cossin01

sincos


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Therefore

821cos

1sin

11sin

0

2

0

2

0

2

0

0

2

0

aaSB

dd

ddd zz


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LAPLACIAN OF A SCALAR

The Laplacian of a scalar field, V written as: V2

Where, Laplacian V is:

zyxzyx zV

yV

xV

zyx

VV

aaaaaa

2


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2

2

2

2

2

22

zV

yV

xVV


2

22

22 11

zVVVV

LAPLACIAN OF A SCALAR (Cont’d)


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LAPLACIAN OF A SCALAR (Cont’d)


2

2

22

22

22

sin1

sinsin11

Vr

Vrr

Vrrr

V


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EXAMPLE 16

Find Laplacian of these scalars:

yxeV z cosh2sin 2cos2zU

cossin10 2rW

(a)

(b)

(c)

Try yourself !!


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yxeV z cosh2sin22

02 U

2cos21cos102 r

W

(a)

(b)

(c)


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THANK YOU


Vector calculus 1st 2

Engineering

Transcript of Vector calculus 1st 2