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Coordinate Systems and Transformations & Vector Calculus By: Hanish Garg 12105017 ECE Branch PEC University of Technology
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Coordinate systems (and transformations) and vector calculus

Jan 27, 2015

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Page 1: Coordinate systems (and transformations) and vector calculus

Coordinate Systems and Transformations

& Vector Calculus

By:Hanish Garg

12105017ECE Branch

PEC University of Technology

Page 2: Coordinate systems (and transformations) and vector calculus

Coordinate Systems

• Cartesian or Rectangular Coordinate System

• Cylindrical Coordinate System

• Spherical Coordinate System

Choice of the system is based on the symmetry of the problem.

Page 3: Coordinate systems (and transformations) and vector calculus

Cartesian Or Rectangular Coordinates

P (x, y, z)

x

y

z

P(x,y,z)

z

y

x

A vector A in Cartesian coordinates can be written as

),,( zyx AAA or zzyyxx aAaAaA

where ax,ay and az are unit vectors along x, y and z-directions.

Page 4: Coordinate systems (and transformations) and vector calculus

Cylindrical CoordinatesP (ρ, Φ, z)

x= ρ cos Φ, y=ρ sin Φ, z=z

z

Φ

z

ρx

y

P(ρ, Φ, z)

z

20

0

A vector A in Cylindrical coordinates can be written as

),,( zAAA orzzaAaAaA

where aρ,aΦ and az are unit vectors along ρ, Φ and z-directions.

zzx

yyx ,tan, 122

Page 5: Coordinate systems (and transformations) and vector calculus

The relationships between (ax,ay, az) and (aρ,aΦ, az)are

zz

y

x

aa

aaa

aaa

cossin

sincos

zz

yx

yx

aa

aaa

aaa

cossin

sincosor

zzyxyx aAaAAaAAA )cossin()sincos(

Then the relationships between (Ax,Ay, Az) and (Aρ, AΦ, Az)are

Page 6: Coordinate systems (and transformations) and vector calculus

zz

yx

yx

AA

AAA

AAA

cossin

sincos

z

y

x

z A

A

A

A

A

A

100

0cossin

0sincos

In matrix form we can write

Page 7: Coordinate systems (and transformations) and vector calculus
Page 8: Coordinate systems (and transformations) and vector calculus

Spherical CoordinatesP (r, θ, Φ)

x=r sin θ cos Φ, y=r sin θ sin Φ, Z=r cos θ

20

0

0

r

A vector A in Spherical coordinates can be written as

),,( AAAr or aAaAaA rr

where ar, aθ, and aΦ are unit vectors along r, θ, and Φ-directions.

θ

Φ

r

z

yx

P(r, θ, Φ)

x

y

z

yxzyxr 1

221222 tan,tan,

Page 9: Coordinate systems (and transformations) and vector calculus

The relationships between (ax,ay, az) and (ar,aθ,aΦ)are

aaa

aaaa

aaaa

rz

ry

rx

sincos

cossincossinsin

sincoscoscossin

yx

zyx

zyxr

aaa

aaaa

aaaa

cossin

sinsincoscoscos

cossinsincossin

or

Then the relationships between (Ax,Ay, Az) and (Ar, Aθ,and AΦ)are

aAA

aAAA

aAAAA

yx

zyx

rzyx

)cossin(

)sinsincoscoscos(

)cossinsincossin(

Page 10: Coordinate systems (and transformations) and vector calculus

z

y

xr

A

A

A

A

A

A

0cossin

sinsincoscoscos

cossinsincossin

In matrix form we can write

cossin

sinsincoscoscos

cossinsincossin

yx

zyx

zyxr

AAA

AAAA

AAAA

Page 11: Coordinate systems (and transformations) and vector calculus
Page 12: Coordinate systems (and transformations) and vector calculus

Cartesian CoordinatesP(x, y, z)

Spherical CoordinatesP(r, θ, Φ)

Cylindrical CoordinatesP(ρ, Φ, z)

x

y

zP(x,y,z)

Φ

z

rx y

z

P(ρ, Φ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Page 13: Coordinate systems (and transformations) and vector calculus

Differential Length, Area and Volume

Differential displacement

zyx dzadyadxadl

Differential area

zyx dxdyadxdzadydzadS

Differential VolumedxdydzdV

Cartesian Coordinates

Page 14: Coordinate systems (and transformations) and vector calculus

Cylindrical Coordinates

ρρ

ρ

ρ

ρρ

ρ

ρ

ρρ

ρ

Page 15: Coordinate systems (and transformations) and vector calculus

Differential Length, Area and Volume

Differential displacement

zdzaadaddl

Differential area

zadddzaddzaddS

Differential Volume

dzdddV

Cylindrical Coordinates

Page 16: Coordinate systems (and transformations) and vector calculus

Spherical Coordinates

Page 17: Coordinate systems (and transformations) and vector calculus

Differential Length, Area and Volume

Differential displacement

adrarddradl r sin

Differential area

ardrdadrdraddrdS r sinsin2

Differential Volume ddrdrdV sin2

Spherical Coordinates

Page 18: Coordinate systems (and transformations) and vector calculus

Line, Surface and Volume Integrals

Line Integral L

dlA.

Surface Integral

Volume Integral

S

dSA.

dvpV

v

Page 19: Coordinate systems (and transformations) and vector calculus

• Gradient of a scalar function is a vector quantity.

• Divergence of a vector is a scalar quantity.

• Curl of a vector is a vector quantity.• The Laplacian of a scalar A

f Vector

A.

A

A2

The Del Operator

Page 20: Coordinate systems (and transformations) and vector calculus

Del Operator

Cartesian Coordinateszyx a

zay

ax

Cylindrical Coordinates

Spherical Coordinates

zazaa

1

a

ra

rar r

sin

11

Page 21: Coordinate systems (and transformations) and vector calculus

VECTOR CALCULUS

GRADIENT OF A SCALAR

DIVERGENCE OF A VECTOR

DIVERGENCE THEOREM

CURL OF A VECTOR

STOKES’S THEOREM

Page 22: Coordinate systems (and transformations) and vector calculus

GRADIENT OF A SCALAR Suppose is the temperature at ,

and is the temperature at

as shown.

zyxT ,,1 zyxP ,,1

2P dzzdyydxxT ,,2

Page 23: Coordinate systems (and transformations) and vector calculus

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

dzdydx ,,

zyx dzdydxd aaaL Ld

However, from differential calculus, the differential temperature:

dzz

Tdy

y

Tdx

x

TTTdT

12

GRADIENT OF A SCALAR (Cont’d)

Page 24: Coordinate systems (and transformations) and vector calculus

But,

z

y

x

ddz

ddy

ddx

aL

aL

aL

So, previous equation can be rewritten as:

Laaa

LaLaLa

dz

T

y

T

x

T

dz

Td

y

Td

x

TdT

zyx

zyx

GRADIENT OF A SCALAR (Cont’d)

Page 25: Coordinate systems (and transformations) and vector calculus

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 z

V

y

V

x

VV aaa

GRADIENT OF A SCALAR (Cont’d)

Page 26: Coordinate systems (and transformations) and vector calculus

For Circular cylindrical coordinate:

zz

VVVV aaa

1

For Spherical coordinate:

aaa

V

r

V

rr

VV r sin

11

GRADIENT OF A SCALAR (Cont’d)

Page 27: Coordinate systems (and transformations) and vector calculus

EXAMPLE

Find gradient of these scalars:

yxeV z cosh2sin

2cos2zU

cossin10 2rW

(a)

(b)

(c)

Page 28: Coordinate systems (and transformations) and vector calculus

SOLUTION TO EXAMPLE

(a) Use gradient for Cartesian coordinate:

zz

yz

xz

zyx

yxe

yxeyxe

z

V

y

V

x

VV

a

aa

aaa

cosh2sin

sinh2sincosh2cos2

Page 29: Coordinate systems (and transformations) and vector calculus

SOLUTION TO EXAMPLE (Cont’d)

(b) Use gradient for Circular cylindrical

coordinate:

z

z

zz

z

UUUU

a

aa

aaa

2cos

2sin22cos2

1

2

Page 30: Coordinate systems (and transformations) and vector calculus

(c) Use gradient for Spherical coordinate:

a

aa

aaa

sinsin10

cos2sin10cossin10

sin

11

2

r

rW

r

W

rr

WW

SOLUTION TO EXAMPLE (Cont’d)

Page 31: Coordinate systems (and transformations) and vector calculus

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

Positive Divergence

Negative Divergence

Zero Divergence

DIVERGENCE OF A VECTOR

Page 32: Coordinate systems (and transformations) and vector calculus

DIVERGENCE OF A VECTOR (Cont’d)

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

v

dS

div s

v

A

A A lim0

Page 33: Coordinate systems (and transformations) and vector calculus

What is ?? s

dSA Vector field A at closed surface S

DIVERGENCE OF A VECTOR (Cont’d)

Page 34: Coordinate systems (and transformations) and vector calculus

Where,

dSdSbottomtoprightleftbackfronts

AA

And, v is volume enclosed by surface S

DIVERGENCE OF A VECTOR (Cont’d)

Page 35: Coordinate systems (and transformations) and vector calculus

For Cartesian coordinate:

z

A

y

A

x

A zyx

A

For Circular cylindrical coordinate:

z

AAA z

11A

DIVERGENCE OF A VECTOR (Cont’d)

Page 36: Coordinate systems (and transformations) and vector calculus

For Spherical coordinate:

A

r

A

rAr

rrr sin

1sin

sin

11 22

A

DIVERGENCE OF A VECTOR (Cont’d)

Page 37: Coordinate systems (and transformations) and vector calculus

Find divergence of these vectors:

zx xzyzxP aa 2

zzzQ aaa cossin 2

aaa coscossincos12

rr

W r

(a)

(b)

(c)

EXAMPLE

Page 38: Coordinate systems (and transformations) and vector calculus

39

(a) Use divergence for Cartesian coordinate:

xxyz

xzzy

yzxx

z

P

y

P

x

P zyx

2

02

P

SOLUTION TO EXAMPLE

Page 39: Coordinate systems (and transformations) and vector calculus

(b) Use divergence for Circular cylindrical

coordinate:

cossin2

cos1

sin1

11

22

Q

zz

z

z

QQQ z

SOLUTION TO EXAMPLE (Cont’d)

Page 40: Coordinate systems (and transformations) and vector calculus

(c) Use divergence for Spherical coordinate:

coscos2

cossin

1

cossinsin

1cos

1

sin

1sin

sin

11

22

22

W

r

rrrr

W

r

W

rWr

rrr

SOLUTION TO EXAMPLE (Cont’d)

Page 41: Coordinate systems (and transformations) and vector calculus

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

Page 42: Coordinate systems (and transformations) and vector calculus

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)

EXAMPLE

Page 43: Coordinate systems (and transformations) and vector calculus

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

topbottomouterinnerS

d DDDDSD

Where,

10)(

)(

2

0

5

01

4

2

0

5

01

3

z

zinner

dzd

dzdD

aa

aa

SOLUTION TO EXAMPLE

Page 44: Coordinate systems (and transformations) and vector calculus

160)(

)(

2

0

5

02

4

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 (Cont’d)

Page 45: Coordinate systems (and transformations) and vector calculus

Therefore

150

0016010

SDS

d

SOLUTION TO EXAMPLE (Cont’d)

Page 46: Coordinate systems (and transformations) and vector calculus

(b) For the right side of Divergence Theorem,

evaluate divergence of D

23 41

D

So,

150

4

5

0

2

0

2

1

4

5

0

2

0

2

1

2

zr

z

dzdddVD

SOLUTION TO EXAMPLE (Cont’d)

Page 47: Coordinate systems (and transformations) and vector calculus

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.

Page 48: Coordinate systems (and transformations) and vector calculus

maxlim0

a

A

A A ns

s s

dl

Curl

Where,

CURL OF A VECTOR (Cont’d)

dldldacdbcabs

AA

Page 49: Coordinate systems (and transformations) and vector calculus

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.

CURL OF A VECTOR (Cont’d)

Page 50: Coordinate systems (and transformations) and vector calculus

For Cartesian coordinate:

zyx

zyx

AAAzyx

aaa

A

zxy

yxz

xyz

y

A

x

A

z

A

x

A

z

A

y

AaaaA

CURL OF A VECTOR (Cont’d)

Page 51: Coordinate systems (and transformations) and vector calculus

z

z

AAAz

aaa

A1

z

zz

AA

z

AA

z

AA

a

aaA

1

1

For Circular cylindrical coordinate:

CURL OF A VECTOR (Cont’d)

Page 52: Coordinate systems (and transformations) and vector calculus

For Spherical coordinate:

ArrAA

rrr

r

sinsin

12

aaa

A

a

aaA

r

rr

A

r

rA

r

r

rAA

r

AA

r

)(1

sin

11sin

sin

1

CURL OF A VECTOR (Cont’d)

Page 53: Coordinate systems (and transformations) and vector calculus

zx xzyzxP aa 2

zzzQ aaa cossin 2

aaa coscossincos12

rr

W r

(a)

(b)

(c)

Find curl of these vectors:

EXAMPLE

Page 54: Coordinate systems (and transformations) and vector calculus

(a) Use curl for Cartesian coordinate:

zy

zyx

zxy

yxz

xyz

zxzyx

zxzyx

y

P

x

P

z

P

x

P

z

P

y

P

aa

aaa

aaaP

22

22 000

SOLUTION TO EXAMPLE

Page 55: Coordinate systems (and transformations) and vector calculus

(b) Use curl for Circular cylindrical coordinate

z

z

zzz

zz

z

z

y

Q

x

QQ

z

Q

z

QQ

aa

a

aa

aaaQ

cos3sin1

cos31

00sin

11

3

2

2

SOLUTION TO EXAMPLE (Cont’d)

Page 56: Coordinate systems (and transformations) and vector calculus

(c) Use curl for Spherical coordinate:

a

aa

a

aaW

22

2

cos)cossin(1

coscos

sin

11cossinsincos

sin

1

)(1

sin

11sin

sin

1

rr

r

r

r

rrr

r

r

W

r

rW

r

r

rWW

r

WW

r

r

r

rr

SOLUTION TO EXAMPLE (Cont’d)

Page 57: Coordinate systems (and transformations) and vector calculus

a

aa

a

aa

sin1

cos2

cossin

sin

2cos

sincossin2

1

cos01

sinsin2cossin

1

3

2

r

rr

rr

r

rr

r

r

r

SOLUTION TO EXAMPLE (Cont’d)

Page 58: Coordinate systems (and transformations) and vector calculus

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

Page 59: Coordinate systems (and transformations) and vector calculus

STOKE’S THEOREM (Cont’d)

Page 60: Coordinate systems (and transformations) and vector calculus

By using Stoke’s Theorem, evaluate

for dlA

aaA sincos

EXAMPLE

Page 61: Coordinate systems (and transformations) and vector calculus
Page 62: Coordinate systems (and transformations) and vector calculus

Stoke’s Theorem,

SL

dSdl AA

where, and

zddd aS Evaluate right side to get left side,

zaA

sin11

SOLUTION TO EXAMPLE (Cont’d)

Page 63: Coordinate systems (and transformations) and vector calculus

941.4

sin11

0

0

60

30

5

2

aA zS

dddS

SOLUTION TO EXAMPLE (Cont’d)

Page 64: Coordinate systems (and transformations) and vector calculus

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

EXAMPLE

Page 65: Coordinate systems (and transformations) and vector calculus

(a) To find LB d

321 LLL

dddd LBLBLBLB

Where,

dd

dzddd z

sincos

sincos

aaaaaLB

SOLUTION TO EXAMPLE

Page 66: Coordinate systems (and transformations) and vector calculus

So

202

1

sincos

2

0

2

0

0

00,0

2

01

LB

zzL

ddd

4cos20

sincos

0

0,200

2

22

LB

zzL

ddd

SOLUTION TO EXAMPLE (Cont’d)

Page 67: Coordinate systems (and transformations) and vector calculus

202

1

sincos

0

2

2

00,0

0

23

r

zzL

ddd

LB

Therefore the closed integral,

8242 LB d

SOLUTION TO EXAMPLE (Cont’d)

Page 68: Coordinate systems (and transformations) and vector calculus

(b) To find SB d

z

z

z

zz

a

aaa

a

aa

aaB

11sin

sinsin1

00

cossin1

0cossin01

sincos

SOLUTION TO EXAMPLE (Cont’d)

Page 69: Coordinate systems (and transformations) and vector calculus

Therefore

82

1cos

1sin

11sin

0

2

0

2

0

2

0

0

2

0

aaSB

dd

ddd zz

SOLUTION TO EXAMPLE (Cont’d)

Page 70: Coordinate systems (and transformations) and vector calculus

LAPLACIAN OF A SCALAR

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

Where, Laplacian V is:

zyxzyx z

V

y

V

x

V

zyx

VV

aaaaaa

2

Page 71: Coordinate systems (and transformations) and vector calculus

For Cartesian coordinate:

2

2

2

2

2

22

z

V

y

V

x

VV

For Circular cylindrical coordinate:

2

22

22 11

z

VVVV

LAPLACIAN OF A SCALAR (Cont’d)

Page 72: Coordinate systems (and transformations) and vector calculus

LAPLACIAN OF A SCALAR (Cont’d)

For Spherical coordinate:

2

2

22

22

22

sin

1

sinsin

11

V

r

V

rr

Vr

rrV

Page 73: Coordinate systems (and transformations) and vector calculus

EXAMPLE

Find Laplacian of these scalars:

yxeV z cosh2sin 2cos2zU

cossin10 2rW

(a)

(b)

(c)

You should try this!!

Page 74: Coordinate systems (and transformations) and vector calculus

SOLUTION TO EXAMPLE

yxeV z cosh2sin22

02 U

2cos21

cos102 r

W

(a)

(b)

(c)

Page 75: Coordinate systems (and transformations) and vector calculus

Thank You !!!