16.360 Lecture 18

• Today

Curl of a vector filed

1. Circulation2. Definition of Curl operator in Cartesian Coordinate 3. Vector identities involving the curl4. Stokes’s Therorem5. Laplacian Operator

16.360 Lecture 18

Circulation of a Vector

,ldBnCirculatio

,0

ˆˆˆˆ

ˆˆˆˆ

00

00

dyyBxdxxBx

dyyBxdxxBxnCirculatio

a

d

d

c

c

b

b

a

16.360 Lecture 18

Circulation of a Vector

,2

ˆ 0

r

IB

,

,ˆ2

ˆ

0

2

0

0

I

dr

IldBnCirculatio

16.360 Lecture 18

Flux in Cartesian Coordinates

sBn

sdz

B

z

Bz

x

B

z

By

z

B

y

Bx

xyz

B

z

Bzzxzy

z

B

y

BxldBn

zxzxyz

zxyz

c

)(ˆ

)](ˆ)(ˆ)(ˆ[

)(ˆ)(ˆ

,)(ˆ

)(ˆ

,

2

,2 2

zyz

B

y

Bx

yByBzBzBx

ldBn

yz

zzzyface yy

zyface

)(ˆ)(ˆ)(ˆz

B

z

Bz

x

B

z

By

z

B

y

BxB zxzxyz

16.360 Lecture 18

Vector identities involving the curl

BABA

)(

0)( A

0)( V

Stokes’s theorem

sdBldBnc

)(

16.360 Lecture 18

Curls in Rectangular, Cylindrical and Spherical Coordinates

zyx BBBzyx

zyx

B

ˆˆˆ

zr ArAAzr

zrr

rB

ˆˆˆ1

ARRAA

R

RRR

RB

R sin

ˆsinˆˆ

sin

12

16.360 Lecture 18

Laplacian Operator of a scalar

zyx AzAyAxz

Vz

y

Vy

x

VxV ˆˆˆˆˆˆ

2

2

2

2

2

2

)(

z

V

y

V

x

V

Az

Ay

Ax

V zyx

2

2

2

2

2

22 )(

z

V

y

V

x

VVV

16.360 Lecture 18

Laplacian Operator of a vector

zyx EzEyExE ˆˆˆ

2

2

2

2

2

2

2

2

2

2

2

22 ˆˆˆ)(

z

Ez

y

Ey

x

ExE

zyxE zyx

)()(2 EEE