derivative of any function f(x,y,z): differential calculus (revisited): gradient of function f

Derivative of any function f(x,y,z):

Differential Calculus (revisited):

dzzf

dyyf

dxxf

df

ldf

dzkdyjdxizf

kyf

jxf

idf

ˆˆˆˆˆˆ

zf

kyf

jxf

ifwhere

ˆˆˆ

Gradient of function f

Gradient of a function

Change in a scalar function f corresponding to a change in position dr

rdfdf

f is a VECTOR

Geometrical interpretation of GradientZ

X

Y

P Qdr

f

Czyxf ),,(

change in f : rdfdf

=0=> f dr

Z

X

Y

P

Q

dr

1Cf

12 CCf

rdfCCdf

12

For a given |dr|, the change in scalar function f(x,y,z) is maximum when:

frd

||

=> f is a vector along the direction of maximum rate of change of the

function

Magnitude: slope along this maximal direction

If f = 0 at some point (x0,y0,z0)

(x0,y0,z0) is a stationary point of f(x,y,z)

=> df = 0 for small displacements about the point (x0,y0,z0)

The Operator

zk

yj

xi

ˆˆˆ

is NOT a vector,

but a VECTOR OPERATORVECTOR OPERATOR

Satisfies: •Vector rules

•Partial differentiation rules

On a scalar function f : f

can act:

GRADIENT

On a vector function F as: . F DIVERGENCE

On a vector function F as: × F CURL

Divergence of a vector

zF

y

F

xF

F zyx

zyx FkFjFiz

ky

jx

iF ˆˆˆˆˆˆ

.F is a measure of how much the vector F spreads out (diverges) from

the point in question.

Divergence of a vector is a scalar.

Physical interpretation of Divergence

Flow of a compressible fluid:

(x,y,z) -> density of the fluid at a point (x,y,z)

v(x,y,z) -> velocity of the fluid at (x,y,z)

Z

X

Y

dy

dxdz

A

DC

B

E F

HG

dydzv xx 0| dydzv dxxx |

dydzdxvx

vx

xx0

(rate of flow in)EFGH

(rate of flow out)ABCD

Net rate of flow out (along- x)

dxdydzvx x

Net rate of flow out through all pairs of surfaces (per unit time):

dxdydzvz

vy

vx zyx

dxdydzv

Net rate of flow of the fluid per unit volume per unit time:

v

DIVERGENCE

Curl

zyx FFF

zyx ///

kji

F

yf

x

fk

xf

zf

jz

f

yf

i xyzxyz ˆˆˆ

Curl of a vector is a vector

×F is a measure of how much the vector F “curls around” the point in question.

Physical significance of Curl

Circulation of a fluid around a loop:

X

Y

00 ,yx

dyyx 00 ,

00 ,ydxx

dyydxx 00 ,

1

4 2

3

yyxx

yyxx

dVdV

dVdV

43

21

Circulation (1234)

))(,()(),(

),(),(

0000

0000

dyyxvdxdyyV

yxv

dydxx

Vyxvdxyxv

yx

x

yyx

dxdyyV

x

V xy

Circulation per unit area = ( × V )|z

z-component of CURL

Curvilinear coordinates:

used to describe systems with symmetry.

Spherical coordinates (r, , Ø)

Cartesian coordinates in terms of spherical coordinates:

cossinrx

sinsinry

cosrz

Spherical coordinates in terms of Cartesian coordinates:

222 zyxr

zyx /tan 221

xy /tan 1

Unit vectors in spherical coordinates

cosˆsinsinˆcossinˆˆ kjir

sinˆsincosˆcoscosˆˆ kji

cosˆsinˆˆ ji r

Z

X

Y

r

Line element in spherical coordinates:

drrddrrld sinˆˆˆ

Volume element in spherical coordinates:

dddrrd sin2

Area element in spherical coordinates:

rddrad ˆsin21

ˆ2 ddrrad

on a surface of a sphere (r const.)

on a surface lying in xy-plane ( const.)

Gradient:

f

rf

rrf

rfsin1ˆ1ˆˆ

F

rF

rFr

rrF r sin

1sin

sin11 2

2

Divergence:

Curl:

r

r

FrFrr

rFr

Fr

FFr

rF

1ˆ

sin11ˆ

sinsin1ˆ

Fundamental theorem for gradient

We know df = (f ).dl

The total change in f in going from a(x1,y1,z1) to b(x2,y2,z2) along any path:

afbfldfb

a

Line integral of gradient of a function is given by the value of the

function at the boundaries of the line.

Corollary 1:

tindependenpathisldfb

a

Corollary 2: 0 ldf

Field from Potential

b

a

ldEaVbV

.

ldVaVbVb

a

From the definition of potential:

From the fundamental theorem of gradient:

E = - V

ldEldVb

a

b

a

Electric DipolePotential at a point due to dipole:

2

0

cos4

1,

r

prV

z

y

x

pr

Electric Dipole

f

rf

rrf

rfsin1ˆ1ˆˆ

ˆsin4

1ˆ13

0 r

pVr

E

rr

pr

rV

Er ˆcos24

1ˆ3

0

E = - V

Recall:

0sin1

V

rE

Electric Dipole

ˆsinˆcos24 3

0

rr

pE

ˆsinˆcos prpp

Using:

]ˆ3[1

41

30

prrpr

rEdip

Fundamental theorem for Divergence

adFdF

The integral of divergence of a vector over a volume is equal to the value of

the function over the closed surface that bounds the volume.

Gauss’ theorem, Green’s theorem

Fundamental theorem for Curl

Stokes’ theorem

ldFadF

Integral of a curl of a vector over a surface is equal to the value of the

function over the closed boundary that encloses the surface.

THE DIRAC DELTA FUNCTION

? F

F

rF

rFr

rrF r sin

1sin

sin11 2

2

Recall:

2

ˆ

r

rFLet

0

dF

0112

22

r

rrr

F

The volume integral of F:

Surface integral of F over a sphere of radius R:

S

AdF

rddRR

r ˆsinˆ 22

4From divergence theorem:

S

AdFdF

4

From calculation of Divergence:

0

dF

By using the Divergence theorem:

4 dF

Note: as r 0; F ∞

0,0

;,0

rat

buteverywhereF

4 dF

And integral of F over any volume containing the point r = 0

The Dirac Delta Function(in one dimension)

0

00

xif

xifx

1

dxxand

Can be pictured as an infinitely high, infinitesimally narrow “spike” with area 1

The Dirac Delta Function(x) NOT a Function

But a Generalized Function OR distribution

Properties: xfxxf 0

00 fdxxfdxxxf

xk

kx ||

1 xx

The Dirac Delta Function(in one dimension)

Shifting the spike from 0 to a;

axif

axifax

0

1

dxaxand

The Dirac Delta Function(in one dimension)

Properties:

axafaxxf

afdxaxxf

The Dirac Delta Function(in three dimension)

zyxr 3

0,0,0

;03

at

buteverywherer

13 drspaceall

From calculation of Divergence:

0ˆ2

d

r

r

By using the Divergence theorem:

The Paradox of Divergence of

4ˆ2

dr

r

2

ˆ

r

r

So now we can write:

rr

r 3

24

ˆ

4ˆ2

dr

rSuch that: