The Fundamental Theorem of Calculus

b

a

afbfdxdxdf

(Integral of a derivative over a region is related to values at the boundary)

Cross Product: determinant of matrix with unit vector

zyx

zyx

BBB

AAA

kji

BA

ˆˆˆ

det

zzyyxx BABABABA

Dot Product: multiply components and add

Scalar Field : a scalar quantity defined at every point of a 2D or 3D space.

),(),( yxfyxS )sin(),( xyxyxS Ex:

0

50

100

0

50

100-10

-5

0

5

10

20 40 60 80 100

10

20

30

40

50

60

70

80

90

100

-4

-2

0

2

4

6

20 40 60 80 100

10

20

30

40

50

60

70

80

90

100

-6

-4

-2

0

2

4

6

EM Fields

0

10

20 05

1015

200

5

10

15

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3D scalar field

3D scatter plot with color giving the field value:

510210310 222

1),,( zyx eeezyxS

Vector Field: a vector quantity defined at every point of a 2D or 3D space.

jyxixyyx ˆˆ)sin(),( S

kVjViVzyx zyxˆˆˆ),,( V

Functions of (x,y,z)

NOT constantsNOT partial derivatives

2D Ex:

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Temperature Map: a scalar field

Wind Map: a vector field

Two Fields

1. Gradient

“the derivative of a scalar field”

y

S

x-2 -1 0 1 2

-2

0

20

5

10

15

20

22 yxS

Derivative (slope) depends on direction!

dyyS

dxxS

dS

kz

jy

ix

ˆˆˆ

Total Differential:

Looks like a dot product: jdyidxjyS

ixS

dS ˆˆˆˆ

ldSdS

“del”

“nabla”

Del is not a vector and it does not multiply a field – it is an operator!

1. The Fundamental Theorem of Gradients

b

a

aSbSldS

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0

2

4

6

8

10

12

0

50

100

0

50

100-10

-5

0

5

10

a

b

(Integral of a derivative over a region is related to values at the boundary)

V

“the creation or destruction of a vector field”

kVjViVkz

jy

ix zyx

ˆˆˆˆˆˆ

zyx Vz

Vy

Vx

(a scalar field!)

jyxixyyx ˆˆ)cos(),( 22 V

yxyy 2)sin( V

2. Divergence

kjyix ˆ0ˆˆ V

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

X

Y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

X

Y

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

kjyix ˆ0ˆˆ V

2 V

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

jyix ˆˆ 22 V

jyix ˆˆ 22 V

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-6

-4

-2

0

2

4

6

yx 22 V

2. The Fundamental Theorem of Divergence

SV

add

VV

volume integral surface integral

(The Divergence Theorem)

(Integral of a derivative over a region is related to values at the boundary)

+ -

+ -

I. Gauss’ Law: relation between a charge distribution and the electric field

point charge

E field lines

0q

dS

aE

0

1 E Gauss’ Law

(differential form)

First Results from a Superconductive Detector for Moving Magnetic Monopoles

Blas Cabrera Physics Department, Stanford University, Stanford, California

94305 Received 5 April 1982

A velocity- and mass-independent search for moving magnetic monopoles is being performed by continuously monitoring the current in a 20-cm2-area superconducting loop. A single candidate event, consistent with one Dirac unit of magnetic charge, has been detected during five runs totaling 151 days. These data set an upper limit of 6.1×10-10 cm-2 sec-1 sr-1 for magnetically charged particles moving through the earth's surface.

PRL 48, p1378 (1982)

The Valentine’s Day Monopole

II. Gauss’ Law for Magnetism: relation between magnetic monopole distribution and the magnetic field

0 B

0S

daB

Cabrera

V

kVjViVkz

jy

ix zyx

ˆˆˆˆˆˆ

“How much a vector field causes something to twist”

zyx

zyx

VVV

kji ˆˆˆ

det

3. Curl

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

kjyix ˆ0ˆˆ V

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

011 V

colorplot = z component of curl(V)

kjyix ˆ0ˆˆ V

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

kjxiyy ˆ0ˆˆsin 22 V

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

kyyyyx ˆ cossin22 2 V

colorplot = z component of curl(V)

kjxiyy ˆ0ˆˆsin 22 V

3. The Fundamental Theorem of Curl

PS

dd lVaV

open surface integral closed perimeter line integral

(Really called Stokes’ Theorem)

V

V

(Integral of a derivative over a region is related to values at the boundary)

Faraday

III. Faraday’s Law: A changing magnetic field induces an electric field.

B

emf

temf B

0

Moving coil in a varying B field.Force on electrons:

BvEF

q

Forces don’t cancel: 0emf

FF

v

v

EE

BvEF

q

0v

EFq

Electric field must be created!

Only left with:

Stationary coil with moving B source:

But we still get an emf …

EE

i

In general:

temf B

SCd

td aBlE

Faraday’s Law(integral form)

Stationary coil and B source, but increasing B strength:

0emf

t B

E

Faraday’s Law(differential form)

IV. Ampere’s Law

iB

enclosedCid 0 lB

More general:

SCdd aJlB

0

J = free current density

MaxwellAmpere

“Something is missing..”

Cd lB

SdaJ

Charging a capacitor

i- +- +

- +- +

- +

Cd lB 0S daJ

i- +- +

- +- +

- +

Charging a capacitor

Maxwell: “…the changing electric field in the capacitor is also a current.”

SCd

td aEJlB

00

“Displacement current”

Ampere-Maxwell Eqn.(Integral Form)

SSd

td aEJaB

00

Get Stoked:

EJB

00

t

Ampere-Maxwell Eqn.(differential form)

Maxwell’s Equations in Free Space with no free charges or currents

0 E

0 B

t B

E

EB

t 00

Faraday

MaxwellAmpere

GaussYour Name Here!