1. Vector Analysis

1.1 Vector Algebra

1.1.1 Vector operations

A scalar has a magnitude (mass, time, temperature, charge).

A vector has a magnitude (its length) and a direction.

Examples: velocity, force, momentum, field strength.

Boldface letters denote vectors.

On the blackboard I use

A .

A

an ˆ,ˆUnit vectors are denoted by

Vectors have no location.

-AA

Vector field A(r)

addition of two vectors: A+B

multiplication by a scalar: aA

dot product (scalar, inner): cosABBA

0

BA

BA

ABBA

AB if parallel

if perpendicular

Example work

)( 12 rrF W

Example 1.1

cross product (vector, outer): nBA ˆsinAB

n̂ is the unit vector perpendicular to the AB-plane.

nBA ˆ,,

nBA ˆ,, form a right-handed system. BAAB

0AA

BA

is the area of the parallelogram.

Example: angular momentum

prL

1.1.2 Component Form

zyxA ˆˆˆ zyx AAA

3,2,1ˆ iAiin 1: x, 2: y, 3: z

components:basis: zyx ˆ,ˆ,ˆ zyx AAA ,,

An iiA

common notation: zyx AAA ,,A

312231123ˆˆˆ

0,0

,ˆˆ

321 cyclic

jiifjiifij

ijji

nnn

nn

Kronecker symbol

Properties of the basis

3

1iiiBABA

iii BA )( BA

3

1

)(i

iiaAa nA

else

if

ijkif

BACor

BBB

AAA

ijk

kjkjijki

zyx

zyx

0

321,213,1321

312,231,1231

ˆˆˆ3

1,

zyx

BAC

Levi-Civitasymbol

Example 1.2

1.1.3 Triple Products

scalar triple product:cyclic)()()( BACACBCBA

volume

zyx

zyx

zyx

CCC

BBB

AAA

)( CBA

vector triple product:

)()()( BACCABCBA

bac - cab rule

Higher order products by repeated bac-cab and symmetriesof the scalar triple product.

1.1.4 Notation

rr

rr

ˆ , : vectorseparation

ˆˆˆ :ntdisplaceme malinfinitesi

ˆˆˆ : vectorposition

rr

zyxl

zyxr

dzdydxd

zyx

1.1.5 How Vectors TransformRotation about the x-axis:

z

y

z

y

A

A

A

A

cossin

sincos

3

1jjiji ARAIn general

3

1

3

1 iii

iii BABABA