Vector Fields - Analytical Mechanics - Lecture Slides, Slides of Applied Mechanics

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Vector

Fields

Time

Derivative

Derivatives

of

vectors

are

by

component.

Derivatives

of

vector

products

use

the

chain

rule.

Scalar

multiplication

Inner

product

Vector

product

db dt

da dt

b

a

d dt

i

i

i

i^

^

da dt s

a

ds dt

sa

d dt

i

i

i^

db dt

a

b

da dt

b a

d dt

j

j

i i

i i^

k j

i

ijk

k j

i

ijk

k j i

ijk

e

db dt

a

e b

da dt

e b a

d dt







Space

Derivative

Spatial

derivatives

depend

on

the

coordinates.

The

partial

derivatives

point

along

coordinate

lines.

Not

the

same

as

the

coordinates.

The

del

operator

is

not

a

vector

but

acts

like

one.

Gradient

changes

scalar

to

vector

i

i^

x

e

y

= const.

x

= const.

y

x

y  

x  

i

i^

s x

e

s

i

i^

x  





Vector

Field

A

vector

field

is

a

vector

that

depends

on

position.

The

differential

operator

is

a

vector

field.

Acts

on

a

scalar

field

Measures

change

From Wolfram’s Mathworld

^

t r a

a

i

i^

x

e

r

Curl

The

vector

product

of

the

del

operator

with

a

vector

is

the

curl.

Vector

result

The

divergence

of

a

curl

is

zero.

Curl

is

related

to

the

inner

product

with

the

tangent

vector

t

^

 a

^

^

j i k

kji

j i k

ijk

j i

ijk

k

a

a

a

a

^ 

^

S

k j i

ijk

S

i i^

dA a

n

ds a t

k j i

ijk

e a

a

but

j

k i

kji

j i k

ijk

a

a

next