Vector Properties

Vector Properties

Vector Properties
Swapnil Sunil Jain
July 18, 2006

Vector Properties

Some Unconventional Syntax:

A

x

⁢

e

^

x

+

A

y

⁢

e

^

y

+

A

z

⁢

e

^

z

=

A

x

⁢

e

^

x

+

A

y

⁢

e

^

y

+

A

ϕ

⁢

e

^

z

d

⁢

x

∧

d

⁢

y

=

d

⁢

x

⁢

d

⁢

y

d

⁢

x

∧

d

⁢

y

∧

d

⁢

z

=

d

⁢

x

⁢

d

⁢

y

⁢

d

⁢

z

∫

t

1

t

2

𝑑

t

⁢

f

⁢

(

t

)

⁢

g

⁢

(

t

)

=

∫

t

1

t

2

f

⁢

(

t

)

⁢

g

⁢

(

t

)

⁢

𝑑

t

∂

x

=

∂

∂

⁡

x

d

⁢

Σ

=

material surface element

f

⁢

(

r

→

)

=

f

⁢

(

x

,

y

,

z

)

𝔼

3

=

ℝ

3

Definitions

:

A

→

∘

B

→

≡

∑

i

=

1

3

A

i

⁢

B

i

A

→

×

B

→

≡

|

𝕖

^

𝕩

𝕖

^

𝕪

𝕖

^

𝕫

A

x

A

y

A

z

B

x

B

y

B

z

|

[

A

→

,

B

→

,

C

→

]

≡

A

→

∘

(

B

→

×

C

→

)

=

(

A

→

×

B

→

)

∘

C

→

=

|

A

x

A

y

A

z

B

x

B

y

B

z

C

x

C

y

C

z

|

Δ

≡

∇

2

(Laplace operator or Laplacian)

□

2

≡

∇

2

-

1

c

2

⁢

∂

t

2

(D’Alembert or wave operator or D’Alembertian)

∇

2

⁡

A

→

≡

∇

⁡

(

∇

∘

A

→

)

-

∇

×

(

∇

×

A

→

)

=

(

∇

2

⁡

A

x

,

∇

2

⁡

A

y

,

∇

2

⁡

A

z

)

=

∂

⁡

A

→

∂

⁡

x

+

∂

⁡

A

→

∂

⁡

y

+

∂

⁡

A

→

∂

⁡

z

(Vector Laplacian)

A

∘

∇

≡

(

a

1

⁢

∂

∂

⁡

x

1

+

a

2

⁢

∂

∂

⁡

x

2

+

…

+

a

n

⁢

∂

∂

⁡

x

n

)

∇

∘

F

→

=

0

⇒

F

→

⁢

is incompressible or solenoidal or divergence-free

∇

×

F

→

=

0

⇒

F

→

⁢

is irrotational or conservative or curl-free

l

⁢

a

⁢

p

⁢

l

⁢

a

⁢

c

⁢

i

⁢

a

⁢

n

⁢

(

F

→

)

≡

d

⁢

i

⁢

v

⁢

(

g

⁢

r

⁢

a

⁢

d

⁢

(

F

→

)

)

D

u

→

⁢

f

≡

∇

⁡

f

∘

u

^

(Directional Derivative)

r

⁢

o

⁢

t

u

→

⁢

F

→

≡

c

⁢

u

⁢

r

⁢

l

⁢

(

F

→

)

⋅

u

^

(Rotational Derivative)

h

⁢

e

⁢

l

⁢

i

⁢

c

⁢

i

⁢

t

⁢

y

⁢

(

F

→

)

≡

c

⁢

u

⁢

r

⁢

l

⁢

(

F

→

)

∘

F

→

∇

∘

F

→

=

lim

δ

⁢

V

→

0

⁡

1

δ

⁢

V

⁢

∯

δ

⁢

S

F

→

∘

n

^

⁢

𝑑

S

(

∇

×

F

→

)

∘

n

→

=

lim

δ

⁢

S

→

0

⁡

1

δ

⁢

S

⁢

∮

δ

⁢

C

F

→

∘

t

^

⁢

𝑑

s

∇

×

F

→

=

lim

δ

⁢

V

→

0

⁡

1

δ

⁢

V

⁢

∯

δ

⁢

S

n

^

×

F

→

⁢

𝑑

S

∇

⁡

f

=

lim

δ

⁢

V

→

0

⁡

1

δ

⁢

V

⁢

∯

δ

⁢

S

n

^

⁢

f

⁢

𝑑

S

∇

≡

∑

i

e

^

i

h

i

⁢

∂

∂

⁡

v

i

∇

∘

≡

∑

i

e

^

i

h

i

∘

∂

∂

⁡

v

i

∇

×

≡

∑

i

e

^

i

h

i

×

∂

∂

⁡

v

i

Gradient

,

Divergence

and Curl in

Curvilinear Coordinates

∇

⁡

f

=

1

h

1

⁢

∂

⁡

f

∂

⁡

u

1

⁢

e

^

1

+

1

h

2

⁢

∂

⁡

f

∂

⁡

u

2

⁢

e

^

2

+

1

h

3

⁢

∂

⁡

f

∂

⁡

u

3

⁢

e

^

3

∇

∘

F

→

=

1

h

1

⁢

h

2

⁢

h

3

⁢

[

∂

∂

⁡

u

1

⁢

(

h

2

⁢

h

3

⁢

F

1

)

+

∂

∂

⁡

u

2

⁢

(

h

1

⁢

h

3

⁢

F

2

)

+

∂

∂

⁡

u

3

⁢

(

h

1

⁢

h

2

⁢

F

3

)

]

∇

2

⁡

f

=

1

h

1

⁢

h

2

⁢

h

3

⁢

[

∂

∂

⁡

u

1

⁢

(

h

2

⁢

h

3

h

1

⁢

∂

⁡

f

∂

⁡

u

1

)

+

∂

∂

⁡

u

2

⁢

(

h

1

⁢

h

3

h

2

⁢

∂

⁡

f

∂

⁡

u

2

)

+

∂

∂

⁡

u

3

⁢

(

h

1

⁢

h

2

h

3

⁢

∂

⁡

f

∂

⁡

u

3

)

]

∇

×

F

→

=

1

h

1

⁢

h

2

⁢

h

3

⁢

|

h

1

⁢

e

^

1

h

2

⁢

e

^

2

h

3

⁢

e

^

3

∂

∂

⁡

u

1

∂

∂

⁡

u

2

∂

∂

⁡

u

3

h

1

⁢

F

1

h

2

⁢

F

2

h

3

⁢

F

3

|

For Cartesian coordinates:

⁢

h

x

=

h

y

=

h

z

=

1

For Cylindrical coordinates:

⁢

h

r

=

1

,

h

ϕ

=

r

,

h

z

=

1

For Spherical coordinates:

⁢

h

r

=

1

,

h

θ

=

r

,

h

ϕ

=

r

⁢

sin

⁡

(

θ

)

∇

⁡

f

=

∑

i

1

h

i

⁢

∂

⁡

f

∂

⁡

q

i

⁢

e

^

i

∇

×

F

→

=

1

Ω

⁢

∑

i

e

^

i

⁢

∑

j

⁢

k

ϵ

i

⁢

j

⁢

k

⁢

h

i

⁢

∂

⁡

(

h

k

⁢

F

k

)

∂

⁡

q

j

∇

⋅

F

→

=

∑

i

1

Ω

⁢

∂

∂

⁡

q

i

⁢

(

Ω

⁢

F

i

h

i

)

∇

2

⁡

f

=

1

Ω

⁢

∑

i

∂

∂

⁡

q

i

⁢

(

Ω

h

i

2

⁢

∂

⁡

f

∂

⁡

q

i

)

where

⁢

Ω

≡

Π

⁢

h

i

Inequalities

:

∥

u

→

+

v

→

∥

≤

∥

u

→

∥

+

∥

v

→

∥

Triangle Inequality

|

u

→

∘

v

→

|

≤

∥

u

→

∥

⁢

∥

v

→

∥

Cauchy-Schwarz Inequality

Product identities:

A

→

∘

(

B

→

×

C

→

)

=

B

→

∘

(

C

→

×

A

→

)

=

C

→

∘

(

A

→

×

B

→

)

Scalar Triple Product

A

→

×

(

B

→

×

C

→

)

=

B

→

⁢

(

A

→

∘

C

→

)

-

C

→

⁢

(

A

→

∘

B

→

)

Vector Triple Product

A

→

×

(

B

→

×

C

→

)

+

B

→

×

(

C

→

×

A

→

)

+

C

→

×

(

A

→

×

B

→

)

=

0

Jacobi Identity

(

A

→

×

B

→

)

∘

(

C

→

×

D

→

)

=

A

→

∘

[

B

→

×

(

C

→

×

D

→

)

]

=

(

A

→

∘

C

→

)

⁢

(

B

→

∘

D

→

)

-

(

B

→

∘

C

→

)

⁢

(

A

→

∘

D

→

)

Scalar Quadruple Product

(

A

→

×

B

→

)

×

(

C

→

×

D

→

)

=

(

A

→

×

B

→

∘

D

→

)

⁢

C

→

-

(

A

→

×

B

→

∘

C

→

)

⁢

D

→

=

[

C

→

,

D

→

,

A

→

]

⁢

B

→

-

[

C

→

,

D

→

,

B

→

]

⁢

A

→

Vector Quadruple Product

A

→

×

[

B

→

×

(

C

→

×

D

→

)

]

=

(

A

→

×

C

→

)

⁢

(

B

→

∘

D

→

)

-

(

B

→

∘

C

→

)

⁢

(

A

→

×

D

→

)

Gradient Identities:

∇

⁡

(

f

+

g

)

=

∇

⁡

f

+

∇

⁡

g

∇

⁡

(

f

⁢

g

)

=

f

⁢

∇

⁡

g

+

g

⁢

∇

⁡

f

∇

⁡

(

A

→

∘

B

→

)

=

A

→

×

(

∇

×

B

→

)

+

B

→

×

(

∇

×

A

→

)

+

(

A

→

∘

∇

)

⁢

B

→

+

(

B

→

∘

∇

)

⁢

A

→

∇

⁡

(

A

→

×

B

→

)

=

(

∇

⁡

A

→

)

×

B

→

-

(

∇

⁡

B

→

)

×

A

→

gradient of vector??

∇

⁡

(

f

⁢

A

→

)

=

(

∇

⁡

f

)

⁢

A

→

+

f

⁢

(

∇

⁡

A

→

)

gradient of vector??

Divergence Identities:

∇

∘

(

A

→

+

B

→

)

=

∇

∘

A

→

+

∇

∘

B

→

∇

∘

(

f

⁢

A

→

)

=

f

⁢

(

∇

∘

A

→

)

+

A

→

∘

(

∇

⁡

f

)

∇

∘

(

A

→

×

B

→

)

=

B

→

∘

(

∇

×

A

→

)

-

A

→

∘

(

∇

×

B

→

)

∇

∘

(

A

→

⁢

B

→

)

=

(

∇

∘

A

→

)

⁢

B

→

+

A

→

∘

(

∇

⁡

B

→

)

=

(

∇

∘

A

→

)

⁢

B

→

+

(

A

→

∘

∇

)

⁢

B

→

Curl Identities:

∇

×

(

A

→

+

B

→

)

=

∇

×

A

→

+

∇

×

B

→

∇

×

(

f

⁢

A

→

)

=

f

⁢

(

∇

×

A

→

)

-

A

→

×

(

∇

⁡

f

)

∇

×

(

A

→

×

B

→

)

=

(

∇

∘

B

→

)

⁢

A

→

-

(

A

→

∘

∇

)

⁢

B

→

+

(

B

→

∘

∇

)

⁢

A

→

-

(

∇

∘

A

→

)

⁢

B

→

∇

×

(

A

→

⁢

B

→

)

=

(

∇

∘

A

→

)

⁢

B

→

-

A

→

×

(

∇

⁡

B

→

)

Laplacian Identities:

∇

2

⁡

(

f

⁢

g

)

=

g

⁢

∇

2

⁡

f

+

2

⁢

∇

⁡

f

∘

∇

⁡

g

+

f

⁢

∇

2

⁡

g

∇

2

⁡

(

f

⁢

A

→

)

=

=

f

⁢

∇

2

⁡

A

→

+

A

→

⁢

∇

2

⁡

f

+

2

⁢

(

∇

⁡

f

∘

∇

)

⁢

A

→

Mixed Identities:

∇

∘

(

∇

×

A

→

)

=

0

∇

×

∇

⁡

f

=

0

∇

×

(

f

⁢

∇

⁡

g

)

=

∇

⁡

f

×

∇

⁡

g

∇

∘

f

⁢

∇

⁡

g

=

f

⁢

∇

2

⁡

g

+

∇

⁡

f

∘

∇

⁡

g

∇

×

(

∇

×

A

→

)

=

∇

⁡

(

∇

∘

A

→

)

-

∇

2

⁡

A

→

only true for rectangular coordinates

Differential

Identities:

d

d

⁢

t

⁢

(

f

⁢

A

→

)

=

f

⁢

d

⁢

A

→

d

⁢

t

+

d

⁢

f

d

⁢

t

⁢

A

→

d

d

⁢

t

⁢

(

A

→

∘

B

→

)

=

A

→

∘

d

⁢

B

→

d

⁢

t

+

B

→

∘

d

⁢

A

→

d

⁢

t

d

d

⁢

t

⁢

(

A

→

×

B

→

)

=

A

→

×

d

⁢

B

→

d

⁢

t

+

B

→

×

d

⁢

A

→

d

⁢

t

Integral

Identities:

Gauss’/Divergence Thm:

Standard Form:

∯

𝒮

F

→

∘

𝑑

A

→

=

∭

𝒱

(

∇

⋅

F

→

)

⁢

𝑑

V

(

𝒮

⁢

encloses

⁢

𝒱

)

Variants:

∯

𝒮

F

→

∘

n

→

⁢

𝑑

S

=

∭

𝒱

(

∇

⋅

F

→

)

⁢

𝑑

V

(

𝒮

⁢

encloses

⁢

𝒱

)

Stokes’ Thm:

Standard Form:

∮

𝒞

F

→

∘

𝑑

l

→

=

∬

𝒮

(

∇

×

F

→

)

∘

n

→

⁢

𝑑

S

(

𝒮

⁢

bounded

⁢

by

⁢

𝒞

)

Variants:

∮

𝒞

F

→

∘

𝑑

l

→

=

∬

𝒮

(

∇

×

F

→

)

∘

𝑑

A

→

(

𝒮

⁢

bounded

⁢

by

⁢

𝒞

)

∮

𝒞

F

→

∘

t

→

⁢

𝑑

s

=

∬

𝒮

(

∇

×

F

→

)

∘

n

→

⁢

𝑑

S

(

𝒮

⁢

bounded

⁢

by

⁢

𝒞

)

∮

𝒞

F

→

∘

t

→

⁢

𝑑

s

=

∬

𝒮

(

∇

×

F

→

)

∘

𝑑

A

→

(

𝒮

⁢

bounded

⁢

by

⁢

𝒞

)

Gradient theorem

:

Standard Form:

∫

𝒞

(

∇

⁡

f

)

∘

𝑑

l

→

=

f

⁢

(

r

2

)

-

f

⁢

(

r

1

)

(

𝒞

⁢

goes

⁢

from

⁢

r

1

⁢

to

⁢

r

2

)

Integration by Parts

for Vectors

∭

𝒱

f

⁢

(

∇

∘

A

→

)

⁢

𝑑

V

=

∬

𝒮

f

⁢

A

→

∘

𝑑

a

→

-

∭

𝒱

A

→

∘

(

∇

⁡

f

)

⁢

𝑑

V

Integral Form of Maxwell’s Equations:

∯

𝒮

D

→

∘

n

^

⁢

d

2

⁢

A

=

Q

f

⁢

r

⁢

e

⁢

e

e

⁢

n

⁢

c

∯

𝒮

B

→

∘

n

^

⁢

d

2

⁢

A

=

0

∮

𝒞

E

→

∘

𝑑

l

→

=

-

d

⁢

Φ

B

d

⁢

t

∮

𝒞

H

→

∘

𝑑

l

→

=

I

f

⁢

r

⁢

e

⁢

e

e

⁢

n

⁢

c

+

d

⁢

Φ

D

d

⁢

t

Φ

B

=

∬

𝒮

B

→

∘

n

^

⁢

d

2

⁢

A

Φ

D

=

∬

𝒮

D

→

∘

n

^

⁢

d

2

⁢

A

Differential Form

of Maxwell’s Equations:

∇

∘

D

→

=

ρ

f

⁢

r

⁢

e

⁢

e

∇

∘

B

→

=

0

∇

×

E

→

=

-

∂

⁡

B

→

∂

⁡

t

∇

×

H

→

=

J

→

f

⁢

r

⁢

e

⁢

e

+

∂

⁡

D

→

∂

⁡

t

Complex Differential Form of Maxwell’s Equations:

∇

∘

M

→

=

i

⁢

ρ

ϵ

o

∇

×

M

→

=

c

⁢

μ

0

⁢

J

→

-

i

c

⁢

∂

⁡

M

→

∂

⁡

t

where

⁢

M

→

≡

c

⁢

B

→

+

i

⁢

E

→

EM Equations:

∇

2

⁡

A

→

-

1

c

2

⁢

∂

2

⁡

A

→

∂

⁡

t

2

=

-

μ

0

⁢

J

→

Vector Poisson’s Equation

∇

2

⁡

V

→

-

1

c

2

⁢

∂

2

⁡

V

∂

⁡

t

2

=

-

ρ

ϵ

Scalar Poisson’s Equation

A

→

⁢

(

r

→

)

=

μ

0

4

⁢

π

⁢

∫

J

→

⁢

(

r

′

→

)

|

r

→

-

r

′

→

|

⁢

d

3

⁢

r

′

→

ϕ

⁢

(

r

→

)

=

1

4

⁢

π

⁢

ϵ

0

⁢

∫

ρ

⁢

(

r

′

→

)

|

r

→

-

r

′

→

|

⁢

d

3

⁢

r

′

→

Title

Vector Properties

Canonical

name

VectorProperties1

Date of creation

2013-03-11 19:25:33

Last modified on

2013-03-11 19:25:33

Owner

swapnizzle (13346)

Last modified by

(0)

Numerical id

1

Author

swapnizzle (0)

Entry type

Definition

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