Keohane & Foy: An Introduction to Classical Electrodynamics 758

Appendix D Vector Calculus

Geometrical Definitions

Vector Algebra Identities

Fundamental Theorems

!F ⋅d!A

surface"∫ =

!∇⋅!F( )dV

volume"∫

!F ⋅d!ℓ

path#∫ =

!∇×!F( ) ⋅d

!A

surface

∫ f !r( )− f !r

0( ) =

!∇f ⋅d

!ℓ

!r0

!r

∫

!∇⋅!F ≡ lim

V→01V

!F ⋅d!A

surface"∫

!∇×!F( )n ≡ limA→0 1A

!F ⋅d!ℓ

path#∫ n

!∇f ≡ lim

δ !r→0

δ fδ !r

!A ⋅!B = AB"

B!

!B

!A

!A ×!B = AB⊥ n

B⊥ !B

!A

⊙ nn

The Divergence

The Gradient

The Curl

∇2 !A =

!∇!∇⋅!A( )−!∇×

!∇×!A( )

∇2 f =!∇⋅!∇f( )

!∇!A ⋅!B( ) = !A ×

!∇×!B( ) + !B ×

!∇×!A( )

+!A ⋅!∇( ) !B +

!B ⋅!∇( ) !A

!∇×

!A ×!B( ) = !B ⋅ !∇( ) !A −

!A ⋅!∇( ) !B

+!A!∇⋅!B( )− !B !∇⋅

!A( )

!∇⋅!A ×!B( ) = !B ⋅ !∇×

!A( )− !A ⋅ !∇×

!B( )

!∇ f g( ) = f

!∇g( )+ g

!∇f( )

!∇⋅ f

!A( ) = f

!∇⋅!A( )+!A ⋅!∇f( )

!∇× f

!A( ) = f

!∇×!A( )−!A×!∇f( )

∇2 f g( ) = f ∇2g( )+ 2

!∇f( ) ⋅

!∇g( )+ g ∇2 f( )

∇2 !A ⋅!B( ) =

!A ⋅ ∇2 !B( )−

!B ⋅ ∇2 !A( )

+ 2!∇⋅

!B ⋅!∇( )!A+!B×!∇×!A( )( )

∇2 f

!A( ) = f ∇2 !A( ) + 2 !∇f ⋅ !∇( ) ⋅ !A +

!A ∇2 f( )

!∇f g( ) = df

dg

!∇g

!∇⋅!A f( ) =

!∇f ⋅ d

df

!A( )

!∇×!A f( ) =

!∇f × d

df

!A( )

∇2 f g( ) = df

dg∇2g+ d2 f

dg2

!∇g

2

!∇×

!∇f( ) = 0

!∇⋅!∇×!A( ) = 0

!A ⋅!B×!C( ) =

!B ⋅!C ×!A( ) =

!C ⋅!A×!B( )

!A×!B×!C( )+

!B×!C ×!A( )+!C ×

!A×!B( ) = 0

!A ⋅!B =!B ⋅!A

!A×!B( ) ⋅

!C ×!D( ) =

!A ⋅!C( )!B ⋅!D( )

−!B ⋅!C( )!A ⋅!D( )

!A×!B( )×

!C ×!D( ) =

!A ⋅!B×!D( )( )!C −

!A ⋅!B×!C( )( )!D

!A ×

!B ×!C( ) = !B !A ⋅ !C( )− !C !A ⋅ !B( )

!A ×

!B ×!C( ) = !B ×

!A ×!C( )− !C ×

!A ×!B( )

!A ×!B ×!C =!A ×

!B ×!C( )

!A ⋅!A = A2

A =!A =

!A ⋅!A

!A×!B = −

!B×!A

!A×!A = 0

Vector Calculus Identities

COMMON CURL AND DIVERGENCE FREE VECTOR FIELDS COMMON VECTOR DERIVATIVES

!∇⋅ r = 2

r

!∇⋅ rn r( ) = n+ 2( )rn−1

!∇×ϕ = z

s =r

r tanθ −θr

!∇⋅!r = 3

!∇⋅ s = 1

s

!∇×θ = ϕ

r

!∇⋅θ = 1

r tanθ

!∇ z = z !∇× f r( ) r = 0

!∇r = r

!∇×!r = 0

!F∝ r

r2

!F∝

3 z ⋅ r( )− zr3

!F∝ ϕ

s

!F∝ s

s

!∇ s = s