Post on 15-Nov-2021
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