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PHY2206 (Electromagnetic Fields) Vector Analysis Formulae
1
Vector Analysis Formulae
Identities
1
• A × B
()
⋅C=A⋅B×C
()
2•A×B×C
()
=BA⋅C
()
−CA⋅B
()
3 A×B
()
⋅C×D
()
=A⋅C
()
B⋅D
()
−A⋅D
()
B⋅C
()
4 A×B
()
×C×D
()
=CA⋅B×D
()
{}
−DA⋅B×C
()
{}
5A×B
()
×C×D
()
=BA⋅C×D
()
{}
−AB⋅C×D
()
{}
6•∇fg
()
=f∇g+g∇f
7 ∇fg
()
=1g
()
∇f−fg
2
()
∇g
8 ∇A⋅B
()
=B⋅∇
()
A+A⋅∇
()
B+B×∇×A
()
+A×∇×B
()
9 ∇⋅ fA
()
=∇f
()
⋅A+f∇⋅A
()
10 • ∇⋅ A×B
()
=B⋅∇×A
()
−A⋅∇×B
()
11 • ∇⋅∇
()
f=∇
2
f
12 • ∇× ∇f
()
=0
13 • ∇⋅ ∇×A
()
=0
14 ∇× fA
()
=∇f
()
×A+f∇×A
()
15 ∇× A×B
()
=B⋅∇
()
A−A⋅∇
()
B+∇⋅B
()
A−∇⋅A
()
B
16a • ∇× ∇×A
()
=∇∇⋅A
()
−∇
2
A
16b ∇
2
A =∇⋅ ∇⋅A
()
−∇× ∇×A
()
17 • ∇ 1/r
()
=−
ˆ
r/r
2
If S is the closed surface that encloses the volume V and C is the closed curve that bounds an open
surface A then:
18
• ∇f
()
⋅dl
a
b
∫
=fb
()
−fa
()
19 ∇f
()
dV
V
∫
=fdS
S
∫
20 ∇×B
()
dV
V
∫
=− B×dS
S
∫
21 • B⋅dS
S
∫
=∇⋅B
()
dV
V
∫
(The Divergence Theorem)
22 • B⋅dl
C
∫
=∇×B
()
⋅dA
A
∫
(Stokes’s Theorem)
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PHY2206 (Electromagnetic Fields) Vector Analysis Formulae
2
© Copyright CDH Williams
Exeter 1996, CW000127/1
Special Coordinate Systems
Cartesian Coordinates x,y,z
()
•∇f=
∂
f
∂
x
ˆ
x+
∂
f
∂
y
ˆ
y+
∂
f
∂
z
ˆ
z
•∇⋅A=
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
• ∇×A=
∂
A
z
∂
y
−
∂
A
y
∂
z
ˆ
x+
∂
A
x
∂
z
−
∂
A
z
∂
x
ˆ
y+
∂
A
y
∂
x
−
∂
A
x
∂
y
ˆ
z
• ∇
2
f =
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
• ∇
2
A=∇
2
A
x
ˆ
x+∇
2
A
y
ˆ
y+∇
2
A
z
ˆ
z=∇∇⋅A
()
−∇× ∇×A
()
Cylindrical Polar Coordinates r,
θ
,z
()
∇f=
∂
f
∂
r
ˆ
r+
1
r
∂
f
∂θ
ˆ
θθ
+
∂
f
∂
z
ˆ
z
∇⋅A=
1
r
∂
∂
r
rA
r
()
+
1
r
∂
A
θ
∂θ
+
∂
A
z
∂
z
∇×A=
1
r
∂
A
z
∂θ
−
∂
A
θ
∂
z
ˆ
r+
∂
A
r
∂
z
−
∂
A
z
∂
r
ˆ
θθ
+
1
r
∂
∂
r
rA
θ
()
−
∂
A
r
∂θ
ˆ
z
∇
2
f=
1
r
∂
∂
r
r
∂
f
∂
r
+
1
r
2
∂
2
f
∂θ
2
+
∂
2
f
∂
z
2
Spherical Polar Coordinates r,
θ
,
ϕ
()
∇f=
∂
f
∂
r
ˆ
r+
1
r
∂
f
∂θ
ˆ
θθ
+
1
rsin
θ
∂
f
∂ϕ
ˆ
ϕϕ
∇⋅A=
1
r
2
∂
∂
r
r
2
A
r
()
+
1
rsin
θ
∂
∂
θ
A
θ
sin
θ
()
+
1
rsin
θ
∂
A
ϕ
∂ϕ
∇×A=
1
rsin
θ
∂
∂
θ
A
ϕ
sin
θ
()
−
∂
A
θ
∂ϕ
ˆ
r+
1
r
1
sin
θ
∂
A
r
∂ϕ
−
∂
∂
r
rA
ϕ
()
ˆ
θ
+
1
r
∂
∂
r
rA
θ
()
−
∂
A
r
∂θ
ˆ
ϕ
∇
2
f=
1
r
2
∂
∂
r
r
2
∂
f
∂
r
+
1
r
2
sin
θ
∂
∂
θ
sin
θ
∂
f
∂θ
+
1
r
2
sin
2
θ
∂
2
f
∂ϕ
2
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PHY2206 (Electromagnetic Fields) Vector Analysis Formulae Vector Analysis Formulae Identities 1
• (A × B)⋅C = A⋅(B × C) 2
• A × (B × C) = B(A⋅C) − C(A⋅B) 3
(A × B)⋅(C × D) = (A⋅C)(B⋅D) − (A⋅D)(B⋅C) 4
(A × B) × (C × D) = C A
{ ⋅(B× D)}− D A { ⋅(B× C)} 5
(A × B) × (C × D) = B A
{ ⋅(C× D)}− A B { ⋅(C× D)} 6
• ∇( fg) = f ∇g + g∇f 7 ∇( f g) = 1
( g)∇f − f g2 ( )∇g 8
∇(A⋅B) = (B⋅∇)A + (A⋅∇)B + B × (∇ × A) + A × (∇ × B) 9
∇⋅( fA) = (∇f )⋅A + f (∇⋅A)
10 • ∇⋅(A × B) = B⋅(∇ × A) − A ⋅(∇ × B)
11 • (∇⋅ ∇) f = ∇2 f
12 • ∇ × (∇f ) = 0
13 • ∇⋅(∇ × A) = 0 14
∇ × ( fA) = (∇f ) × A + f (∇ × A) 15
∇ × (A × B) = (B⋅∇)A − (A ⋅∇)B + (∇ ⋅ B)A − (∇ ⋅ A)B
16a • ∇ × (∇ × A) = ∇(∇ ⋅ A) − ∇2A 16b
∇2A = ∇ ⋅(∇ ⋅ A) − ∇ × (∇ × A) 17 • ∇ 1
( / r) = −ˆr / r2 cuu duong than cong . com
If S is the closed surface that encloses the volume V and C is the closed curve that bounds an open surface A then: b 18
• ∫ (∇f )⋅ dl = f(b) − f(a) a 19
∫ (∇f) dV = f dS ∫ V S 20
∫ (∇ × B)dV = − B× dS ∫ V S 21 • B⋅ dS ∫
= ∫ (∇⋅B) dV (The Divergence Theorem) S V 22 • B⋅ dl ∫
= ∫ (∇ × B)⋅dA (Stokes’s Theorem) C A 1 CuuDuongThanCong.com
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PHY2206 (Electromagnetic Fields) Vector Analysis Formulae Special Coordinate Systems
Cartesian Coordinates (x, y, z)
• ∇f = ∂f ˆ ˆ ˆ
∂ x + ∂f y + ∂f z x ∂y ∂z ∂ • ∇ ⋅ A A = ∂Ax + y + ∂Az ∂x ∂y ∂z  ∂  ∂ •   ∇ × A   A A = ∂Az − y − ∂Az y − ∂Ax  ∂y
∂z  ˆx + ∂Ax  ∂z
∂x  ˆy +  ∂x ∂y  ˆz • ∂2 ∂2 ∂2 ∇2 f f f f = + + ∂x2 ∂y2 ∂z2
• ∇2A = ∇2A ˆx + ∇2A ˆy + ∇2A ˆz = ∇(∇ ⋅ A) − ∇ × (∇ × A) x y z
Cylindrical Polar Coordinates (r,θ, z) ∇ ∂f f = ∂f ˆ ˆθ + ∂f ˆ ∂ r + 1 z r r ∂θ ∂z ∇⋅ ∂ ∂A A = 1
(rA )+ 1 θ + ∂Az r ∂r r r ∂θ ∂z ∇ ×  ∂A    ∂  A = 1 z − ∂Aθ ˆ ( )   − ∂Az θ + 1 rAθ − ∂Ar r ∂θ
∂z  ˆr + ∂Ar  ∂z ∂r r ∂r ∂θ ˆz ∇ ∂  ∂ ∂2 2 f f f = 1 r + ∂2 f
r ∂r  ∂r  + 1 r2 ∂θ 2 ∂z2
Spherical Polar Coordinates (r,θ,ϕ) ∇ ∂f ∂f f = ∂f ˆ ˆθ + 1 ˆ ϕ ∂ r + 1 r r ∂θ r sin θ ∂ϕ cuu duong than cong . com ∂ ∇ ⋅ ∂ ∂ Aϕ A = 1 (r2A )+ 1 (Aθsinθ) + 1 r2 ∂r r r sin θ ∂θ r sin θ ∂ϕ ∇ ×  ∂   1 ∂A   ∂  A = 1 A ( r ϕ sin θ ) − ∂Aθ − ∂ rA ( ϕ) (rAθ)− ∂Ar r sin θ ∂θ
∂ϕ ˆr + 1r sin  θ ∂ϕ ∂r
ˆθ + 1r ∂r ∂θ  ˆϕ ∇ ∂  ∂ ∂  ∂ ∂2 2 f f f f = 1 r2 sin θ r2 ∂r  ∂r + 1 r2 sin θ ∂θ  ∂θ  + 1 r2 sin2 θ ∂ϕ 2 © Copyright CDH Williams Exeter 1996, CW000127/1 2 CuuDuongThanCong.com
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