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
https://fb.com/tailieudientucntt
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
https://fb.com/tailieudientucntt