Introduction to Electrodynamics 5th Edition – David Griffiths (2023) - eTextbook

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, VECTOR DERIVATIVES

Cartesian dl = dx x̂ + dy ŷ + dz ẑ, dτ = dx dy dz
∂t ∂t ∂t
Gradient: ∇t = x̂ + ŷ + ẑ
∂x ∂y ∂z

∂vx ∂vy ∂vz
Divergence: ∇·v= + +
∂x ∂y ∂z
 ∂vz ∂vy   ∂vx ∂vz   ∂vy ∂vx 
Curl: ∇×v= − x̂ + − ŷ + − ẑ
∂y ∂z ∂z ∂x ∂x ∂y

∂2t ∂2t ∂2t
Laplacian: ∇2 t = 2
+ 2+ 2
∂x ∂y ∂z

Spherical dl = dr r̂ + r dθ θ̂ + r sin θ dφ φ̂, dτ = r2 sin θ dr dθ dφ
∂t 1 ∂t 1 ∂t
Gradient: ∇t = r̂ + θ̂ + φ̂
∂r r ∂θ r sin θ ∂φ

1 ∂ 2 1 ∂ 1 ∂vφ
Divergence: ∇·v= (r vr ) + (sin θ vθ ) +
r2 ∂r r sin θ ∂θ r sin θ ∂φ

1  ∂ ∂vθ 
Curl: ∇×v= (sin θ vφ ) − r̂
r sin θ ∂θ ∂φ

1  1 ∂vr ∂  1 ∂ ∂vr 
+ − (rvφ ) θ̂+ (rvθ ) − φ̂
r sin θ ∂φ ∂r r ∂r ∂θ

1 ∂  2 ∂t  1 ∂  ∂t  1 ∂2t
Laplacian: ∇2 t = r + sin θ +
r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂φ2

Cylindrical dl = ds ŝ + s dφ φ̂ + dz ẑ, dτ = s ds dφ dz
∂t 1 ∂t ∂t
Gradient: ∇t = ŝ + φ̂ + ẑ
∂s s ∂φ ∂z

1 ∂ 1 ∂vφ ∂vz
Divergence: ∇·v= (svs ) + +
s ∂s s ∂φ ∂z
 1 ∂vz ∂vφ   ∂vs ∂vz 
Curl: ∇×v= − ŝ + − φ̂
s ∂φ ∂z ∂z ∂s

1 ∂ ∂vs 
+ (svφ ) − ẑ
s ∂s ∂φ

1 ∂  ∂t  1 ∂ 2 t ∂2t
Laplacian: ∇2 t = s + 2 +
s ∂s ∂s s ∂φ2 ∂z2

, VECTOR IDENTITIES

Triple Products
(1) A · (B × C) = B · (C × A) = C · (A × B)
(2) A × (B × C) = B(A · C) − C(A · B)
Product Rules
(3) ∇(fg) = f (∇g) + g(∇f )
(4) ∇(A · B) = A × (∇ × B) + B × (∇ × A) + (A · ∇)B + (B · ∇)A
(5) ∇ · (f A) = f (∇ · A) + A · (∇f )
(6) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
(7) ∇ × (f A) = f (∇ × A) − A × (∇f )
(8) ∇ × (A × B) = (B · ∇)A − (A · ∇)B + A(∇ · B) − B(∇ · A)
Second Derivatives
(9) ∇ · (∇ × A) = 0
(10) ∇ × (∇f ) = 0
(11) ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A
FUNDAMENTAL THEOREMS

 b
Gradient Theorem: (∇f ) · dl = f (b) − f (a)
a 
Divergence Theorem: (∇ · A) dτ = A · da
 
Curl Theorem: (∇ × A) · da = A · dl

, Introduction to Electrodynamics
Fifth Edition



David J. Griffiths is Emeritus Professor of Physics at Reed College, Oregon,
where he has taught for over 30 years. He received his BA and Ph.D.
from Harvard University, where he studied elementary particle theory.
He has published three widely-used textbooks on the topics of quantum
mechanics, electrodynamics, and particle physics, and a lower-level text
titled Revolutions in Twentieth-Century Physics (Cambridge, 2012).