Summary Electromagnetic Filed

Coordinate system
Cartesian Cylindrical Spherical




Graph




Points 𝒙, 𝒚, 𝒛 𝝆, ∅, 𝒛 𝒓, 𝜽, ∅
−∞ ≤ 𝒙 ≤ ∞ 0≤𝜌≤∞ 0≤r≤∞
range −∞ ≤ 𝑦 ≤ ∞ 0 ≤ ∅ ≤ 2π 0 ≤ ∅ ≤ 2π
−∞ ≤ z ≤ ∞ −∞ ≤ z ≤ ∞ 0≤θ≤π
Length element dx, dy, dz 𝑑𝜌 , 𝜌𝑑∅, 𝑑𝑧 𝑑𝑟, 𝑟𝑑𝜃, 𝑟 𝑠𝑖𝑛𝜃 𝑑∅
∬ 𝑑𝑥 𝑑𝑦 ∬ 𝜌 𝑑𝜌 𝑑∅ ∬ 𝑟 𝑑𝑟 𝑑𝜃

Area (S) ∬ 𝑑𝑥 𝑑𝑧 ∬ 𝑑𝜌 𝑑𝑧 ∬ 𝑟 𝑠𝑖𝑛𝜃 𝑑𝑟 𝑑∅

∬ 𝑑𝑦 𝑑𝑧 ∬ 𝜌𝑑∅ 𝑑𝑧 ∬ 𝑟 2 sin 𝜃 𝑑𝜃 𝑑∅

Volume (V) ∭ 𝑑𝑥 𝑑𝑦 𝑑𝑧 ∭ 𝜌 𝑑𝜌 𝑑∅ 𝑑𝑧 ∭ 𝑟 2 𝑠𝑖𝑛𝜃 𝑑𝑟 𝑑𝜃 𝑑∅
𝑥̂ = 𝑎̂𝑥 𝜌̂ = 𝑎
̂𝜌 𝑟̂ = 𝑎
̂𝑟
Unit vector (𝑛̂) 𝑦̂ = 𝑎̂𝑦 ̂=𝑎
∅ ̂∅ 𝜃̂ = 𝑎̂𝜃
𝑧̂ = 𝑎
̂𝑧 𝑧̂ = 𝑎
̂𝑧 ̂=𝑎
∅ ̂∅
𝑥̂ ∙ 𝑥̂ = 𝑦̂ ∙ 𝑦̂ = 𝑧̂ ∙ 𝑧̂ = 1 𝜌̂ ∙ 𝜌̂ = ∅̂∙ ∅̂ = 𝑧̂ ∙ 𝑧̂ = 1 ̂∙ ∅
𝑟̂ ∙ 𝑟̂ = 𝜃̂ ∙ 𝜃̂ = ∅ ̂=1
Dot product ̂ = 𝑧̂ ∙ ∅
̂ = 𝑧̂ ∙ 𝜌̂ = 0 ̂ ∙ 𝜃̂ = ∅
̂ ∙ 𝑟̂ = 0
𝑥̂ ∙ 𝑦̂ = 𝑧̂ ∙ 𝑦̂ = 𝑧̂ ∙ 𝑥̂ = 0 𝜌̂ ∙ ∅ 𝑟̂ ∙ 𝜃̂ = ∅
𝑥̂ × 𝑥̂ = 𝑦̂ × 𝑦̂ = 𝑧̂ × 𝑧̂ = 0 ̂
𝜌̂ × 𝜌̂ = ∅ × ∅ ̂ = 𝑧̂ × 𝑧̂ = 0 ̂ ̂ ̂
𝑟̂ × 𝑟̂ = 𝜃 × 𝜃 = ∅ × ∅ ̂=0
𝑥̂ × 𝑦̂ = 𝑧̂ 𝜌̂ × ∅̂ = 𝑧̂ ̂
𝑟̂ × 𝜃̂ = ∅
𝑧̂ × 𝑦̂ = −𝑥̂ ̂ = −𝜌̂
𝑧̂ × ∅ ̂ × 𝜃̂ = −𝑟̂
∅
𝑥̂ × 𝑧̂ = 𝑦̂ ̂
𝜌̂ × 𝑧̂ = ∅ 𝑟̂ × ∅̂ = 𝜃̂
Cross product
Z X Z r

Y