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Rectangular Coordinates
Rectangular Position Vector:
𝑑𝑙
𝑟⃗ = 𝑥𝑎̂ + 𝑦𝑎̂ + 𝑧𝑎̂
𝑎̂
Basis is Cyclic under Cross Product:
𝑑𝑙
𝑎̂
𝑎̂
𝑎̂ × 𝑎̂ = 𝑎̂ ,
{𝑖𝑗𝑘} = {𝑥𝑦𝑧, 𝑧𝑥𝑦, 𝑦𝑧𝑥}
𝑟⃗
Rotating the xy-plane by 𝜙:
𝑥
𝑦
=
cos 𝜙
sin 𝜙
−sin 𝜙
cos 𝜙
Line segment parallel to axis:
𝑑𝑙 = 𝑑𝑥 𝑎̂
𝑦, 𝑧 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑥
𝑦
z
𝑑𝑆
Given 𝑓 = 𝑓(𝑥, 𝑦, 𝑧) and 𝐹 ⃗ = 𝐹 𝑎̂ + 𝐹 𝑎̂ + 𝐹 𝑎̂ :
𝜕 𝑓 𝜕 𝑓 𝜕 𝑓
∇ 𝑓=
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
Scalar In,
Scalar Out
𝜕𝐹
𝜕𝐹
𝜕𝐹
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
Vector In,
Scalar Out
∇ ∙ 𝐹⃗ =
∇ × 𝐹⃗ =
Scalar In,
Vector Out
𝜕𝑓
𝜕𝑓
𝜕𝑓
𝑎̂ +
𝑎̂ +
𝑎̂
𝜕𝑥
𝜕𝑦
𝜕𝑧
∇𝑓 =
𝜕𝐹
𝜕𝐹
−
𝜕𝑦
𝜕𝑧
𝑎̂ +
𝜕𝐹
𝜕𝐹
−
𝜕𝑧
𝜕𝑥
𝑎̂ +
Line in constant-z plane:
𝑑𝑙 = 𝑑𝑥 𝑎̂ + 𝑑𝑦 𝑎̂
𝑦 = 𝑚𝑡 + 𝑏 , 𝑥 = 𝑡
∴ 𝑑𝑙 = 1𝑎̂ + 𝑚𝑎̂ 𝑑𝑡
𝜕𝐹
𝜕𝐹
−
𝜕𝑥
𝜕𝑦
𝑎̂
Vector In,
Vector Out
𝑑𝑆
𝑑𝑆
y
x
𝑑𝑆 = 𝑑𝑦𝑑𝑧 𝑎̂
𝑥 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑑𝑆 = 𝑑𝑥𝑑𝑧 𝑎̂
𝑦 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑑𝑆 = 𝑑𝑥𝑑𝑦 𝑎̂
𝑧 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Differential Line Element:
z
𝑑𝑙 = 𝑑𝑥𝑎̂ + 𝑑𝑦𝑎̂ + 𝑑𝑧𝑎̂
dz
𝑑𝑙
In terms of one degree of freedom:
dy
dx
𝑑𝑙 =
y
𝑑𝑥
𝑑𝑦
𝑑𝑡 𝑎̂ +
𝑑𝑡 𝑎̂
𝑑𝑡
𝑑𝑡
𝑑𝑧
+
𝑑𝑡 𝑎̂
𝑑𝑡
x
𝑑𝑉 = 𝑑𝑥𝑑𝑦𝑑𝑧
1
Cylindrical Coordinates
Rectangular Position Vector:
z
𝑑𝑙
𝑟⃗ = 𝜌 𝑐𝑜𝑠 𝜙 𝑎̂ + 𝜌 𝑠𝑖𝑛 𝜙 𝑎̂ + 𝑧𝑎̂
𝑎̂
𝑑𝑙
Basis is Cyclic under Cross Product:
𝑎̂
Fixed 𝜙 in xy-plane:
𝑑𝑙 = 𝑑𝜌 𝑎̂
𝜙 , 𝜙 , 𝑧 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑎̂ × 𝑎̂ = 𝑎̂ ,
{𝑖𝑗𝑘} = {𝜌𝜙𝑧, 𝑧𝜌𝜙, 𝜙𝑧𝜌}
𝑎̂
𝑟⃗
ϕ
𝑑𝑙
y
ρ
x
Fixed 𝜌 in xy-plane:
𝑑𝑙 = 𝜌𝑑𝜙 𝑎̂
𝜌 , 𝜌 , 𝑧 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Basis in terms of Rectangular Basis:
𝑎̂ = cos 𝜙 𝑎̂ + sin 𝜙 𝑎̂
𝑎̂ = − sin 𝜙 𝑎̂ + cos 𝜙 𝑎̂
𝑎̂ = 𝑎̂
Logarithmic Spiral:
𝑑𝑙 = 𝑑𝜌𝑎̂ + 𝜌𝑑𝜙𝑎̂ ,
𝑑𝑙 = 𝑒 𝑎̂ + 𝑒 𝑎̂ 𝑑𝜙
𝑑𝑆
Given 𝑓 = 𝑓(𝜌, 𝜙, 𝑧) and 𝐹 ⃗ = 𝐹 𝑎̂ + 𝐹 𝑎̂ + 𝐹 𝑎̂ :
grad(𝑓) =
𝜕𝑓
1 𝜕𝑓
𝜕𝑓
𝑎̂ +
𝑎̂ +
𝑎̂
𝜕𝜌
𝜌 𝜕𝜙
𝜕𝑧
1 𝜕
𝜕𝑓
1
𝛻 𝑓=
𝜌
+
𝜌 𝜕𝜌
𝜕𝜌
𝜌
div 𝐹 ⃗ =
curl 𝐹 ⃗ =
𝜕𝐹
1 𝜕𝐹
−
𝜌 𝜕𝜙
𝜕𝑧
1
𝜌
𝜕
𝜌𝐹
𝜕𝜌
𝑎̂ +
+
𝑑𝑆
𝑑𝑆
𝜕 𝑓
𝜕 𝑓
+
𝜕𝜙
𝜕𝑧
1 𝜕𝐹
𝜌 𝜕𝜙
𝜕𝐹
𝜕𝐹
−
𝜕𝑧
𝜕𝜌
+
𝑎̂ +
𝑑𝑆 = 𝜌𝑑𝜙𝑑𝑧 𝑎̂
𝜌 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜕𝐹
𝜕𝑧
𝜕𝐹
1 𝜕
(𝜌𝐹 ) −
𝜌 𝜕𝜌
𝜕𝜙
𝑑𝑆 = 𝑑𝜌𝑑𝑧 𝑎̂
𝜙 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑑𝑆 = 𝜌𝑑𝜌𝑑𝜙 𝑎̂
𝑧 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑎̂
Differential Line Element:
z
𝑑𝑙 = 𝑑𝜌𝑎̂ + 𝜌𝑑𝜙𝑎̂ + 𝑑𝑧𝑎̂
𝑑𝑙
In terms of one degree of freedom:
dz
𝑑𝑙 =
dρ
y
dϕ
𝑑𝜌
𝑑𝜙
𝑑𝑡𝑎̂ + 𝜌
𝑑𝑡𝑎̂
𝑑𝑡
𝑑𝑡
𝑑𝑧
+
𝑑𝑡𝑎̂
𝑑𝑡
𝑑𝑉 = 𝜌𝑑𝜌𝑑𝜙𝑑𝑧
x
2
Spherical Coordinates
Rectangular Position Vector:
z
𝑎̂
θ
𝑑𝑙
Basis is Cyclic under Cross Product:
𝑎̂
𝑟⃗
Fixed 𝜙 in xy-plane:
𝑑𝑙 = 𝑑𝑟 𝑎̂
𝜙 , 𝜙 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜃 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝜋/2)
𝑎̂ × 𝑎̂ = 𝑎̂ ,
{𝑖𝑗𝑘} = {𝑟𝜃𝜙, 𝜙𝑟𝜃, 𝜃𝜙𝑟}
y
ϕ
Basis in terms of Rectangular Basis:
𝑎̂ = 𝑠𝜃 𝑐𝜙 𝑎̂ + 𝑠𝜃 𝑠𝜙 𝑎̂ + 𝑐𝜃 𝑎̂
𝑎̂ = 𝑐𝜃 𝑐𝜙 𝑎̂ + 𝑐𝜃 𝑠𝜙 𝑎̂ − 𝑠𝜃 𝑎̂
𝑎̂ = −𝑠𝜙 𝑎̂ + 𝑐𝜙 𝑎̂
x
grad(𝑓) =
div 𝐹 ⃗ =
curl 𝐹 ⃗ =
1
𝑟
𝜕
(𝑟 𝐹 )
𝜕𝑟
+
1
𝜕
𝜕𝐹
𝐹 sin 𝜃 −
𝑟 sin 𝜃 𝜕𝜃
𝜕𝜙
𝜕
𝜕𝑓
𝑠in 𝜃
𝜕𝜃
𝜕𝜃
1
𝑟 sin 𝜃
𝑎̂ +
𝜕
(𝐹 sin 𝜃)
𝜕𝜃
+
z
z
𝑑𝑆
x
𝑑𝑆 = 𝑟 sin 𝜃 𝑑𝜃𝑑𝜙 𝑎̂
𝑟 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜕𝐹
1
𝑟 sin 𝜃 𝜕𝜙
1 1 𝜕𝐹
𝜕
1 𝜕
𝜕𝐹
− (𝑟𝐹 ) 𝑎̂ +
(𝑟𝐹 ) −
𝑟 sin 𝜃 𝜕𝜙 𝜕𝑟
𝑟 𝜕𝑟
𝜕𝜃
𝑑𝑙 = 𝑒 𝑎̂ + 𝑒 𝑎̂ 𝑑𝜙
z
x
1
𝜕 𝑓
𝑟 sin 𝜃 𝜕𝜙
+
Logarithmic Spiral:
𝑑𝑙 = 𝑑𝑟𝑎̂ + 𝑟sinθ 𝑑𝜙𝑎̂ ,
𝑑𝑆
𝜕𝑓
1 𝜕𝑓
1
𝜕𝑓
𝑎̂ +
𝑎̂ +
𝑎̂
𝜕𝑟
𝑟 𝜕𝜃
𝑟 sin 𝜃 𝜕𝜙
1 𝜕
𝜕𝑓
1
𝑟
+
𝑟 𝜕𝑟
𝜕𝑟
𝑟 sin 𝜃
Fixed 𝑟 in xy-plane:
𝑑𝑙 = 𝑟 sin 𝜃 𝑑𝜙 𝑎̂
𝑟 , 𝑟 , 𝜃 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑑𝑆
Given 𝑓 = 𝑓(𝑟, 𝜃, 𝜙) and 𝐹 ⃗ = 𝐹 𝑎̂ + 𝐹 𝑎̂ + 𝐹 𝑎̂ :
𝛻 𝑓=
𝑑𝑙
𝑟 = 𝑟 sin𝜃 cos 𝜙 𝑎̂ + 𝑟 sin𝜃 sin 𝜙 𝑎̂
+ 𝑟 cos𝜃𝑎̂
𝑎̂
r
𝑑𝑙
𝑑𝑆 = 𝑟 sin 𝜃 𝑑𝑟𝑑𝜙 𝑎̂
𝜃 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑑𝑆 = 𝑟𝑑𝑟𝑑𝜃 𝑎̂
𝜙 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑎̂
Differential Line Element:
dr
dθ
𝑑𝑙 = 𝑑𝑟𝑎̂ + 𝑟𝑑𝜃𝑎̂ + 𝑟 sin 𝜃 𝑑𝜙𝑎̂
In terms of one degree of freedom:
𝑑𝑙
y
dϕ
x
𝑑𝑙 =
𝑑𝑟
𝑑𝜃
𝑑𝑡𝑎̂ + 𝑟
𝑑𝑡𝑎̂
𝑑𝑡
𝑑𝑡
𝑑𝜙
+ 𝑟 sin 𝜃
𝑑𝑡𝑎̂
𝑑𝑡
𝑑𝑉 = 𝑟 sin 𝜃 𝑑𝑟𝑑𝜃𝑑𝜙
3