Coordinate Systems and Vector Derivatives Formula Sheet
Rectangular (Cartesian) Coordinates (x, y, z )
Line element: d~
`= ˆx dx + ˆy dy + ˆz dz
Volume element: dτ =dx dy dz
Gradient: ~
∇f=∂f
∂x ˆx+∂f
∂y ˆy+∂f
∂z ˆz
Divergence: ~
∇ · ~v =∂vx
∂x +∂vy
∂y +∂vz
∂z
Curl: ~
∇ × ~v =µ∂vz
∂y −∂vy
∂z ¶ˆx+µ∂vx
∂z −∂vz
∂x ¶ˆy+µ∂vy
∂x −∂vx
∂y ¶ˆz
Laplacian: ∇2f=∂2f
∂x2+∂2f
∂y2+∂2f
∂z2
Spherical Coordinates (r, θ, φ)
Relations to rectangular (Cartesian) coordinates and unit vectors:
x=rsin θcos φ
y=rsin θsin φ
z=rcos θ
ˆx= ˆrsin θcos φ+ˆ
θcos θcos φ−ˆ
φsin φ
ˆy= ˆrsin θsin φ+ˆ
θcos θsin φ+ˆ
φcos φ
ˆz= ˆrcos θ−ˆ
θsin θ
r=px2+y2+z2
θ= tan−1(px2+y2/z)
φ= tan−1(y/x)
ˆr= ˆxsin θcos φ+ ˆysin θsin φ+ ˆzcos θ
ˆ
θ= ˆxcos θcos φ+ ˆycos θsin φ−ˆzsin θ
ˆ
φ=−ˆxsin φ+ ˆycos φ
Line element: d~
`= ˆr dr +ˆ
θ r dθ +ˆ
φ r sin θ dφ
Volume element: dτ =r2sin θ dr dθ dφ
Gradient: ~
∇f=∂f
∂r ˆr+1
r
∂f
∂θ ˆ
θ+1
rsin θ
∂f
∂φ ˆ
φ
Divergence: ~
∇ · ~v =1
r2
∂
∂r (r2vr) + 1
rsin θ
∂
∂θ (sin θ vθ) + 1
rsin θ
∂vφ
∂φ
Curl: ~
∇ × ~v =1
rsin θ·∂
∂θ (sin θ vφ)−∂vθ
∂φ ¸ˆr+1
r·1
sin θ
∂vr
∂φ −∂
∂r (r vφ)¸ˆ
θ
+1
r·∂
∂r (r vθ)−∂vr
∂θ ¸ˆ
φ
Laplacian: ∇2f=1
r
∂2
∂r2(rf) + 1
r2sin θ
∂
∂θ µsin θ∂f
∂θ ¶+1
r2sin2θ
∂2f
∂φ2
