Summary of Vector Relations
Ax xˆ + Ay yˆ + Az zˆ
Cylindrical (r,φ, z)
r,φ , z
Ar rˆ + Aφ φˆ + Az zˆ
Spherical (R, θ, φ)
R, θ , φ
ˆ + A θˆ + A φˆ
A R
Ax + Ay + Az
Ar + Aφ + Az
AR + Aθ + Aφ
Cartesian (x,y,z)
x, y , z
Variables

A=
Magnitude, A =
Basis vector
properties
Dot
 
product, A ⋅ B =
Cross product,
 
A×B =
2
2
2
2
2
2
xˆ ⋅ xˆ = yˆ ⋅ yˆ = zˆ ⋅ zˆ = 1
xˆ ⋅ yˆ = yˆ ⋅ zˆ = zˆ ⋅ xˆ = 0
xˆ × yˆ = zˆ
yˆ × zˆ = xˆ
rˆ ⋅ rˆ = φˆ ⋅ φˆ = zˆ ⋅ zˆ = 1
rˆ ⋅ φˆ = φˆ ⋅ zˆ = zˆ ⋅ rˆ = 0
zˆ × xˆ = yˆ
zˆ × rˆ = φˆ
Ax B x + Ay B y + Az B z
Ar Br + Aφ Bφ + Az B z
xˆ
Ax
Bx

Diff. length, d l =
Differential area
yˆ
Ay
By
rˆ × φˆ = zˆ
φˆ × zˆ = rˆ
rˆ
Ar
Br
zˆ
Az
Bz
θ
R
2
φ
2
2
ˆ ⋅R
ˆ = θˆ ⋅ θˆ = φˆ ⋅ φˆ = 1
R
ˆ ⋅ θˆ = θˆ ⋅ φˆ = φˆ ⋅ R
ˆ =0
R
ˆ × θˆ = φˆ
R
ˆ
θˆ × φˆ = R
φˆ
Aφ
Bφ
zˆ
Az
Bz
ˆ = θˆ
φˆ × R
AR BR + Aθ Bθ + Aφ Bφ
ˆ
R
Ar
Br
θˆ
Aθ
Bθ
φˆ
Aφ
Bφ
xˆ dx + yˆ dy + zˆ dz

d s x = xˆ dydz

d s y = yˆ dxdz

d s z = zˆ dxdy
rˆ dr + φˆ r dφ + zˆ dz

d s r = rˆ r dφ dz

d sφ = φˆ dr dz

d s z = zˆ r dr dφ
ˆ R dθ + φ
ˆ dR + θ
ˆ R sin θ dφ
R

ˆ R 2 sin θ dθ dφ
dsR = R

d sθ = θˆ R sin θ dR dφ

d sφ = φˆ R dr dθ
dx dy dz
r dr dφ dz
R 2 sin θ dR dθ dφ
Diff. volume,
dv =
Gradient, Divergence, Curl and Laplacian Operations
∇V =

∇⋅A =

∇×A =
∇ 2V =
Cartesian (x,y,z)
∂V
∂V
∂V
xˆ +
yˆ +
zˆ
∂x
∂y
∂z
∂
∂
∂
Ax +
Ay + Az
∂x
∂y
∂z
xˆ
∂
∂x
Ax
yˆ
∂
∂y
Ay
zˆ
∂
∂z
Az
∂ 2V ∂ 2V ∂ 2V
+
+
∂x 2 ∂y 2 ∂z 2
Cylindrical (r,φ, z)
∂V
∂V
∂V
rˆ +
φˆ +
zˆ
∂r
∂z
r∂φ
1 ∂
1 ∂Aφ ∂Az
+
(rAr ) +
∂z
r ∂r
r ∂φ
rˆ
1 ∂
r ∂r
Ar
r φˆ
∂
∂φ
r Aφ
zˆ
∂
∂z
Az
1 ∂  ∂V  1 ∂ 2V ∂ 2V
+
+
r
r ∂r  ∂r  r 2 ∂φ 2 ∂z 2
Spherical (R, θ, φ)
1 ∂V
∂V ˆ
∂V ˆ
R+
θ+
φˆ
∂R
R ∂θ
R sin θ ∂φ
∂
1 ∂
1
( Aθ sin θ )
( R 2 AR ) +
2
R sin θ ∂θ
R ∂R
1 ∂Aφ
+
R sin θ ∂φ
ˆ
R θˆ
R sin θ φˆ
R
∂
∂
∂
1
2
∂φ
R sin θ ∂R ∂θ
Ar R Aθ ( R sin θ ) Aφ
1 ∂  2 ∂V 
1
∂ 
∂V 

 sin θ
+ 2
R
2
∂θ 
∂R  R sin θ ∂θ 
R ∂R 
∂ 2V
1
+ 2
R sin 2 θ ∂φ 2
Coordinate Transforms
Coordinate variables
Cartesian to
cylidrical
Cylindrical to
Cartesian
Catersian to
spherical
r = x2 + y2
φ = tan −1 ( y / x)
z=z
x = r cos φ
y = r sin φ
z=z
R = x2 + y2 + z2
θ = tan −1 ( x 2 + y 2 / z )
φ = tan −1 ( y / x)
Spherical to
Cartesian
Cylindrical to
spherical
Spherical to
cylindrical
x = R sin θ cos φ
y = R sin θ sin φ
z = R cos θ
R = r2 + z2
θ = tan −1 (r / z )
φ =φ
r = R sin θ
φ =φ
z = R cos θ
Unit vectors
rˆ = xˆ cos φ + yˆ sin φ
φˆ = −xˆ sin φ + yˆ cos φ
zˆ = zˆ
xˆ = rˆ cos φ − φˆ sin φ
yˆ = rˆ sin φ + φˆ cos φ
zˆ = zˆ
ˆ = xˆ sin θ cos φ
R
+ yˆ sin θ sin φ + zˆ cos θ
θˆ = xˆ cos θ cos φ
+ yˆ cos θ sin φ − zˆ sin θ
φˆ = − xˆ sin φ + yˆ cos φ
ˆ sin θ cos φ
xˆ = R
+ θˆ cos θ cos φ − φˆ sin φ
ˆ sin θ sin φ
yˆ = R
+ θˆ cos θ sin φ + φˆ sin φ
ˆ cos θ − θˆ sin θ
zˆ = R
ˆ = rˆ sin θ + zˆ cos θ
R
Vector components
Ar = Ax cos φ + Ay sin φ
Aφ = − Ax sin φ + Ay cos φ
Az = Az
Ax = Ar cos φ − Aφ sin φ
Ay = Ar sin φ + Aφ cos φ
Az = Az
AR = Ax sin θ cos φ
+ Ay sin θ sin φ + Az cos θ
Aθ = Ax cos θ cos φ
+ Ay cos θ sin φ − Az sin θ
Aφ = − Ax sin φ + Ay cos φ
Ax = AR sin θ cos φ
+ Aθ cos θ cos φ − Aφ sin φ
Ay = AR sin θ sin φ
+ Aθ cos θ sin φ + Aφ cos φ
Az = AR cos θ − Aθ sin θ
AR = Ar sin θ + Az cos θ
θˆ = rˆ cos θ − zˆ sin θ
φˆ = φˆ
ˆ sin θ + θˆ cos θ
rˆ = R
Aθ = Ar cos θ − Az sin θ
φˆ = φˆ
ˆ cos θ − θˆ sin θ
zˆ = R
Aφ = Aφ
Aφ = Aφ
Ar = AR sin θ + Aθ cos θ
Az = AR cos θ − Aθ sin θ