E&M at ESG Spring 2005

Vector Operators

in

Cylindrical and Spherical Coordinates

Tables such as these abound. Here’s another. One page is used for each of cylindrical and spherical coordniate systems, for clarity and for printing only what’s neded.

The notation is a combination of that of Jackson’s Classical Electrodynamics, Simmons’s Calculus and Analtyic Geometry and Purcell’s Electricity and Magnetism. ψ is any sufficiently differentiable scalar function and A is any sufficiently differentiable vector function. Subscripts represent components of A, not partial differentiation, as in A = A_{x} x̂ + A_{y} ŷ + A_{z} ẑ. In spherical coordinates, “physics” notation, with z = r cos θ, is used.

For completeness, the familiar expressions in Cartesian coordinates are included.

Cartesian

Following Purcell, the unit vectors are denoted {x̂, ŷ, ẑ}, x̂ × ŷ = ẑ.

∇ψ = \frac{∂ψ}{∂x} x̂ + \frac{∂ψ}{∂y} ŷ + \frac{∂ψ}{∂z} ẑ

∇ · A = \frac{∂A_{x}}{∂x} + \frac{∂A_{y}}{∂y} + \frac{∂A_{z}}{∂z}

∇ × A = (\frac{∂A_{z}}{∂y} − \frac{∂A_{y}}{∂z} ) x̂ + (\frac{∂A_{x}}{∂z} − \frac{∂A_{z}}{∂x} ) ŷ + (\frac{∂A_{y}}{∂x} − \frac{∂A_{x}}{∂y} ) ẑ

∇^{2}ψ = \frac{∂^{2}ψ}{∂x^{2}} + \frac{∂^{2}ψ}{∂y^{2}} + \frac{∂^{2}ψ}{∂z^{2}}

Cylindrical

Unit vectors {r̂, θ̂, ẑ}, r̂×θ̂ = ẑ, A = A_{r} r̂ + A_{θ}θ̂ + A_{z} ẑ.

\begin{aligned}x & = r cos θ \\ y & = r sin θ \\ z & = z \end{aligned}  \begin{aligned}r & = \sqrt{x^{2} + y^{2}} \\ tan θ & = \frac{y}{x} \\ z & = z \end{aligned}

∇ψ = \frac{∂ψ}{∂r} r̂ + \frac{1}{r} \frac{∂ψ}{∂θ} θ̂ + \frac{∂ψ}{∂z} ẑ

∇ · A = \frac{1}{r} \frac{∂}{∂r} (r A_{r}) + \frac{1}{r} \frac{∂A_{θ}}{∂θ} + \frac{∂A_{z}}{∂z}

∇ × A = (\frac{1}{r} \frac{∂A_{z}}{∂θ} − \frac{∂A_{θ}}{∂z} ) r̂ + (\frac{∂A_{r}}{∂z} − \frac{∂A_{z}}{∂r} )θ̂ + \frac{1}{r} (\frac{∂}{∂r} (r A_{θ}) − \frac{∂A_{r}}{∂θ} ) ẑ

Note that

∇ × A = \frac{1}{r} | \begin{matrix}r̂ & rθ̂ & ẑ \\ \frac{∂}{∂r} & \frac{∂}{∂θ} & \frac{∂}{∂z} \\ A_{r} & r A_{θ} & A_{z} \end{matrix} | .

∇^{2}ψ = \frac{1}{r} \frac{∂}{∂r} (r \frac{∂ψ}{∂r} ) + \frac{1}{r^{2}} \frac{∂^{2}ψ}{∂θ^{2}} + \frac{∂^{2}ψ}{∂z^{2}}

Spherical

Unit vectors {r̂, θ̂, ϕ̂}, r̂ ×θ̂ =ϕ̂, A = A_{r} r̂ + A_{θ}θ̂ + A_{ϕ}ϕ̂.

x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = r cos θ

r = \sqrt{x^{2} + y^{2} + z^{2}}_{cos θ = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}}_{tan ϕ = \frac{y}{x}}

0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2 π

∇ ψ = \frac{∂ψ}{∂r} r̂ + \frac{1}{r} \frac{∂ψ}{∂θ} θ̂ + \frac{1}{r sin θ} \frac{∂ψ}{∂ϕ} ϕ̂

∇ · A = \frac{1}{r^{2}} \frac{∂}{∂r} (r^{2} A_{r}) + \frac{1}{r sin θ} \frac{∂}{∂θ} (sin θ A_{θ}) + \frac{1}{r sin θ} \frac{∂A_{ϕ}}{∂ϕ}

∇ × A = \frac{1}{r sin θ} [\frac{∂}{∂θ} (sin θ A_{ϕ}) − \frac{∂A_{θ}}{∂ϕ} ] r̂
+ [\frac{1}{r sin θ} \frac{∂A_{r}}{∂ϕ} − \frac{1}{r} \frac{∂}{∂r} (r A_{ϕ})]θ̂ + \frac{1}{r} [\frac{∂}{∂r} (r A_{θ}) − \frac{∂A_{r}}{∂θ} ]ϕ̂

Note that

∇ × A = \frac{1}{r^{2} sin θ} | \begin{matrix}r̂ & rθ̂ & r sin θϕ̂ \\ \frac{∂}{∂r} & \frac{∂}{∂θ} & \frac{∂}{∂ϕ} \\ A_{r} & r A_{θ} & r sin θ A_{ϕ} \end{matrix} | .

∇^{2}ψ = \frac{1}{r^{2}} \frac{∂}{∂r} (r^{2} \frac{∂ψ}{∂r} ) + \frac{1}{r^{2} sin θ} \frac{∂}{∂θ} (sin θ \frac{∂ψ}{∂θ} ) + \frac{1}{r^{2} sin^{2} θ} \frac{∂^{2}ψ}{∂ϕ^{2}}

[Note that \frac{1}{r^{2}} \frac{∂}{∂r} (r^{2} \frac{∂ψ}{∂r} ) = \frac{1}{r} \frac{∂^{2}}{∂r^{2}} (r ψ) .]