Coördinate system

Cylindrical coördinates

Cylindrical coördinates are an orthogonal coördinate system useful for problems involving cylindrical symmetry, eg. about a pipe or solenoid. They are a natural extension of Polar coördinates

They can be defined by the transformation

with the signature ⃗𝐫 :(0,∞) ×[0,2𝜋) ×[0,∞) ↣ℝ3 ∖{(0,0,𝑧) :𝑧 ∈ℝ}. Any point on the 𝑧-axis cannot be given a unique coördinate., since any 𝜙 yields the same point

Calculus

The following differential quantities may be useful¹

1. 𝑑⃗ℓ=𝑑𝑠ˆ𝐬+𝑟𝑑𝜙ˆ𝜙+𝑑𝑧ˆ𝐳

\begin{align*}

d\tau = s\,ds\,d\phi\,dz
\end{align*}
$$

3. grad⁡𝐹=𝜕𝐹𝜕𝑠ˆ𝐬+1𝑠𝜕𝐹𝜕𝜙ˆ𝜙+𝜕𝐹𝜕𝑧ˆ𝐳 4. Δ𝐹=1𝑠𝜕𝜕𝑠(𝑠𝜕𝐹𝜕𝑠)+1𝑠2𝜕2𝐹𝜕𝜙2+𝜕2𝐹𝜕𝑧2 5. div⁡⃗𝐅=1𝑠𝜕𝜕𝑠(𝑠𝐹𝑠)+1𝑠𝜕𝐹𝜙𝜕𝜙+𝜕𝐹𝑧𝜕𝑧 6. curl⁡⃗𝐅=(1𝑠𝜕𝐹𝑧𝜕𝜙−𝜕𝐹𝜙𝜕𝑧)ˆ𝐬+(𝜕𝐹𝑠𝜕𝑧−𝜕𝐹𝑧𝜕𝑠)ˆ𝜙+1𝑠(𝜕𝜕𝑠(𝑠𝐹𝜙)−𝜕𝐹𝑠𝜕𝜙)ˆ𝐳

#state/tidy | #lang/en | #SemBr