Cylindrical CoordinatesEdit
Cylindrical coordinates provide a natural language for describing space when circular symmetry around an axis is at play. They pair a radial distance with an angle in the plane and a vertical height, making many problems in physics and engineering cleaner to formulate and solve. In practice, this system often leads to neater expressions for boundaries, integrals, and differential operators than Cartesian coordinates, especially when the geometry resembles a cylinder or a tube. In the horizontal plane, the coordinates reduce to the familiar polar system, while the vertical direction remains a straightforward linear axis.
From a pragmatic standpoint, cylindrical coordinates are a staple tool for problems with axisymmetry. They let you exploit symmetry to simplify boundary conditions, reduce dimensionality when possible, and compute quantities like volumes, masses, fluxes, and field strengths with less algebraic clutter than you’d see in a purely rectangular setup. This makes them indispensable in disciplines ranging from electromagnetism to fluid dynamics. For readers who want to connect the horizontal cross-section to a standard 2D system, the connection to polar coordinates is immediate, while the full three-dimensional description extends these ideas along the vertical axis.
Definition and coordinates
The cylindrical coordinate triple is (r, θ, z) with r ≥ 0, θ in [0, 2π), and z ∈ ℝ. The point described by these coordinates corresponds to the Cartesian point x = r cos θ, y = r sin θ, z = z, so the axial coordinate z remains the same as in Cartesian coordinates.
In the horizontal plane, (r, θ) are just the familiar polar coordinates, while z adds the vertical dimension. The inverse relations are r = sqrt(x^2 + y^2) and θ = atan2(y, x) (the angle corresponding to the direction from the axis to the point).
The axis of the cylinder is the z-axis, and the coordinate system is naturally adapted to problems with circular cross-sections and axial symmetry.
Jacobian and volume element
When converting volume integrals from Cartesian to cylindrical coordinates, the Jacobian determinant for (r, θ, z) → (x, y, z) is r. Accordingly, the differential volume element is dV = r dr dθ dz.
This r factor has practical consequences: for axisymmetric regions (where integrands do not depend on θ), one can often perform the θ-integration immediately, yielding a factor of 2π.
A simple check is the volume of a cylinder with radius R and height H: ∫∫∫ dV = ∫0^H ∫0^{2π} ∫0^R r dr dθ dz = (π R^2) H.
Differential operators in cylindrical coordinates
The fact that the basis vectors er and eθ depend on θ changes how derivatives are taken. For a vector field A = Ar er + Aθ eθ + Az ez, the standard differential operators in cylindrical coordinates are:
Gradient: ∇f = (∂f/∂r) er + (1/r)(∂f/∂θ) eθ + (∂f/∂z) ez.
Divergence: ∇·A = (1/r) ∂(r Ar)/∂r + (1/r) ∂Aθ/∂θ + ∂Az/∂z.
Curl: ∇×A = (1/r) [∂Az/∂θ − ∂Aθ/∂z] er + [∂Ar/∂z − ∂Az/∂r] eθ + (1/r) [∂(r Aθ)/∂r − ∂Ar/∂θ] ez.
Laplacian for a scalar field f: ∇^2 f = (1/r) ∂/∂r ( r ∂f/∂r ) + (1/r^2) ∂^2 f/∂θ^2 + ∂^2 f/∂z^2.
These formulas reflect the θ-dependence of the basis and are standard tools in vector calculus and physical applications. They’re especially convenient when a problem has rotational symmetry around the z-axis or when the boundary surfaces are cylinders or cones.
Applications and examples
Mass or charge distribution: If density ρ is a function of r, θ, and z, the total mass or charge is found by integrating ρ over dV = r dr dθ dz. For a uniform density ρ on a cylinder of radius R and height H, the mass is ρ π R^2 H.
Electromagnetism and Gauss’s law: A long straight wire or a coaxial cable naturally leads to cylindrical symmetry. Using cylindrical coordinates simplifies the calculation of electric and magnetic fields, as the field components often depend only on r. For a line charge with linear density λ, Gauss’s law gives a radial field E(r) ∝ 1/r, with the flux through a cylinder of radius r and height L being proportional to L and r.
Fluid dynamics in pipes and ducts: Steady, axisymmetric flow in a cylindrical pipe is elegantly described in cylindrical coordinates. The velocity field often has only a z-component that varies with r, simplifying the continuity equation and the Navier–Stokes equations in the axisymmetric form. Mass flow rate through a cross-section is ∫∫ ρ v_z dA = ∫0^R 2π r ρ v_z(r) dr.
Boundary-value problems in heat conduction and acoustics: Cylindrical coordinates match the geometry of many problems, from heated rods to acoustic tubes, reducing partial differential equations to forms that are easier to solve with separation of variables.
Relationship to other coordinate systems
The coordinate change to Cartesian coordinates is explicit, with x = r cos θ, y = r sin θ, z = z, and the inverse r = sqrt(x^2 + y^2), θ = atan2(y, x). This makes it straightforward to translate boundary conditions and sources between systems.
The cross-section in cylindrical coordinates is governed by polar coordinates in the plane, while the vertical direction remains linear. When the problem lacks axial symmetry, one may still choose cylindrical coordinates for partial benefit or combine with boundary conditions to simplify certain terms.
For some problems with strong curvature or three-dimensional geometry that doesn’t align with an axis, switching to Spherical coordinates or a coordinate-free, vector-field formulation can be advantageous. The choice of coordinates is a balancing act between symmetry, boundary geometry, and computational convenience.