EEI

Appendix A: Coordinate systems

Electrical Engineering I · Module 92315 · Appendix A

A reference for Cartesian, cylindrical and spherical coordinates: how each describes a point, its unit vectors, how to convert between them, and the path, area and volume elements that every field integral in the course is built from. Keep it open next to the exercises.

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A.1 Three ways to locate a point

Each system gives a point three numbers. Cartesian coordinates measure along three fixed directions. Cylindrical coordinates give the distance ρ from the z-axis, the angle φ around it (measured from the x-axis) and the height z. Spherical coordinates give the distance r from the origin, the angle ϑ down from the z-axis and the same angle φ.

SystemCoordinates and rangesUnit vectors
Cartesianx, y, z, each from −∞ to ∞ex, ey, ez
Cylindrical0 ≤ ρ < ∞, 0 ≤ φ < 2π, z from −∞ to ∞eρ, eφ, ez
Spherical0 ≤ r < ∞, 0 ≤ ϑ ≤ π, 0 ≤ φ < 2πer, eϑ, eφ
Cylindrical to Cartesianx = ρ cos φy = ρ sin φz = zRead it as: “x is rho cos phi, y is rho sin phi, and z stays z”.Going back: ρ = √x2 + y2, and φ from tan φ = y/x, choosing the quadrant from the signs of x and y.
Spherical to Cartesianx = r sin ϑ cos φy = r sin ϑ sin φz = r cos ϑRead it as: “x is r sin theta cos phi, y is r sin theta sin phi, z is r cos theta”.r sin ϑ is the point’s distance from the z-axis, so it plays the part of ρ. Going back: r = √x2 + y2 + z2, cos ϑ = z/r, and φ as for cylindrical coordinates.

A.2 Unit vectors and cross products

In every system the three unit vectors are perpendicular to each other and have length 1, so a dot product of two different ones is 0. They form a right-handed set in the order listed: crossing one with the next gives the third, and the order runs round in a circle. Crossing in the opposite order flips the sign.

Only the Cartesian unit vectors are the same everywhere. eρ, eφ, er and eϑ change direction from point to point, which matters as soon as you add or integrate vectors (A.6).

Cylindrical. P lies on the cylinder of radius ρ. Its unit vectors point away from the axis (eρ), around it in the direction of growing φ (eφ) and up (ez). The shaded patch is the wall element dA.
Spherical. P lies on the sphere of radius r. er points away from the origin, eϑ down the line of longitude (growing ϑ) and eφ along the line of latitude. The shaded patch is the sphere element dA.
Cartesianex × ey = ezey × ez = exez × ex = eyRead it as: “e x cross e y is e z”, and so on round the circle x, y, z.Reversed order: ey × ex = −ez.
Cylindricaleρ × eφ = ezeφ × ez = eρez × eρ = eφThe circle runs ρ, φ, z. At φ = 0 the set is just ex, ey, ez.
Sphericaler × eϑ = eφeϑ × eφ = ereφ × er = eϑThe circle runs r, ϑ, φ. eφ is the same vector in cylindrical and spherical coordinates.

A.3 Converting unit vectors

Each line below writes one unit vector in another system’s unit vectors. The coefficients are the cosines of the angles between the two directions, so the same numbers appear in both directions of a conversion. Use these to turn a field given in one system into another, or to add vectors that sit at different points.

Cylindrical unit vectors in Cartesian oneseρ = ex cos φ + ey sin φeφ = −ex sin φ + ey cos φez = ezCheck at φ = 90°: eρ = ey and eφ = −ex, a quarter turn on from ex, ey.
Cartesian unit vectors in cylindrical onesex = eρ cos φ − eφ sin φey = eρ sin φ + eφ cos φThe same table read the other way: the coefficient of eφ in ex equals the coefficient of ex in eφ.
Spherical unit vectors in Cartesian oneser = ex sin ϑ cos φ + ey sin ϑ sin φ + ez cos ϑeϑ = ex cos ϑ cos φ + ey cos ϑ sin φ − ez sin ϑeφ = −ex sin φ + ey cos φer is the position of P divided by r. At the north pole (ϑ = 0) er = ez; on the equator (ϑ = 90°) eϑ = −ez.
Cartesian unit vectors in spherical onesex = er sin ϑ cos φ + eϑ cos ϑ cos φ − eφ sin φey = er sin ϑ sin φ + eϑ cos ϑ sin φ + eφ cos φez = er cos ϑ − eϑ sin ϑEach Cartesian unit vector takes its coefficients from one column of the card above.
Between cylindrical and sphericaleρ = er sin ϑ + eϑ cos ϑez = er cos ϑ − eϑ sin ϑBoth systems share φ and eφ, so only a rotation by ϑ in the plane through the z-axis is left.

A.4 Position vector and its length

The position vector r points from the origin to the point. In cylindrical and spherical coordinates the angles do not appear as terms of their own: they are hidden in the directions of eρ and er, which depend on where the point is.

Position vectorr = x ex + y ey + z ezr = ρ eρ + z ezr = r erNever write φ eφ into a position vector: an angle is not a length.
Distance from the origin|r| = √x2 + y2 + z2 = √ρ2 + z2 = rThe distance between two points is the length of the difference of their position vectors. Form that difference in Cartesian components, because eρ at one point is not eρ at the other.

A.5 Path, area and volume elements

Change one coordinate a little and the point moves a short distance. In Cartesian coordinates that distance is just dx, dy or dz. An angle step must be turned into a length: moving by dφ on a circle of radius ρ covers ρ dφ, and moving by dφ on a sphere covers r sin ϑ dφ, because the circle of latitude has radius r sin ϑ. Every element below is built from these three step lengths.

Connected topics: line integrals for the potential (§1.8) use dr, charge densities (§1.6) use dV, and flux and Gauss’s law (§1.10) use dA.

Path elementdr = ex dx + ey dy + ez dzdr = eρ dρ + eφ ρ dφ + ez dzdr = er dr + eϑ r dϑ + eφ r sin ϑ dφRead it as: “dr is the sum of the three step lengths, each along its unit vector”.On a path where only one coordinate changes, keep only that term: on a circle round the z-axis, dr = eφ ρ dφ.
Volume elementdV = dx dy dzdV = ρ dρ dφ dzdV = r2 sin ϑ dr dϑ dφThe product of the three step lengths. Check: the ball of radius R has V = ∫02π ∫0π ∫0R r2 sin ϑ dr dϑ dφ = 4πR3/3.

An area element is a vector: its size is the product of the two step lengths that lie in the surface, and its direction is the unit vector of the coordinate that stays fixed on the surface (outward for a closed surface).

The Cartesian volume element. Each face is a surface on which one coordinate is fixed. Its patch dA is the product of the two edges in the face, and its normal is the unit vector of the fixed coordinate.
Area elements, Cartesianx fixed: dA = ex dy dzy fixed: dA = ey dx dzz fixed: dA = ez dx dyOn the opposite face of a box the normal points the other way, for example −ex dy dz.
Area elements, cylindricalwall, ρ fixed: dA = eρ ρ dφ dztop or base, z fixed: dA = ±ez ρ dρ dφhalf-plane, φ fixed: dA = eφ dρ dzCheck: the wall of a cylinder of radius a and height h has area 2πa h; each end disk has πa2.
Area elements, sphericalsphere, r fixed: dA = er r2 sin ϑ dϑ dφcone, ϑ fixed: dA = eϑ r sin ϑ dr dφhalf-plane, φ fixed: dA = eφ r dr dϑCheck: ∫02π ∫0π R2 sin ϑ dϑ dφ = 4πR2, the area of a sphere.

A.6 Choosing a system, and three common slips

Pick the system in which the boundary of the problem is a surface where one coordinate is fixed. The integration limits then become constants, and a symmetric field has only one component.

The problem hasUseTypical example
Flat faces, plates, a boxCartesianPlate capacitor, a charged slab
An axis: wires, coaxial cables, cylindersCylindricalLine charge, coaxial capacitor
A centre: point charges, spheres, shellsSphericalPoint charge, charged sphere
  • Forgetting the length factor. dV is ρ dρ dφ dz, not dρ dφ dz; the sphere’s dA carries r2 sin ϑ. Check the units: every dV must come out in m3 and every dA in m2.
  • Moving a varying unit vector out of an integral. eρ depends on φ, so ∫ eρ dφ over a full circle is 0, not 2πeρ. Write it in Cartesian unit vectors first (A.3) and integrate the components.
  • Mixing up the two angles. ϑ runs only from 0 to π (pole to pole), while φ runs from 0 to 2π (once round). A full sphere is not covered by letting both run to 2π.

Personal reference sheet based on the course’s coordinate-system formula collection (Lehrstuhl für Elektromagnetische Felder), rearranged and explained in my own words; the original table and figures are not reproduced. Not an official FAU publication; check details against the current course material. Made by Thanate Amornjiraporn (Focus).