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Coordinate Systems and the Differential Element

How you slice space into small pieces decides whether an integral is easy or hard. You learn to pick the coordinate system that fits the symmetry and to use its small piece, the volume element.

Pick the volume element that fits the piece of space

The gold piece is a region cut out of a sphere. Drag to turn it around and lay the small cells of the three coordinate systems over it one by one; only one of them fills the gold piece without gaps.

Drag to rotate; tap a tab to switch coordinate systems.
Differential volume of the chosen system
dV = dx · dy · dz
A box with three straight edges. All three edges are just the coordinate increments, so there is no scale factor.
Misaligned

Cartesian — a straight cube

The three axes are mutually perpendicular and the grid spacing is the same everywhere. So the volume element is simply the product of the three edges, and since no edge bends, every scale factor is 1.

ObservedV = dx dy dz
The three Cartesian edges are multiplied as they are.

Cylindrical — an angle becomes an arc

In cylindrical coordinates, the same angle dφ sweeps a longer distance the farther you are from the axis, that is, the larger ρ is. So the length of the φ edge is not dφ but ρ dφ. This ρ, which turns an angle into a length, is the scale factor, while the other two edges, dρ and dz, stay straight.

ObservedV = dρ ρ dφ dz
Only the angular edge picks up ρ to become a length.
ChoosedV = dρ (?) dz
Only the angular edge picks up ρ to become a length.

Spherical — two arcs

Spherical coordinates have two curved edges. The edge along a meridian is an arc of a circle of radius r, so its length is r dθ. The edge along a parallel is an arc of a smaller circle whose radius is not r but r sinθ, so its length is r sinθ dφ; the parallels shrink as you approach the poles. Multiplying the two scale factors gives r² sinθ, and the radial edge dr stays straight.

ChoosedV = dr (?) ( ? )
The meridian arc has radius r, the radius of the sphere itself.
Fill indV = dr (r dθ) (?)
The parallel arc has radius r sinθ, which shrinks near the poles.
On your owndV = ? dr dθ dφ
Gathering the two scale factors gives r² sinθ.

Surface elements follow the same logic

The area of a small piece on a sphere of radius r is found the same way. Multiplying the two arcs gives dA = r² sinθ dθ dφ. The surface integral of Gauss's law in the coming units is exactly the job of adding up these pieces over the whole sphere.

Back to the first screen

What filled the gold piece without gaps was the spherical volume element. Its three edges (dr, r dθ, r sinθ dφ) grow in the same directions as the piece's symmetry. The Cartesian box could not follow the curved surface, and the cylindrical wedge has flat top and bottom faces that miss the sphere. Choosing a coordinate system ultimately means matching the shape of the small piece to the symmetry of the problem.

A volume element is the product of three edge lengths. Straight edges are the coordinate increments themselves, but curved edges are multiplied by a scale factor to become arc lengths. Cartesian dV = dx dy dz, cylindrical dV = ρ dρ dφ dz, spherical dV = r² sinθ dr dθ dφ.
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Coordinate systems and small elements are the handles of every field integral. With this tool equipped, the next units open up. Scalar and vector fields (EM-02), electric flux and divergence (EM-05), and the surface integral of Gauss's law (EM-06) are all computed on the small pieces chosen here.