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Scalar Fields and Vector Fields

A field is a function that attaches a value to every point in space. You learn to tell them apart: if each point carries one number, it is a scalar field; if each point carries an arrow with magnitude and direction, it is a vector field.

What do we attach to the same points in space?

Drag to look around the evenly spaced points in space and switch tabs: what kind of field you have depends on what you attach to the same points. It starts with bare points, moves to gold glows whose brightness shows a number, and then to arrows.

Drag to rotate; tap a tab to switch representations.
What one point carries
One point = location only, 0 values attached
Each point has only a position; no quantity is attached yet.
No value

Scalar field — one number per point

A scalar field pairs each point in space with one number. The temperature in a room T(x, y, z), the altitude of a mountain h(x, y), and the potential V around a charge are all scalar fields. A value with magnitude but no direction does not change at a point no matter how you rotate the coordinate axes.

ObserveT : ℝ³ → ℝ
A scalar field assigns one real number to a point.

Vector field — an arrow per point

A vector field pairs each point with an arrow. An arrow carries magnitude and direction together. The velocity v of flowing air and the electric field E made by a charge are examples. The arrow at a point can be written as three components along the coordinate axes, (E_x, E_y, E_z). The components, however, are numbers that depend on the choice of axes, so they change when you rotate the axes, while the arrow itself stays the same.

ChooseE : ℝ³ → ?
A vector field assigns a bundle of three real numbers to a point.
Fill inE = Ex x̂ + Ey ? + Ez ẑ
The three components ride on the unit vectors of the axes, one each.

Reassembling into magnitude and direction

The length of the arrow is its magnitude, found by squaring the three components, adding them, and taking the square root. The direction is separated out as a unit vector, the same arrow shrunk to length 1. Even with three components, the arrow is still one per point, a single piece that carries a direction.

On your own|E| = ?
The magnitude is the square root of the sum of the squared components.

Scalar or vector?

Temperature, mass, potential, and potential difference are scalars; velocity, force, electric field, and magnetic field are vectors. An easy test is to ask the quantity which way it points. If the question makes sense, it is a vector; if not, a scalar. A surer test is whether it adds like arrows. Current in a wire has a sense of flow, but at a junction the currents simply add, so it is a scalar; the matching vector is the current density, the current per unit area. The electric field is not fully stated until its direction is given (EM-03), while the potential is a single value with no direction (EM-08).

Back to the first screen

On the first screen, the bare points filled first with numbers and then with arrows. Filled with numbers, they form a scalar field; filled with arrows, a vector field. The two read the same space as different quantities. An arrow points to the same place no matter how you rotate the coordinate axes, so even when a vector field is written as three components, at heart it remains a single piece with a direction.

A field is a function that assigns a value to every point in space. If the value is one number, it is a scalar field T(x, y, z); if it has magnitude and direction, it is a vector field E = Ex x̂ + Ey ŷ + Ez ẑ. A scalar field holds 1 component per point, a vector field 3.
Tool equipOnce you equip this tool

The notation of scalar and vector fields is the vocabulary of all field theory. With this notation equipped, the next units open up. The electric field E (EM-03) reads directly as a vector field and the potential V (EM-08) as a scalar field, and flux and divergence (EM-05) and Gauss's law (EM-06) are all defined on this vector field.