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EM-02 · Electrostatics toolkitTool equip

Scalar Fields and Vector Fields

A field is a function that attaches a value to every point of space. Learn to tell a scalar field (one number per point) from a vector field (an arrow per point).

What do you attach to the same space?

Drag to look around the space. What you attach to the same points decides the kind of field. Go from bare points to numbers, then to arrows.

Drag to orbit. Tap to switch representation.
What one point carries
One point = location only, 0 values attached
We see space as points only. No quantity is attached yet.
No value

Scalar field — one number per point

A scalar field assigns one number to each point of space. The temperature in a room T(x, y, z), the altitude on a mountain h(x, y), the potential V near a charge — all are scalar fields. The value has only size and no direction, so rotating the coordinate axes never changes the number at a point.

ObserveT : ℝ³ → ℝ
A scalar field sends each point to one real number.

Vector field — an arrow per point

A vector field assigns an arrow to each point. The arrow carries both magnitude and direction. The velocity of air v, the electric field E made by a charge — these are vector fields. The arrow at a point splits into three components, so a vector field can be read as three scalar fields (E_x, E_y, E_z) bundled together at each point.

ChooseE : ℝ³ → ?
A vector field sends each point to a triple of real numbers.
Fill inE = Ex x̂ + Ey ? + Ez ẑ
Three components ride their own unit vectors.

Reassembling into magnitude and direction

The length of the arrow is its magnitude, found as the root of the sum of the squared components. The direction is peeled off as a unit vector — the same arrow shrunk to length one. Even with three components, the arrow is still one per point: a single directed whole.

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

Scalar or vector?

Temperature, mass, potential, and voltage are scalars. Velocity, force, electric field, and magnetic field are vectors. The test is simple: if asking "which direction?" makes sense for the quantity, it is a vector; if not, a scalar. The electric field is incomplete until you give a direction (EM-03); the potential is a single directionless value (EM-08).

Back to the first screen

On the first screen, the bare points filled first with numbers, then with arrows. Filling with numbers gives a scalar field; filling with arrows gives a vector field. The two are the same space read through different quantities. Because an arrow keeps pointing the same way however you rotate the axes, a vector field — even split into three components — stays at heart one directed whole.

A field is a function that maps every point of space to a value. 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 carries 1 component per point, a vector field 3.
Tool equipOnce you equip this tool

The notation for scalar and vector fields is the vocabulary of all field theory. Once you equip it, the next units open up: the electric field E (EM-03) reads as a vector field, the potential V (EM-08) as a scalar field. Flux and divergence (EM-05) and the Gauss law (EM-06) are all defined on this vector field.