What Is a Tensor?

Press your thumb into a lump of clay. The clay pushes back — but in which direction? Straight out along your thumb? Not quite: squeeze a block of rubber and the internal forces on a slanted slice point off at an angle to that slice. To describe the force on a surface you must supply the surface's orientation (a direction) and get back a force (another direction). A single number can't do that. A single arrow can't either. You need a machine that eats one direction and returns another, linearly. That machine is a tensor.

You already know the two simplest tensors under other names. A scalar — temperature, mass — is just a number. A vector — velocity, force — is a number-with-a-direction. Tensors are the natural continuation of that sequence: objects that package together several directions at once. Stress, the moment of inertia of a spinning top, the electromagnetic field, the curvature of spacetime — none of these fit into a scalar or a vector. They are tensors, and this page is about what that word actually means.

View 1: a tensor is a multilinear machine

The cleanest definition lives in the language of vector spaces. A tensor is a multilinear map: a function that takes in some vectors (and covectors) and outputs a number, and is linear in every one of its slots separately.

The machine. A tensor T of rank k takes k vector arguments and returns a scalar: T(\mathbf{u}, \mathbf{v}, \dots) \in \mathbb{R}.
Linear in each slot. Double an input, double the output; add two inputs in one slot, add the outputs — with all other slots held fixed: T(a\mathbf{u}+b\mathbf{u}', \mathbf{v}) = a\,T(\mathbf{u},\mathbf{v}) + b\,T(\mathbf{u}',\mathbf{v}).

A rank-2 tensor, fed one vector, hasn't finished — it still has a slot open, so it hands back a function-waiting-for-one-more-vector, which is itself a covector. That is exactly the "eats a direction, returns a direction" behaviour of stress: the stress tensor \boldsymbol{\sigma} takes the normal \mathbf{n} of a surface and returns the traction (force per area) \mathbf{t} = \boldsymbol{\sigma}\,\mathbf{n} acting on it — generally pointing a different way from \mathbf{n}.

Seeing a rank-2 tensor act

A rank-2 tensor is a linear map: feed it a direction and it returns a vector. The figure below does exactly that. Drag the angle slider to sweep the input unit vector \mathbf{n} (the shorter arrow) around the circle; the tensor T maps it to the output T\mathbf{n} (the longer arrow), whose tips trace the grey ellipse — the image of the unit circle.

Notice the key feature: for a general direction, the output T\mathbf{n} points a different way from the input \mathbf{n} — the surface pushes back at an angle. Only along two special directions (the ellipse's axes) do input and output line up. Those are the tensor's eigenvectors, the "principal axes" — for the inertia tensor they are the axes a body spins about cleanly, and for the stress tensor they are the directions of pure push or pull.

View 2: a tensor is what transforms like a tensor

Physicists more often meet the other definition, in components. Pick a basis and a rank-2 tensor becomes an array of numbers T_{ij}, written with indices and the Einstein summation convention. But — and this is the whole point — not every array of numbers is a tensor. It is a tensor only if its components change under a change of basis in the one specific way that keeps the underlying geometric object fixed:

T'_{ij} = \frac{\partial x^k}{\partial x'^{i}}\,\frac{\partial x^l}{\partial x'^{j}}\,T_{kl}.

One factor of the transformation (a Jacobian) per index. An index that transforms this way — "down stairs" — is covariant; an index that transforms with the inverse factor — "upstairs", T^{ij} — is contravariant. The rank (or order) of the tensor is simply the number of indices, and a rank- k tensor in n dimensions has n^k components.

The two views are the same idea: a coordinate-free machine (view 1), and its components together with the rule that keeps the machine the same when you change coordinates (view 2).

The ladder of ranks

Rank 0 — a scalar. No indices, n^0 = 1 component. Same number in every frame: temperature, charge, mass.
Rank 1 — a vector (or covector). One index, n components: velocity, force, the electric field.
Rank 2 — the "matrix-like" tensors. Two indices, n^2 components: the stress tensor \sigma_{ij}, the moment-of-inertia tensor I_{ij} (which relates angular velocity to angular momentum, L_i = I_{ij}\,\omega_j), and the metric tensor g_{ij} that measures lengths.
Rank 3, 4, … More slots: the elasticity tensor C_{ijkl} (rank 4) links stress to strain; the Riemann curvature tensor (rank 4) encodes the bending of spacetime.

A handy special tensor is the Kronecker delta \delta^i_{\,j} — 1 when i=j, else 0 — the identity map, and the only rank-2 tensor whose components are the same in every basis.

Worked examples

Example 1 — counting components. How many components has a rank-3 tensor in 3-dimensional space? It is n^k = 3^3 = 27.

Example 2 — inertia in action. A rank-2 tensor relates two vectors linearly. For a spinning body, L_i = I_{ij}\,\omega_j (summed over j). Because I_{ij} is generally not diagonal, the angular momentum \mathbf{L} need not be parallel to the angular velocity \boldsymbol{\omega} — which is exactly why an unbalanced wheel wobbles.

Example 3 — is it a tensor? Write down the array A_{ij} whose entries are your three favourite numbers arranged in a grid. Is it a tensor? Only if those numbers were defined so that they obey the transformation law under every change of basis. A grid of arbitrary constants that you leave unchanged when you rotate the axes is not a tensor — it fails the rule.

Example 4 — a spacetime count. In 4-dimensional spacetime, how many components does the rank-2 metric tensor have? n^k = 4^2 = 16 (of which symmetry leaves 10 independent).

No — and the difference is the whole subject. A rank-2 tensor can be written as a grid of numbers once you fix a basis, but a grid of numbers is only a tensor if those numbers transform correctly when you change basis (view 2). Scribble down nine arbitrary numbers and freeze them regardless of coordinates and you have a matrix that is not a tensor. The array is just the tensor's shadow in one coordinate system; the tensor is the coordinate-free object casting it.

A second, sneakier trap: "rank" means two different things. A tensor's rank is its number of indices (a stress tensor is "rank 2" with a full 3\times3 of components). A matrix's rank, from linear algebra, counts its independent rows. They are unrelated — a rank-2 tensor can have matrix-rank 1, 2 or 3. Keep the two senses of the word apart.

Because of a deep principle: the laws of physics cannot depend on the coordinate system you happened to choose. Your choice of axes is human bookkeeping; the universe doesn't know about it. The magic of the tensor transformation law is that if a tensor equation like \mathbf{T} = 0 holds in one coordinate system, it holds in every one — because both sides transform the same way, the equality survives the change intact.

This is called covariance, and it is the reason Einstein wrote general relativity entirely in tensors: G_{\mu\nu} = 8\pi\,T_{\mu\nu} is guaranteed to mean the same physics to every observer, accelerating or not. Writing a law as a tensor equation is a promise that it is real physics and not an artefact of your grid. That is why "is it a tensor?" is the first question a physicist asks of any new quantity.