Basic tensors

Understand multi-directional quantities such as inertia, strain, or stress.

What it is

A second-order tensor is a mathematical object that relates two vectors, or more generally, that needs two indices to fully describe a physical quantity. Unlike a scalar (no indices) or a vector (one index), a second-order tensor such as the inertia tensor or the stress tensor responds differently depending on which direction it is "asked about" and in which direction the response is measured.

In practice, a second-order tensor in a finite-dimensional space is represented by a matrix, but with an important conceptual difference: a tensor has an intrinsic physical meaning independent of the chosen basis, while its numerical components change if the coordinate system is changed, following specific transformation rules.

Why it appears in physics

It appears because many physical quantities do not behave the same way in every direction. The inertia tensor relates the angular velocity of a rigid body to its angular momentum, and in general the two vectors are not parallel unless the body rotates about a principal axis. The stress tensor in a material relates the orientation of an internal surface to the force per unit area acting on it, and that force can have components both perpendicular and tangential to the surface.

The strain tensor in elasticity also appears, as does the electric polarization tensor in anisotropic media, and in relativity, the metric tensor and the stress-energy tensor, which generalize these ideas to a context where spacetime itself has a nontrivial geometric structure.

Visual or geometric idea

The visual idea is this: imagine asking "what force per unit area acts on this surface?" while pointing in different directions. A scalar would give the same answer in every direction (like pressure in a fluid at rest); a stress tensor, instead, gives a different vector answer depending on the orientation of the chosen surface, and that answer depends linearly on that orientation.

Another useful image is the inertia ellipsoid: if you plot how the moment of inertia of a rigid body varies with the chosen rotation axis, you get an ellipsoidal surface whose principal axes are precisely the directions where the tensor acts as a simple scalar (the axes where angular velocity and angular momentum are parallel).

Minimal symbolic language

A second-order tensor T is represented by its components T_ij in a given basis, and it acts on a vector v to produce another vector w through w_i = Σ_j T_ij v_j, exactly like the matrix-vector product. If T is symmetric (T_ij = T_ji), like the inertia tensor or the stress tensor in the absence of internal torques, it has real eigenvalues and orthogonal eigenvectors, called principal axes.

When changing basis through a rotation matrix R, the tensor's components transform as T' = R T R^T, a different rule from how a simple vector transforms (v' = Rv), which reflects that the tensor "connects" two directions at once.

What the result means

The result of applying a tensor to a vector means a new vector physical quantity that depends linearly on the input direction, but which in general is not parallel to it. For example, the angular momentum resulting from applying the inertia tensor to an arbitrary angular velocity does not point in the same direction as that angular velocity, unless it coincides with a principal axis.

The principal axes and their associated eigenvalues (the same eigenvalues and eigenvectors from the corresponding leaf) give the simplest interpretation of the tensor: along those special directions, the tensor acts as a simple scale factor, and the complicated behavior along arbitrary directions can be understood as a combination of those simple behaviors.

When not to use it

Do not treat a tensor as if it were a simple number or a vector when the problem explicitly depends on the relative orientation between two directions: reducing the inertia tensor to a single number (the scalar moment of inertia) is valid only if the rotation occurs exactly about a known fixed axis, not in the general case.

Also do not confuse the numerical components of a tensor in a particular basis with the tensor itself: the tensor represents a physical quantity with meaning independent of the basis, and its components change predictably when the coordinate system changes, not arbitrarily.

Operational summary

Identify which two directions the physical tensor of the problem relates (for example, angular velocity and angular momentum, or surface orientation and force), represent the tensor as a matrix of components in a specific basis, apply the matrix-vector product to obtain the resulting quantity, and if possible, find the principal axes to simplify the physical interpretation.

Procedure

Before calculating

Identify which two vector quantities the tensor of the physical problem relates (angular velocity and angular momentum, surface orientation and force, strain and stress) and in which basis its numerical components are given.

Check whether the tensor is symmetric, since this guarantees real eigenvalues and orthogonal principal axes, which greatly simplifies the later interpretation.

Step 1: identify the mathematical object

Write the tensor as a matrix of components T_ij in the given basis. If the problem gives you the matrix directly, check its symmetry by comparing T_ij with T_ji.

If the problem asks for the principal axes, recognize that this is equivalent to finding the eigenvalues and eigenvectors of the tensor, treated as a symmetric matrix.

Step 2: choose the operation

If you need the tensor's response to a specific direction, apply the matrix-vector product w_i = Σ_j T_ij v_j. If you need the principal axes, diagonalize the matrix following the procedure from the eigenvalues and eigenvectors leaf.

If you need to express the tensor in another basis (for example, after rotating the reference frame), apply the transformation rule T' = R T R^T, not the transformation rule for a simple vector.

Step 3: preserve units and signs

Check that the tensor's units are consistent with the two quantities it relates: for example, the inertia tensor has units of mass times length squared, and the stress tensor has units of force per area.

Preserve the signs of the off-diagonal components, since they indicate how the different directions are coupled; zeroing them out by mistake destroys the physical anisotropy of the problem.

Step 4: calculate

Calculate the matrix-vector product component by component, or solve the eigenvalue problem if you are looking for the principal axes, following exactly the same algebraic procedure as for any symmetric matrix.

If you need to transform the tensor to another basis, calculate R T first and then multiply by R^T, respecting the order of operations.

Step 5: interpret the result

Interpret the resulting vector as the physical response of the system to the given input direction: angular momentum, force per unit area, or an analogous quantity depending on the context. Compare its direction with that of the input vector to decide whether they are parallel (special case, principal axis) or not (general case).

If you found the principal axes, interpret each eigenvalue as the value the tensor would take if it acted as a simple scalar along that specific direction.

Self-check checklist

Check that you correctly identified the two vector quantities the tensor relates, that you applied the matrix-vector product with the indices in the correct order, that you respected the transformation rule T' = R T R^T when changing basis, and that you can explain why the result is not generally parallel to the input vector.

Guided example

Physical situation

An asymmetric rigid body has inertia tensor I = [[3, 0, 0], [0, 2, 0], [0, 0, 1]] in kg·m² with respect to its principal axes. It rotates with angular velocity ω = (1, 1, 0) rad/s, which does not coincide with any principal axis. The task is to calculate its angular momentum.

Mathematical translation

The angular momentum is obtained by applying the inertia tensor to the angular velocity: L = I ω, a direct matrix-vector product because the tensor is already expressed in its principal-axis basis.

Step-by-step calculation

L_x = 3·1 + 0·1 + 0·0 = 3, L_y = 0·1 + 2·1 + 0·0 = 2, L_z = 0·1 + 0·1 + 1·0 = 0. So L = (3, 2, 0) kg·m²/s.

Reading the result

The vector ω = (1, 1, 0) has the same magnitude along the x and y axes, but the resulting angular momentum L = (3, 2, 0) does not: its x component is larger than its y component, in the same proportion as the principal moments of inertia I_x = 3 and I_y = 2. This confirms that L and ω are not parallel, because ω does not coincide with any principal axis of the body.

Common error avoided

This avoids treating the moment of inertia as a simple scalar and calculating L = I_scalar · ω, which would only work if ω coincided exactly with a principal axis. In the general case, the full tensor must be applied component by component.

Inverse example

Given result

An inertia tensor has all three of its eigenvalues (principal moments of inertia) equal to each other.

What physical question it can answer

This allows one to conclude that the body is a spherically symmetric rotor in terms of its mass distribution: any axis passing through its center of mass behaves as a principal axis, and the angular momentum is always parallel to the angular velocity, regardless of the chosen rotation direction. A physical example is a homogeneous sphere or a homogeneous cube about its center.

Common errors

Error 1: treating a tensor as a scalar

It seems reasonable to summarize a tensor quantity as a single number, as is often done with mass or charge. It fails because a second-order tensor depends on two directions simultaneously, and reducing it to a scalar is only valid in particular, highly symmetric cases. Detect it by checking whether the result depends on the chosen direction when applying the tensor. Correct it by always applying the full matrix-vector product unless the symmetric case has been explicitly verified. Mini-example: the "moment of inertia" of an asymmetric body is not a single number, but a tensor with three distinct principal moments.

Error 2: confusing a tensor's transformation with a vector's

It seems reasonable to apply the same rule v' = Rv when changing basis for both vectors and tensors. It fails because a second-order tensor transforms as T' = R T R^T, with two multiplications by the rotation matrix, not one. Detect it by checking whether the applied transformation preserves the symmetry of the original tensor. Correct it by always applying the double rule for second-order tensors. Mini-example: rotating from only one side (T' = RT) generally does not give a physically correct, or even symmetric, result.

Error 3: assuming the result is always parallel to the input

It seems reasonable to expect that, when applying the tensor to a vector, the result points in the same direction. It fails in general, unless the input vector coincides with a principal axis of the tensor. Detect it by explicitly comparing the direction of the resulting vector with that of the input vector. Correct it by remembering that parallelism is the exception, not the rule, for a tensor not proportional to the identity. Mini-example: the angular velocity and angular momentum of an asymmetric body rotating off a principal axis are not parallel.

Error 4: ignoring the combined units of a physical tensor

It seems reasonable to assign generic units to a tensor without checking that they correctly combine the two quantities it relates. It fails because the inertia tensor, for example, has units of mass times length squared, not just mass, and the stress tensor has units of force per area, not force. Detect it by checking that the units of the matrix-vector product result match those expected for the output physical quantity. Correct it by reviewing the units of each tensor component before operating. Mini-example: multiplying the inertia tensor by an angular velocity in rad/s must give units of angular momentum, kg·m²/s.

Error 5: neglecting the tensor's symmetry when building its matrix

It seems reasonable to write any set of components in a square matrix and call it a tensor. It fails because many relevant physical tensors (inertia, stress, strain) are symmetric by physical construction, and an asymmetric matrix would indicate an error in the setup or an additional physical phenomenon not considered, such as internal torques. Detect it by comparing T_ij with T_ji for each pair of indices. Correct it by reviewing the physical definition of the tensor if an unexpected asymmetry appears. Mini-example: an asymmetric stress tensor in a material in static equilibrium with no internal torques indicates a calculation error, not a real physical phenomenon.

Quick detection rule

If the result of applying a tensor turns out to always be parallel to the input without having verified it, if a vector's transformation rule was used for a tensor, or if the units of the result do not match the expected physical quantity, the error is probably in the tensor setup, not in the final arithmetic.

Physical applications

1. Rigid-body rotation

It appears in the inertia tensor, which relates angular velocity to angular momentum. It is calculated by applying the tensor to a specific angular velocity vector. The result predicts whether a body rotates stably or exhibits precession. The dangerous error is assuming that angular momentum is always parallel to angular velocity. As a final control, the learner must explain why rotating about a non-principal axis produces an angular momentum not aligned with the rotation axis.

2. Elasticity and internal stresses

It appears in the stress tensor, which relates the orientation of an internal surface to the force per unit area acting on it. It is calculated by applying the tensor to a vector normal to the surface of interest. The result predicts whether a material will fail and in which direction. The dangerous error is confusing normal stress with shear stress by not correctly decomposing the resulting vector. As a final control, the learner must explain what the tangential component of the resulting force represents physically.

3. Material deformation

It appears in the strain tensor, which describes how a material stretches, compresses, or distorts under load. It is calculated by relating the tensor to the relative displacements between nearby points in the material. The result predicts changes in shape and volume. The dangerous error is confusing strain (dimensionless) with displacement (with units of length). As a final control, the learner must explain the physical difference between the two concepts.

4. Anisotropic media in optics and electromagnetism

It appears in the electric permittivity tensor or the conductivity tensor, which relate the applied electric field to the resulting polarization or current in materials that do not respond the same way in every direction. It is calculated by applying the tensor to the field vector. The result predicts phenomena such as birefringence in crystals. The dangerous error is using a scalar constant when the material is clearly anisotropic. As a final control, the learner must explain why light can travel at different speeds depending on its polarization in an anisotropic crystal.

5. General relativity

It appears in the metric tensor, which describes the geometry of spacetime, and in the stress-energy tensor, which describes how matter and energy curve that geometry. It is calculated by relating both tensors through Einstein's field equations. The result predicts how spacetime curves in the presence of mass or energy. The dangerous error is treating gravity as a simple vector force instead of as a manifestation of the geometry described by tensors. As a final control, the learner must explain why the trajectory of a freely falling object depends on the tensorial structure of spacetime, not on a directly applied force.

Practice

Level 1: recognition

Identify, for three given physical quantities (pressure in a fluid, inertia tensor, stress tensor), whether they are described by a scalar, a vector, or a second-order tensor, and justify the choice.

Exercise: explain why the moment of inertia of an asymmetric body cannot be described by a single number.

Level 2: direct application

Apply a simple diagonal tensor (expressed along its principal axes) to a given vector and calculate the resulting vector component by component.

Exercise: given a diagonal inertia tensor and an angular velocity with two nonzero components, calculate the resulting angular momentum.

Level 3: choosing the tool

Decide whether the problem requires applying the tensor directly to a vector, finding its principal axes, or transforming it to another basis, and justify the choice based on the question asked.

Exercise: given a stress tensor and a surface with a specific orientation, decide what operation you need to find the force per unit area on that surface.

Level 4: physical interpretation

Interpret whether the result of applying a tensor to a vector is parallel to the input vector or not, and explain what that means physically in each case.

Exercise: given a non-diagonal inertia tensor and an arbitrary angular velocity, explain why the resulting angular momentum does not point in the same direction as the angular velocity.

Level 5: intentional error

Analyze a calculation where a tensor was treated as a simple scalar, or where a vector's transformation rule was applied instead of a tensor's. Detect the error, correct it, and explain the physical consequence.

Exercise: invent a plausible error when transforming a stress tensor to a new basis, show why it seems reasonable, correct it, and state the incorrect result it would have produced.