Tensor algebra and curvilinear coordinates
A tensor generalises vectors and matrices to an object with any number of indices that transforms in a fixed, predictable way under a change of coordinates; curvilinear systems like polar or spherical coordinates need this machinery because their basis vectors change direction from point to point, so gradient, divergence and curl each pick up extra terms that vanish in flat Cartesian coordinates.
Practise this conceptA vector is a tensor with one index, a scalar is a tensor with zero indices, and a matrix that represents a genuine linear map is a tensor with two — the pattern generalises to any number of indices. What actually earns something the name "tensor" is not how many indices it carries but how its components change when you switch coordinate systems: a tensor's components transform by a specific, predictable rule (contracting with the Jacobian of the coordinate change, once per index) that keeps the underlying geometric object — a length, a flux, a stress — the same regardless of which coordinates you used to describe it.
Curvilinear coordinates make this machinery unavoidable rather than optional. In Cartesian coordinates the basis vectors point the same way everywhere, so the usual formulas for gradient, divergence and curl need no correction. In polar or spherical coordinates the basis vectors themselves rotate as you move — the "radial" direction at one point is not parallel to the "radial" direction at another — so distances, areas and derivative operators all pick up extra factors that account for that rotation. The metric tensor is exactly the bookkeeping device that captures this: in polar coordinates, ds² = dr² + r²dθ² says a small step in the angular direction covers more actual ground the farther out from the origin you are.
Raising and lowering indices — turning a "covariant" component into a "contravariant" one and back — is a real operation performed by contracting with the metric tensor, not a notational nicety. In an orthonormal Cartesian basis the metric is just the identity, so the distinction is invisible and both kinds of component happen to have the same numerical value; the moment the basis is curvilinear (or genuinely curved), raising and lowering an index changes the number, and treating the two as interchangeable produces a wrong answer.
Key ideas
Requires: Vectors in space, The dot product, The cross product, Vector fields and the gradient, Divergence and curl
Formulas
| Symbol | Name | Unit |
|---|---|---|
| radius | 1 (dimensionless) | |
| displacement | 1 (dimensionless) | |
| angle | 1 (dimensionless) | |
| length | 1 (dimensionless) |
Common misconceptions
- Believes a tensor is just a matrix with a fancier name, so any rectangular array of numbers qualifies — missing that what makes something a tensor is its transformation law under a change of coordinates, not its shape.
- Thinks the metric tensor is only needed in curved (non-Euclidean) space, missing that flat space already needs it the moment the coordinates are curvilinear — the polar metric ds² = dr² + r²dθ² carries a nontrivial metric even though the underlying plane is perfectly flat.
- Believes raising or lowering an index is a cosmetic notational choice with no numerical consequence, rather than an operation performed by contracting with the metric tensor that genuinely changes the components' values whenever the basis isn't orthonormal.
Resources
Part of these UT courses
This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.
- Mathematical physicsLTFY.04.013