Tensor Calculus

Prerequisites: Linear Algebra, Vector Calculus. Full path (3 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

A tensor is a relation between one vector and another. If you start with one vector, such as a force, and mathematically apply it to a tensor, then you get another vector. That vector might be, for example, the stress caused by the force.

That’s the simplest thing they do. Tensors do other things too; for example, the metric tensor represent the geometry of space. A tensor can represent the energy-momentum density. A tensor can represent a combination of electric and magnetic fields in a way that some of Maxwell’s equations greatly simplify. But the simplest and most basic connection is the vector to vector one.

A function relates one number (a “scalar”) to another one. A tensor does that for vectors.http://qr.ae/TUTNzc

Concrete

In tensor calculus a tensor is described by its components in a coordinate system $x^\mu$, together with the rule for how these components change when we switch to new coordinates $x'^\mu(x)$. We use the Einstein convention: an index that appears once up and once down is summed over.

Transformation laws

A contravariant vector (upper index) transforms like the coordinate differentials $dx^\mu$, a covariant vector (lower index) like the gradient $\partial_\mu f$:

$$V'^\mu = \frac{\partial x'^\mu}{\partial x^\nu} V^\nu , \qquad W'_\mu = \frac{\partial x^\nu}{\partial x'^\mu} W_\nu .$$

A tensor of type $(r,s)$ has $r$ upper and $s$ lower indices and picks up one such factor per index, for example

$$T'^{\mu}{}_{\nu} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x^\beta}{\partial x'^\nu} T^{\alpha}{}_{\beta} .$$

The point of this rule is that an equation of the form $A^{\mu}{}_{\nu} = B^{\mu}{}_{\nu}$ that holds in one coordinate system holds in all of them. Writing physical laws as tensor equations guarantees that they do not depend on the choice of coordinates.

Algebraic operations

• Products: $V^\mu W_\nu$ is a tensor of type $(1,1)$.
• Contraction: summing an upper with a lower index gives a tensor with two fewer indices. $V^\mu W_\mu$ is a scalar, the same number in every coordinate system, because the two Jacobian factors are inverse to each other.
• The metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$ raise and lower indices: $V_\mu = g_{\mu\nu} V^\nu$. Lengths and angles come from $g_{\mu\nu} V^\mu V^\nu$. See also covariant and contravariant coordinates.

Covariant derivative

The partial derivative of a vector field is not a tensor. Differentiating the transformation law gives

$$\partial'_\nu V'^\mu = \frac{\partial x^\beta}{\partial x'^\nu}\frac{\partial x'^\mu}{\partial x^\alpha}\, \partial_\beta V^\alpha + \frac{\partial x^\beta}{\partial x'^\nu}\frac{\partial^2 x'^\mu}{\partial x^\beta \partial x^\alpha} V^\alpha ,$$

and the second term spoils the tensor law whenever the transformation is not linear. The fix is the covariant derivative

$$\nabla_\nu V^\mu = \partial_\nu V^\mu + \Gamma^\mu{}_{\nu\lambda} V^\lambda , \qquad \nabla_\nu W_\mu = \partial_\nu W_\mu - \Gamma^\lambda{}_{\nu\mu} W_\lambda ,$$

where the connection coefficients $\Gamma$ transform in exactly the way that cancels the extra term. With a metric, the standard choice is the Christoffel symbols

$$\Gamma^\lambda{}_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma}\left( \partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu} \right) .$$

Example: in the plane with polar coordinates, $ds^2 = dr^2 + r^2 d\varphi^2$, the nonzero Christoffel symbols are $\Gamma^r{}_{\varphi\varphi} = -r$ and $\Gamma^\varphi{}_{r\varphi} = \Gamma^\varphi{}_{\varphi r} = 1/r$. They are nonzero even though the plane is flat, because the coordinate basis vectors change from point to point.

In physics

The stress tensor $\sigma_{ij}$ of a material gives the force per area $t_i = \sigma_{ij} n_j$ acting on a surface with unit normal $n_j$. The electromagnetic field strength $F_{\mu\nu}$ combines $\vec E$ and $\vec B$, and the metric $g_{\mu\nu}$, the Riemann tensor and the energy-momentum tensor $T_{\mu\nu}$ are the ingredients of general relativity.

Recommended Books

• A Student's Guide to Vectors and Tensors by Fleisch - a nice student-friendly introduction
• Introduction to Tensor Calculus by Kees Dullemond & Kasper Peeters (free)
• A Brief on Tensor Analysis by James Simmonds - a concise but great introductory text.

Abstract

Let $M$ be a smooth manifold with tangent spaces $T_pM$ and cotangent spaces $T_p^*M$. A tensor of type $(r,s)$ at $p$ is an element of

$$T^{(r,s)}_p M = \underbrace{T_pM \otimes \dots \otimes T_pM}_{r} \otimes \underbrace{T_p^*M \otimes \dots \otimes T_p^*M}_{s} ,$$

equivalently (in finite dimensions) a multilinear map $(T_p^*M)^r \times (T_pM)^s \to \mathbb{R}$. A tensor field is a smooth section of the vector bundle $T^{(r,s)}M$. The components of the Concrete layer are the coefficients in the basis $\partial_{\mu_1} \otimes \dots \otimes dx^{\nu_s}$, and the transformation law is the change of basis induced by a change of chart. Nothing in the definition refers to coordinates; the transformation law is a consequence.

A useful criterion: a map that takes $s$ vector fields and $r$ one-forms to a smooth function is a tensor field if and only if it is multilinear over $C^\infty(M)$, that is $T(\dots, fX, \dots) = f\,T(\dots, X, \dots)$ for all functions $f$. This is how one checks that, for example, the torsion and the curvature of a connection are tensors while the connection itself is not.

Which operations need extra structure

• Tensor product and contraction are defined on any manifold.
• The Lie derivative $\mathcal{L}_X T$ along a vector field is defined on any manifold, using the flow of $X$.
• The exterior derivative $d$ acts on differential forms, the totally antisymmetric covariant tensors, without extra structure.
• Differentiating a general tensor field in a direction requires a connection $\nabla$ on $TM$, which extends uniquely to all $T^{(r,s)}M$ by requiring the Leibniz rule and commutation with contractions.
• Raising and lowering indices requires a metric $g$. On a (pseudo-)Riemannian manifold there is a unique connection that is torsion-free and satisfies $\nabla g = 0$, the Levi-Civita connection, whose coefficients in a chart are the Christoffel symbols.

Curvature

The curvature tensor of a connection is defined by $R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$. For a torsion-free connection, in components and with the convention $R(\partial_\mu, \partial_\nu)\partial_\sigma = R^\rho{}_{\sigma\mu\nu}\partial_\rho$,

$$[\nabla_\mu, \nabla_\nu] V^\rho = R^\rho{}_{\sigma\mu\nu} V^\sigma .$$

Sign and index-order conventions for $R$ differ between textbooks, so they should be checked before comparing formulas. The curvature measures the failure of parallel transport around small loops to return a vector to itself, and it vanishes identically if and only if the connection is locally flat.

Why is it interesting?

Tensor calculus is a technique that can be regarded as a follow-up on linear algebra. It is a generalization of classical linear algebra. In classical linear algebra one deals with vectors and matrices. Tensors are generalizations of vectors and matrices. Introduction to Tensor Calculus by Kees Dullemond & Kasper Peeters

Path to this page

1. Calculus
2. Linear Algebra
3. Vector Calculus

Leads to: Covariant and Contravariant Coordinates.