Leon Bilton

Notation

Last modified on May 17, 2022

This article outlines the mathematical notation conventions adopted throughout the website.

  1. Arbitrary quantities
  2. Conventional quantities and operators
  3. Examples

Arbitrary quantities

An arbitrary set will be represented by an uppercase letter without an index, e.g. \(X\). Arbitrary elements of a set will be denoted as lowercase letters. If the elements require indices, they will be enclosed in braces for distinction from tensor components, e.g. \(\begin{Bmatrix} x_{i}\end{Bmatrix} \in X\).

An arbitrary tensor quantity of rank 1 (vector) or above will be represented by a boldface letter, e.g. \(\bm{V}\).

If a coordinate system is established (or implied), Tensor components will be represented by indices (lowercase letters) attached to a letter, e.g. \(\delta^{i}_{\phantom{i}j}\). Covariant tensor components will be represented by lower indices, e.g. \(g_{ij}\). Contravariant tensor components will be represented by upper indices, e.g. \(C^{i}\).[1] The distinction is irrelevant in the case of an Euclidean metric.

Wherever the same index appears in an upper and lower position in a product, summation over the index is implied, i.e.

\[ a_{i} {b}^{i}_{\phantom{i}j} c^{j} ≡ \sum_{ij} a_{i}b_{ij}c_{j} \]

Matrix or array entries will remain in boldface to distinguish them from more general tensor components. They will always have all indices as subscripts, e.g. \(\bm{A}_{12}\) or \(\bm{A}_{1i}\). Summation over array entries is never implied, and will always be explicitly written.

A hat will be used to denote unit vectors, e.g. \(\bm{\hat{e}}\) with components \(\hat{e}_{i}\).

Conventional quantities and operators

The number sets are defined conventionally as \(\begin{Bmatrix} ℝ,ℕ,ℤ,ℚ,ℂ,𝕀\end{Bmatrix}\), which correspond to real, natural, integer, raitonal, complex and imaginary numbers, respectively. Euclidean charts in \(n\) dimensions are denoted \(ℝ^n\). Their standard basis vectors (which are usually left implicit) are denoted \(\{\bm{\hat{e}}_{x_{i}}\}\).

The uppercase letters \(D\), \(J\), and \(H\) will be reserved for the material derivative, Jacobian and Hessian of a tensor (field). See also the provided examples.

The zero tensor (all components are zero) will be denoted with the boldface \(\bm{0}\), even in component-form equations (the indices are dropped, they have no meaning).

The following symbols refer to the indicated tensors unless otherwise clarified:

\(ε \quad\) (with components \(ε_{ijk}\) in a basis) is the permutation symbol (Levi-Civita symbol),

\(δ \quad\) (with components \(δ_{ij}\) in a basis) is the Kronecker delta (discrete delta function), and

\(g \quad\) (with components \(g_{ij}\) in a basis) is a (positive-definite) metric tensor (field).

Examples

Here are some examples of common operations, shown using both matrix and tensor notation.

OperationMatrix notationTensor notation
Inner (scalar) product\(a = \bm{u} ⋅ \bm{v}\)\(a = g_{ij}u^{i}v^{j}\)
Cross product in \(ℝ^3\)\(\bm{w} = \bm{u} × \bm{v}\)\(w^{i} = ε^{i}_{\phantom{i}jk}u^{j}v^{k}\)
Matrix-vector product\(\bm{u} = \bm{A}\bm{v}\)\(u^{i} = g_{jk}A^{ik}v^{j}\)
Matrix-matrix product\(\bm{C} = \bm{A}\bm{B}\)\(C^{ij} = g_{kl}A^{il}B^{kj}\)
Outer (tensor) product\(\bm{A} = \bm{u} ⊗ \bm{v}\)\(A^{ij} = u^{i}v^{j}\)

Continuum mechanics quantities will be frequently used. These examples show notation conventions for an arbitrary field \(f:ℝ^{n} → ℝ^{m}\) evaluated at position \(\bm{x} ∈ ℝ^{n}\) to give the tensor \(\bm{F} ≔ [f]_{\bm{x}}\).

QuantityMatrix notationTensor notation
Gradient (spatial)\([∇\bm{F}] ≡ [∇f]_{\bm{x}} ≔ \begin{bmatrix} [∂_{x_{1}} \circ f]_{\bm{x}}\\ ⋮\\ [∂_{x_{n}} \circ f]_{\bm{x}}\end{bmatrix}\)\(∂^{i}F^{jk} ≔ [g^{ij}∂_{j}f]_{\bm{x}}\)
Jacobian\([J\bm{F}] ≔ \begin{bmatrix} [∂_{x_{1}} \circ f]_{\bm{x}} & ⋯ & [∂_{x_{n}} \circ f]_{\bm{x}}\end{bmatrix}\)\(J(F)^{ij}\)
Hessian\([H\bm{F}] ≔ \begin{bmatrix} [∂^{2}_{x_{1}} \circ f]_{\bm{x}} & ⋯ & [∂^{2}_{x_{n}} \circ f]_{\bm{x}}\end{bmatrix}\)\(H(F)^{ij}\)
Divergence (spatial)\([∇ ⋅ \bm{F}] ≡ [∇ ⋅ f]_{\bm{x}}\)\(∇_{i}F^{ij}\)
Curl (spatial)\([∇ × \bm{F}] ≡ [∇ × f]_{\bm{x}}\)\(ε^{ijk}∂_{i}F_{lj}\)
Derivative (temporal)\(\dot{\bm{F}} ≡ [∂_{t}f]_{\bm{x}} ≔ \begin{bmatrix} [∂_{t} \circ f]_{\bm{x}} \cdot \bm{\hat{e}}_{x_{1}}\\ ⋮\\ [∂_{t} \circ f]_{\bm{x}} \cdot \bm{\hat{e}}_{x_{n}}\end{bmatrix}\)\(\dot{F}^{ij}\)
Material derivative\([D\bm{F}] ≔ \dot{\bm{F}} + \begin{bmatrix} d_{t}\bm{x} \cdot \bm{\hat{e}}_{x_{1}}\\ ⋮\\ d_{t}\bm{x} \cdot \bm{\hat{e}}_{x_{n}}\end{bmatrix} \cdot ∇\bm{F}\)\(D(F)^{ij}\)
Derivative (with respect to \(ξ\))\(d_{ξ}\bm{F} ≡ [∂_{ξ}f]_{\bm{x}}\)\(d_{ξ}(F)^{ij}\)

[1] Unfortunately, I have had to resort to tedious manual usage of \phantom to get index ordering to work. No \(\LaTeX\) web engine that I am aware of can render tensor notation properly. MathML, the new XML-based (yuck) mathematics markup specification for the web does have a multiscripts definition, but web browser support is predictably poor. I may eventually switch to that if it becomes standard.