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

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 \(\hat{e}_{i}\).

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).

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

Operation | Matrix notation | Tensor 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 \(\bm{F} ≔ [f]_{\bm{x}}\).

Quantity | Matrix notation | Tensor 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. |

© Leon Bilton. Last modified: May 17, 2022. Website built with Franklin.jl and the Julia programming language.