Leon Bilton

# Notation

## Arbitrary quantities

A set will be represented by an uppercase letter without an index, e.g. $$X$$.

A sequence will be enclosed in parentheses to distinguish it from tensor components. The notation will often be further abbreviated as $$(x_{n}) ≡ (x_{n})_{n∈ℕ} ≡ (x_{1}, x_{2}, …)$$.

A 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. $$C^{i}$$. 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.

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

However, the summation should be explicitly clarified in case of potential ambiguity, e.g. $$\sqrt{\sum_{i}x_{i}x^{i}} ≠ \sum_{i}\sqrt{x_{i}x^{i}}$$.

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 are also defined according to convention:

$$ε \quad$$ (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 (assuming that the matrix is used to represent a tensor). Explicit use of the metric in the tensor notation will often be omitted, as shown by the equivalences in the table, when it is implied that all operations are being performed within Euclidean space.

OperationMatrix notationTensor notation
Inner (scalar) product$$a = \bm{u} ⋅ \bm{v}$$$$a = g_{ij}u^{i}v^{j} ≡ u_{i}v^{j}$$
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$$\bm{u} = \bm{A}\bm{v}$$$$u^{i} = g_{jk}A^{ik}v^{j} ≡ A^{i}_{\phantom{i}j}v^{j}$$
Matrix-matrix product$$\bm{C} = \bm{A}\bm{B}$$$$C^{ij} = g_{kl}A^{il}B^{kj} ≡ A^{i}_{\phantom{i}k}B^{kj}$$
Outer (tensor) product$$\bm{A} = \bm{u} ⊗ \bm{v}$$$$A^{ij} = u^{i}v^{j}$$
Trace of a square matrix$$a = \mathrm{trace\,}A$$$$a = g_{ij}A^{ij} ≡ A^{i}_{\phantom{i}i}$$

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 $$\bm{F} ≔ [f]_{\bm{x}}$$.

QuantityGeneral notationComponent notation
Gradient (spatial)$$[∇\bm{F}] ≡ [∇f]_{\bm{x}}$$$$∇^{i}F^{jk}$$
Jacobian$$[J\bm{F}]$$$$J(F)^{ij}$$
Hessian$$[H\bm{F}]$$$$H(F)^{ij}$$
Divergence (spatial)$$[∇ ⋅ \bm{F}] ≡ [∇ ⋅ f]_{\bm{x}}$$$$∇_{i}F^{ij}$$
Curl (spatial, $$ℝ^{3}$$)$$[∇ × \bm{F}] ≡ [∇ × f]_{\bm{x}}$$$$ε^{ijk}∇_{i}F_{lj}$$
Derivative (with respect to $$ξ$$)$$∂_{ξ}\bm{F} ≡ [∂_{ξ}f]_{\bm{x}}$$$$∂_{ξ}(F)^{ij}$$
Derivative (temporal)$$\dot{\bm{F}} ≡ [∂_{t}f]_{\bm{x}}$$$$\dot{F}^{ij}$$
Material derivative$$[D\bm{F}] ≔ \dot{\bm{F}} + \begin{bmatrix} ∂_{t}\bm{x} \cdot \bm{\hat{e}}_{x_{1}}\\ ⋮\\ ∂_{t}\bm{x} \cdot \bm{\hat{e}}_{x_{n}}\end{bmatrix} \cdot ∇\bm{F}$$$$D(F)^{ij}$$

  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.