Leon Bilton

# Notation

## 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. $$C^{i}$$. 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 $$ξ$$)$$d_{ξ}\bm{F} ≡ [∂_{ξ}f]_{\bm{x}}$$$$d_{ξ}(F)^{ij}$$

  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.