# Notation

Two types of notation are used:
Indicial notation
Equations of continuum mechanics are usually written in this form.
Matrix notation
Used for equations pertinent to the finite element implementation.

## Index Notation

Components of tensors and matrices are given explicitly. A vector, which is a first order tensor, is denoted in indicial notation by ${x}_{i}$. The range of the index is the dimension of the vector.

To avoid confusion with nodal values, coordinates will be written as $x$ , $y$ or $z$ rather than using subscripts. Similarly, for a vector such as the velocity ${v}_{i}$ , numerical subscripts are avoided so as to avoid confusion with node numbers. So, ${x}_{1}=x,\text{ }{x}_{2}=y,\text{ }{x}_{3}=z$ and ${v}_{1}={v}_{x}\text{\hspace{0.17em}},\text{ }{v}_{2}={v}_{y}\text{\hspace{0.17em}}$ and ${v}_{3}={v}_{2}$ .

Indices repeated twice in a list are summed. Indices which refer to components of tensors are always written in lower case. Nodal indices are always indicated by upper case Latin letters. For instance, ${v}_{iI}$ is the i-component of the velocity vector at node I. Upper case indices repeated twice are summed over their range.

A second order tensor is indicated by two subscripts. For example, ${E}_{ij}$ is a second order tensor whose components are ${E}_{xx},{E}_{xy}$ , ...

## Matrix Notation

Matrix notation is used in the implementation of finite element models. For instance, equation(1)
${r}^{2}={x}_{i}\cdot {x}_{i}={x}_{1}\cdot {x}_{1}+{x}_{2}\cdot {x}_{2}+{x}_{3}\cdot {x}_{3}$
is written in matrix notation as:(2)
${r}^{2}={x}^{T}x$

All vectors such as the velocity vector $\nu$ will be denoted by lower case letters. Rectangular matrices will be denoted by upper case letters.