Equations

You will also need to write mathematical expressions on your exercise sheets. Please be precise when formulating equations and avoid possible ambiguity. In particular, it is important to recognize which mathematical objects are hidden behind a symbol.

Vectors and Tensors

The symbol \(x\) denotes a scalar here, unless you explicitly tell us that it is a vector, \(x\in\mathbb{R}^3\). In engineering and natural sciences, however, it is more common to indicate the type of object by the symbol. Vectors, for example, are represented by arrows, \(\vec{x}\). An alternative is bold symbols, \(\mathbf{x}\). Components of a vector, \(x_\alpha\), do not have an arrow because these are numbers (scalars).

The learning material uses arrows \(\vec{x}\) for vectors and underlines \(\underline{K}\) for matrices. If you deviate from this notation, you must explain your notation in the exercise sheets. We use arrows and underscores because, unlike bold symbols, they can be easily realized when writing on a blackboard.

Inner and outer products

An inner product (scalar product) is expressed by a dot \(\cdot\), \(\vec{a}\cdot\vec{b}\). The expression \(\vec{a}\vec{b}\) is not a scalar product but the outer product. We recommend using a specific symbol for the outer product to avoid ambiguity, e.g. \(\vec{a}\otimes\vec{b}\). Each dot (in the scalar product or the double contraction \(\underline{A}:\underline{B}\)) represents a sum.