Equations

You will regularly write down mathematical expressions for solving the problems we assign to you and for passing the exam. Please pay attention to precision when formulating equations and avoid possible ambiguity. In particular, it is important to identify which mathematical objects are hidden behind a symbol.

Vectors and tensors

The symbol \(x\) here denotes a scalar, unless you explicitly tell us that it is a vector, for example, \(x\in\mathbb{R}^3\). In engineering and the natural sciences, however, it is more common to indicate what kind of object we are dealing with by the type of symbol. Vectors, for example, are represented by arrows, \(\vec{x}\). Bold symbols, \(\mathbf{x}\), are an alternative. Components of a vector, \(x_\alpha\), have no arrow because these are numbers (scalars).

The lecture material uses arrows \(\vec{x}\) for vectors and underscores \(\underline{K}\) for matrices. If you deviate from this notation, you will need to explain your notation in the exercise sheets or your solution to an exam question. We use arrows and underscores because, unlike bold symbols, they can be easily implemented in the blackboard notation.

Inner and outer products

An inner product (scalar product) is expressed by a point \(\cdot\), \(\vec{a}\cdot\vec{b}\). 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 point (in the scalar product or the double contraction \(\underline{A}:\underline{B}\)) represents a sum.