Notation

This module will involve quite a lot of reading through equations and implementing them. Thus, it is important that we clearly define the notation we will be using throughout this module.

For denoting scalar value variables, we will use lowercase letters such as $x$ and $y$. When variables refer to vectors, we will use lowercase bold letters, such as $\mathbf{x}$ and $\mathbf{y}$.

For denoting scalar valued functions (i.e. functions whose values are just scalars), we will use lowercase letters, such as $f$, $g$ and $h$, followed by their their arguments enclosed between parenthesis. For instance, $f(x)$ is a scalar valued function of a single variable x, which is also a scalar.

When evaluating a function $f(x)$ at the point where $x=a$, we will use the notation $f(x)|_{x=a}$. For instance, the derivative of a function $f(x)$ evaluated at $x=0$ will be denoted as $f'(x)|_{x=0}$

In the first sections of this module, we will use $f'(x)$ for denoting the derivative of a scalar valued function $f$.

In subsequent sections, we will use Leibniz's notation, whereby the first derivative of $f$ with respect to $x$ is denoted by $\frac{df}{dx}$ or $\frac{d}{dx}f$. The $n$-th order derivative of $f$ with respect to $x$ is denoted by $\frac{d^nf}{dx^n}$