# Unit 1: Thinking about multivariable functions

## 1. What are multivariable functions?

• A function is called multivariable if its input is made up of multiple numbers. $\underbrace{f(x, y)}_{\text{Multiple numbers in the input} } = x^2 y$
• If the output of a function consists of multiple numbers are called vector-valued functions. $f(x) = \begin{bmatrix} \cos(x) \\ \sin(x) \end{bmatrix} \gets \text{Multiple numbers in output}$

### 1.1. Parametric Equation

• A parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface.
• For example: $f(x) = \begin{bmatrix} t \cos(t) \\ t \sin(t)\end{bmatrix}$

## 2. Five different visualizations

• Graphs: single-variable functions and multivariable functions with a two-dimensional input and a one-dimensional output.
• Contour maps. Contour maps only show the input space and are useful for functions with a two-dimensional input and a one-dimensional output.
• Parametric curves/surfaces. Parametric curves and surfaces only show the output space and are used for functions whose output space has more dimensions than the input space.
• Vector fields. These apply to functions whose input space and output space have the same number of dimensions.
• Transformations. These have the benefit of applying to any function, no matter the dimension of the input and output space. However, the downside is that they are impractical for representing the function precisely.