# 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.
- Finding a parametric function that describes a curve is called
**parameterizing** that curve.
**Note:** When you are parameterizing a curve, you must not only specify the parametric function, but also the range of input values that will draw the curve.
- 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.
- For example: $f(x) = \begin{bmatrix} \cos(t) - 3 + \frac{1}{2\pi} t\\ t \sin(t)\end{bmatrix}, t \in [0, 12 \pi]$

- 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.
- https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/ways-to-represent-multivariable-functions/a/visualizing-multivariable-functions

## 3. Words

**parametric** [,pærə'metrik] adj. [数][物] 参数的；[数][物] 参量的
**contour** ['kɔntuə] n. 轮廓；等高线；周线；电路；概要 vt. 画轮廓；画等高线