# 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. $\underbrace{f(x, y)}_{\text{Multiple numbers in the input} } = x^2 y$ - A function with single-number inputs and a single-number outputs is called a
**single-variable function**. - If the output of a function consists of multiple numbers are called
**vector-valued functions**. $f(x) = \begin{bmatrix} \cos(x) \\ \sin(x) \end{bmatrix} \gets \text{Multiple numbers in output}$ - functions with a single number as their output are called either
**scalar-valued**, as is common in engineering, or**real-valued**, as is common is pure math (real as in real number).

### 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