Introduction to Probability

Multivariable Calculus

Algorithms: Part II

Algorithms: Part I

Introduction to Software Design and Architecture

Calculus Two: Sequences and Series

LAFF Linear Algebra

Stanford Machine Learning

Calculus One

Computational Thinking

Effective Thinking Through Mathematics

CS50 Introduction to Computer Science


Unit 1: Thinking about multivariable functions

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).

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)=[tcos(t)tsin(t)]f(x) = \begin{bmatrix} t \cos(t) \\ t \sin(t)\end{bmatrix}

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)=[cos(t)3+12πttsin(t)],t[0,12π]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.


  • 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