Dave4Math / Courses / Functions of Several Variables

Functions of Several Variables

Master functions of several variables. In this course, you'll learn about multivariable limits, partial derivatives, and the extrema of multiple variable functions.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Suddenly you’re asking yourself; maybe I should take another math course? Introducing Dave’s online math course: Functions of Several Variables. Many students ask themselves this question after taking a course with Dave. What is else is better than feeling confident in something that used to make you so frustrated?

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in functions of several variables.
  • Students who are enrolled in calculus 3 and want to improve their grade.
  • Anyone interested in getting prepared for linear algebra.
  • Anyone wanting to learn about limits and continuity of multivariable functions, partial derivatives, directional derivatives, tangent planes, extrema of functions, and Lagrange multipliers.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Functions of Several Variables

  • Functions of Two Variables
  • Graphs of Functions of Two Variables
  • Computer Graphics
  • Level Curves
  • Functions of Three Variables
  • Level Surfaces
  • An Intuitive Definition of a Limit
  • Continuity of a Function of Two Variables
  • Continuity on a Set
  • Functions of Three or More Variables
  • A Formal Definition of a Limit
  • Partial Derivatives with Two Variables
  • Computing Partial Derivatives
  • Implicit Differentiation
  • Partial Derivatives with More Variables
  • Higher-Order Derivatives
  • Increments
  • The Total Differential
  • Error in Approximating
  • Differentiability with Two Variables
  • Differentiability and Continuity
  • Functions of Three or More Variables
  • The Chain Rule for Functions Involving One Independent Variable
  • The Chain Rule for Functions Involving Two Independent Variables
  • The General Chain Rule
  • Implicit Differentiation
  • The Gradient of a Function of Two Variables
  • Properties of the Gradient
  • Geometric Interpretation of the Gradient
  • Tangent Planes and Normal Lines
  • Using the Tangent Plane to Approximate a Surface
  • Relative and Absolute Extrema
  • Critical Points: Candidates for Relative Extrema
  • The Second Derivative Test for Relative Extrema
  • Finding the Absolute Extremum Values of a Continuous Function on a Closed Set
  • Constrained Maxima and Minima
  • The Method of Lagrange Multipliers
  • Optimizing a Function Subject to Two Constraints

Course Description

In this exciting course, we study functions of several variables. For most students taking a calculus course, this is their first look at functions of more than one variable. So before we begin with the calculus, we explain what multivariable functions are and how to graph them using level curves, level surfaces, and computer graphics. After that, we introduce calculus. We give a thorough introduction to the limits of multivariable functions and discuss continuity as well.

One of the most exciting parts of calculus is learning partial derivatives. We illustrate what partial derivatives are both formally and graphically. We then work through many examples, including implicit differentiation, higher-order partial derivatives, and Clairaut’s Theorem. After that, we examine differentials and the total derivative. These topics lead us to understand what differentiability means for multivariable functions; in particular, we discuss tangent planes.

Now enters the chain rule for multivariable functions. We work through different stages and go through several examples demonstrating how the chain rule works, including the Implicit Differentiation Theorem. Next, partial derivatives are generalized by studying the directional derivative and gradient vectors. This study includes a discussion using the tangent plane to approximate a surface locally.

Towards the end of the course, we explore some applications, including finding the extrema of multivariable functions and using Lagrange multipliers. In both cases, we work through many examples and provide formal verification of the theorems.

Recommended Prerequisites for Functions of Several Variables

I recommend the prerequisite course Vectors and the Geometry of Space. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 3 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.