Multivariable Calculus

Calculus of many variables, from vectors to volume.

Back

Overview

Change is an essential part of our world, and calculus helps us quantify it. The change that most interests us happens in systems with more than one variable: weather depends on time of year and location on the Earth, economies have several sectors, important chemical reactions have many reactants and products.

Multivariable calculus continues the story of calculus. Learn how tools like the derivative and integral generalize to functions depending on several independent variables, and discover some of the exciting new realms in physics and pure mathematics they unlock.

Topics covered

• Chain Rule
• Contour Maps & Level Sets
• Extreme Value Theorem
• Hessian Test for Critical Points
• Jacobians & Variable Changes
• Lagrange Multipliers
• Limits and Continuity
• Multiple Integrals
• Multivariable Functions & Graphs
• Partial Derivatives
• Vectors and Matrices

44

475+

1. 1

Introduction

Double the variables, double the fun.

1. Many Variables in a Nutshell

Take a lightning tour of calculus with several variables.

1
2. Finding Extrema

Learn how partial derivatives can solve important real-world problems.

2
3. Coordinates in 3D

Explore new ways to navigate in three dimensions.

3
4. 3D Volumes

Bridge the gap between geometry and multiple integrals with Riemann sums.

4
2. 2

Vector Bootcamp

Master vectors, the basic building blocks of multivariable calculus.

1. Vector Arithmetic

Work hands-on with vectors, the building blocks of multivariable calculus.

5
2. Vector Properties

Continue to build your vector intuition by approaching it geometrically!

6
3. Equations of Lines

Apply your vector knowledge to the motion of heavenly bodies.

7
4. Dot Product Definition and Properties

Use geometry to make the dot product, an essential way of multiplying vectors.

8
5. Matrices

Transform vectors with matrices and find out what they have in common.

9
6. Determinants

Is it ever OK to divide by a matrix?

10
7. The Cross Product

Apply the determinant to find a second vector multiplication rule.

11
3. 3

Multivariable Functions

Take the first step into higher dimensions.

1. Multivariable Functions

Explore functions of several variables and discover what they're good for.

12
2. Function Domains

Connect multivariable functions with set geometry.

13
3. Basic Graphing

Learn to capture the most important qualities of a function with a 3D picture.

14
4. Graphs by Slices

Develop skills to visualize the shape of a function and to think in higher dimensions.

15
5. Contour Maps

What do graphing and mountain climbing have in common?

16
6. Level Sets

Find out how to compress a complicated function down into a handy 2D map.

17
4. 4

Limits with Many Variables

Uncover unexpected function properties with limits.

1. Searching Square Lake

Begin to uncover the mysteries of limits with the search for a mythical beast.

18
2. Multivariable Limits

Connect limits with many variables to limits with just one.

19
3. Shock Waves and Discontinuities

Learn to visualize limits and apply them to the real world.

20
4. Extreme Value Theorem (Part I)

Get a bird's-eye view of a crucial calculus theorem.

21
5. Extreme Value Theorem (Part II)

Apply limits like a mathematician and prove the extreme value theorem.

22
5. 5

Derivatives

Measuring rates of change is just the beginning...

1. Basic Partial Derivatives

Master the mechanics of multivariable rates of change.

23
2. Higher-Order Partials

Learn about the uses of a derivative's derivative, like the wave equation.

24
3. Under the Microscope: Tangent Planes

Zoom in on a function's graph and see its tangent planes up close.

25
4. Directional Derivatives

Dive beneath Square Lake to develop directional rates of change.

26

Build the gradient, the source for everything there's to know about how quickly a function changes.

27
6. Chain Rule of Several Variables

Find out what the gradient looks like in different coordinate systems.

28
6. 6

Optimization

Put derivatives to work finding and classifying extreme values.

1. Local Maxima and Minima

Use gradient geometry to find the highs and lows of a graph.

29
2. Back Under the Microscope: Quadrics

Dust off your function microscope and see the basic shape of a graph near a critical point.

30
3. Back Under the Microscope: Hessian Test

Use the microscope to come up with a simple test to classify local extrema.

31
4. Boundary Extrema

Discover how functions can achieve extreme values on exotic shapes.

32
5. Method of Lagrange

Develop a simple means for finding constrained extrema using gradient geometry.

33
6. Application: Lagrange Multipliers

Apply Lagrange's Method to a fun real-world example.

34
7. Global Maxima and Minima

Practice all of your extrema-hunting strategies here.

35
8. More Hessians! (Optional)

Extremize functions of more than two variables with linear algebra.

36
7. 7

Multiple Integrals

Become a master of multivariable integration.

1. Double Integrals (Part I)

Gain double integral intuition through Riemann sums.

37
2. Double Integrals (Part II)

Evaluate simple double integrals with geometric reasoning.

38
3. Fubini's Theorem (Part I)

Break difficult double integrals down into bite-sized pieces.

39
4. Fubini's Theorem (Part II)

Master the art of integral domain slicing.

40
5. Multiple Integrals

What does it mean to integrate a function with more than two variables?

41
6. Multiple Integrals Applications

Discover why multiple integrals are so useful.

42
7. Change of Variables

Reshape a multiple integral into something easier through coordinate transformations.

43
8. Cylindrical & Spherical Integrals

Practice on real-world applications with symmetry.

44