Multivariable Calculus

Calculus of many variables, from vectors to volume.

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.

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1. 1

Introduction

Double the variables, double the fun.

1. Many Variables in a Nutshell

Take a lightning tour of calculus with several variables.

2. Finding Extrema

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

3. Coordinates in 3D

Explore new ways to navigate in three dimensions.

4. 3D Volumes

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

2. 2

Vector Bootcamp

Master vectors, the basic building blocks of multivariable calculus.

1. Included with

Vector Arithmetic

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

2. Included with

Vector Properties

Continue to build your vector intuition by approaching it geometrically!

3. Included with

Equations of Lines

Apply your vector knowledge to the motion of heavenly bodies.

4. Included with

Dot Product Definition and Properties

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

3. 3

Multivariable Functions

Take the first step into higher dimensions.

1. Included with

Multivariable Functions

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

2. Included with

Function Domains

Connect multivariable functions with set geometry.

3. Included with

Basic Graphing

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

4. Included with

Graphs by Slices

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

4. 4

Limits with Many Variables

Uncover unexpected function properties with limits.

1. Included with

Searching Square Lake

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

2. Included with

Multivariable Limits

Connect limits with many variables to limits with just one.

3. Included with

Shock Waves and Discontinuities

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

4. Included with

Extreme Value Theorem (Part I)

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

5. 5

Derivatives

Measuring rates of change is just the beginning...

1. Included with

Basic Partial Derivatives

Master the mechanics of multivariable rates of change.

2. Included with

Higher-Order Partials

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

3. Included with

Under the Microscope: Tangent Planes

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

4. Included with

Directional Derivatives

Dive beneath Square Lake to develop directional rates of change.

6. 6

Optimization

Put derivatives to work finding and classifying extreme values.

1. Included with

Local Maxima and Minima

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

2. Included with

Back Under the Microscope: Quadrics

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

3. Included with

Back Under the Microscope: Hessian Test

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

4. Included with

Boundary Extrema

Discover how functions can achieve extreme values on exotic shapes.

7. 7

Multiple Integrals

Become a master of multivariable integration.

1. Included with

Double Integrals (Part I)

Gain double integral intuition through Riemann sums.

2. Included with

Double Integrals (Part II)

Evaluate simple double integrals with geometric reasoning.

3. Included with