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# 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
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#### Vector Arithmetic

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

2. Included with
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#### Vector Properties

Continue to build your vector intuition by approaching it geometrically!

3. Included with
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#### Equations of Lines

Apply your vector knowledge to the motion of heavenly bodies.

4. Included with
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#### 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
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#### Multivariable Functions

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

2. Included with
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#### Function Domains

Connect multivariable functions with set geometry.

3. Included with
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#### Basic Graphing

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

4. Included with
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#### 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
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#### Searching Square Lake

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

2. Included with
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#### Multivariable Limits

Connect limits with many variables to limits with just one.

3. Included with
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#### Shock Waves and Discontinuities

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

4. Included with
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#### 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
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#### Basic Partial Derivatives

Master the mechanics of multivariable rates of change.

2. Included with
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#### Higher-Order Partials

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

3. Included with
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#### Under the Microscope: Tangent Planes

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

4. Included with
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#### 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
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#### Local Maxima and Minima

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

2. Included with
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#### 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
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#### Back Under the Microscope: Hessian Test

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

4. Included with
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#### 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
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#### Double Integrals (Part I)

Gain double integral intuition through Riemann sums.

2. Included with
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#### Double Integrals (Part II)

Evaluate simple double integrals with geometric reasoning.

3. Included with
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#### Fubini's Theorem (Part I)

Break difficult double integrals down into bite-sized pieces.

4. Included with
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#### Fubini's Theorem (Part II)

Master the art of integral domain slicing.