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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.

Interactive
quizzes

44

Concepts and
exercises

505+
  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.

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      Vector Arithmetic

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

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      Vector Properties

      Continue to build your vector intuition by approaching it geometrically!

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      Equations of Lines

      Apply your vector knowledge to the motion of heavenly bodies.

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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.

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      Multivariable Functions

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

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      Function Domains

      Connect multivariable functions with set geometry.

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      Basic Graphing

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

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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.

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      Searching Square Lake

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

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      Multivariable Limits

      Connect limits with many variables to limits with just one.

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      Shock Waves and Discontinuities

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

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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...

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      Basic Partial Derivatives

      Master the mechanics of multivariable rates of change.

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      Higher-Order Partials

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

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      Under the Microscope: Tangent Planes

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

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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.

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      Local Maxima and Minima

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

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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.

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      Back Under the Microscope: Hessian Test

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

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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.

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      Double Integrals (Part I)

      Gain double integral intuition through Riemann sums.

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      Double Integrals (Part II)

      Evaluate simple double integrals with geometric reasoning.

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      Fubini's Theorem (Part I)

      Break difficult double integrals down into bite-sized pieces.

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      Fubini's Theorem (Part II)

      Master the art of integral domain slicing.