Vector Calculus

Complete the multivariable calculus saga with vector fields.

Change is deeply rooted in the natural world. Fluids, electromagnetic fields, the orbits of planets, the motion of molecules; all are described by vectors and all have characteristics depending on where we look and when.

In this course, you'll learn how to quantify such change with calculus on vector fields. Go beyond the math to explore the underlying ideas scientists and engineers use every day.

Interactive
quizzes

35

Concepts and
exercises

405+
  1. 1

    Introduction

    Vector fields, surface integrals, div and curl.

    1. Vector Calculus in a Nutshell

      Explore the possibilities that come from combining calculus and vectors.

    2. Calculus of Motion

      Look at the world in motion through the lens of vector calculus.

    3. Introducing Surface Integrals

      Mix vectors with integrals to uncover an essential tool for applications.

    4. Introducing the Divergence Theorem

      See one of the most important theorems in vector calculus at work.

  2. 2

    Vector-valued Functions

    Force fields, motion through space, and much, much more...

    1. Included with
      Brilliant Premium

      Space Curves

      Fly along curves through three dimensions.

    2. Included with
      Brilliant Premium

      Integrals and Arc Length

      Exactly how long is a space curve?

    3. Included with
      Brilliant Premium

      Frenet Formulae

      Measure the shape of space curves with vector calculus.

    4. Included with
      Brilliant Premium

      Parametric Surfaces

      Expand your gallery of shapes to include a few exotic cases.

  3. 3

    Surface Integrals and Divergence

    Part I of the essential vector calculus toolbox.

    1. Included with
      Brilliant Premium

      Flux (Part I)

      Experiment with charges moving in electric field and discover the concept of flux.

    2. Included with
      Brilliant Premium

      Flux (Part II)

      Use flux to uncover surface integrals and see how they're used to solve important problems.

    3. Included with
      Brilliant Premium

      Surface Integrals

      Master integrals of functions on parametrized surfaces.

    4. Included with
      Brilliant Premium

      Divergence (Part I)

      Construct a crucial vector derivative through a fundamental law of nature.

  4. 4

    Line Integrals & Curl

    Part II of the essential vector calculus toolbox.

    1. Included with
      Brilliant Premium

      Work (Part I)

      Explore an important physics application of vector calculus.

    2. Included with
      Brilliant Premium

      Work (Part II)

      Unveil a new kind of integral by delving into a familiar physics concept.

    3. Included with
      Brilliant Premium

      Line Integrals

      Learn how to integrate along space curves and why it's so useful.

    4. Included with
      Brilliant Premium

      Path Independence

      Journey to where calculus and topology meet to discover a crucial property of vector fields.

  5. 5

    Advanced Applications

    Solve important real-world problems with vector calculus.

    1. Included with
      Brilliant Premium

      The Laplacian

      Apply a new derivative to problems in electrostatics and fluid dynamics.

    2. Included with
      Brilliant Premium

      Gaussian Integrals (Part I)

      Detour into the world of multivariable calculus to compute an integral crucial for applications.

    3. Included with
      Brilliant Premium

      Gaussian Integrals (Part II)

      Use linear algebra and a change of variables to find the most general Gaussian integral.

    4. Included with
      Brilliant Premium

      Fourier Transform

      Apply Gaussian integrals to understand the Fourier transform, a powerful way to solve pdes.