Advanced Math · Level 3

3.2 Multivariable Calculus

Continue your calculus journey into the realm of multiple variables, exploring everything from vectors to volume.

Searching Square Lake

Multivariable Limits

Shock Waves and Discontinuities

Extreme Value Theorem (Part I)

Extreme Value Theorem (Part II)

Basic Partial Derivatives

Higher-Order Partials

Under the Microscope: Tangent Planes

Directional Derivatives

The Gradient

Chain Rule of Several Variables

Local Maxima and Minima

Back Under the Microscope: Quadrics

Back Under the Microscope: Hessian Test

Boundary Extrema

Method of Lagrange

Application: Lagrange Multipliers

Global Maxima and Minima

More Hessians! (Optional)

Double Integrals (Part I)

Double Integrals (Part II)

Fubini's Theorem (Part I)

Fubini's Theorem (Part II)

Multiple Integrals

Multiple Integrals Applications

Change of Variables

Cylindrical & Spherical Integrals

Course description

Change is an essential part of our world, and calculus helps us quantify it — 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
  • Extreme Value Theorem
  • Gradient
  • Hessian Test for Critical Points
  • Jacobians & Variable Changes
  • Lagrange Multipliers
  • Limits and Continuity
  • Multiple Integrals
  • Multivariable Functions & Graphs
  • Partial Derivatives

Prerequisites and next steps

You need to be familiar with multivariable functions, vectors and vector operations, and the basic ideas of calculus, including limits, derivatives, and integrals.


  • Calculus Fundamentals
  • Integral Calculus

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Advanced Math · Level 4

4.1 Introduction to Linear Algebra

Delve into the abstract depths of linear algebra: vector spaces, determinants, eigenvalues, wedge products, and more.

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