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Multivariable Calculus

via Brilliant


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.


  • Limits with Many Variables: Uncover unexpected function properties with limits.
    • Searching Square Lake: Begin to uncover the mysteries of limits with the search for a mythical beast.
    • Multivariable Limits: Connect limits with many variables to limits with just one.
    • Shock Waves and Discontinuities: Learn to visualize limits and apply them to the real world.
    • Extreme Value Theorem (Part I): Get a bird's-eye view of a crucial calculus theorem.
    • Extreme Value Theorem (Part II): Apply limits like a mathematician and prove the extreme value theorem.
  • Derivatives: Measuring rates of change is just the beginning...
    • Basic Partial Derivatives: Master the mechanics of multivariable rates of change.
    • Higher-Order Partials: Learn about the uses of a derivative's derivative, like the wave equation.
    • Under the Microscope: Tangent Planes: Zoom in on a function's graph and see its tangent planes up close.
    • Directional Derivatives: Dive beneath Square Lake to develop directional rates of change.
    • The Gradient: Build the gradient, the source for everything there's to know about how quickly a function changes.
    • Chain Rule of Several Variables: Find out what the gradient looks like in different coordinate systems.
  • Optimization: Put derivatives to work finding and classifying extreme values.
    • Local Maxima and Minima: Use gradient geometry to find the highs and lows of a graph.
    • Back Under the Microscope: Quadrics: Dust off your function microscope and see the basic shape of a graph near a critical point.
    • Back Under the Microscope: Hessian Test: Use the microscope to come up with a simple test to classify local extrema.
    • Boundary Extrema: Discover how functions can achieve extreme values on exotic shapes.
    • Method of Lagrange: Develop a simple means for finding constrained extrema using gradient geometry.
    • Application: Lagrange Multipliers: Apply Lagrange's Method to a fun real-world example.
    • Global Maxima and Minima: Practice all of your extrema-hunting strategies here.
    • More Hessians! (Optional): Extremize functions of more than two variables with linear algebra.
  • Multiple Integrals: Become a master of multivariable integration.
    • Double Integrals (Part I): Gain double integral intuition through Riemann sums.
    • Double Integrals (Part II): Evaluate simple double integrals with geometric reasoning.
    • Fubini's Theorem (Part I): Break difficult double integrals down into bite-sized pieces.
    • Fubini's Theorem (Part II): Master the art of integral domain slicing.
    • Multiple Integrals: What does it mean to integrate a function with more than two variables?
    • Multiple Integrals Applications: Discover why multiple integrals are so useful.
    • Change of Variables: Reshape a multiple integral into something easier through coordinate transformations.
    • Cylindrical & Spherical Integrals: Practice on real-world applications with symmetry.


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