Many Variables in a Nutshell
Take a lightning tour of calculus with several variables.
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.
Double the variables, double the fun.
Take a lightning tour of calculus with several variables.
Learn how partial derivatives can solve important real-world problems.
Explore new ways to navigate in three dimensions.
Bridge the gap between geometry and multiple integrals with Riemann sums.
Master vectors, the basic building blocks of multivariable calculus.
Work hands-on with vectors, the building blocks of multivariable calculus.
Continue to build your vector intuition by approaching it geometrically!
Apply your vector knowledge to the motion of heavenly bodies.
Use geometry to make the dot product, an essential way of multiplying vectors.
Transform vectors with matrices and find out what they have in common.
Is it ever OK to divide by a matrix?
Apply the determinant to find a second vector multiplication rule.
Take the first step into higher dimensions.
Explore functions of several variables and discover what they're good for.
Connect multivariable functions with set geometry.
Learn to capture the most important qualities of a function with a 3D picture.
Develop skills to visualize the shape of a function and to think in higher dimensions.
What do graphing and mountain climbing have in common?
Find out how to compress a complicated function down into a handy 2D map.
Uncover unexpected function properties with limits.
Begin to uncover the mysteries of limits with the search for a mythical beast.
Connect limits with many variables to limits with just one.
Learn to visualize limits and apply them to the real world.
Get a bird's-eye view of a crucial calculus theorem.
Apply limits like a mathematician and prove the extreme value theorem.
Measuring rates of change is just the beginning...
Master the mechanics of multivariable rates of change.
Learn about the uses of a derivative's derivative, like the wave equation.
Zoom in on a function's graph and see its tangent planes up close.
Dive beneath Square Lake to develop directional rates of change.
Build the gradient, the source for everything there's to know about how quickly a function changes.
Find out what the gradient looks like in different coordinate systems.
Put derivatives to work finding and classifying extreme values.
Use gradient geometry to find the highs and lows of a graph.
Dust off your function microscope and see the basic shape of a graph near a critical point.
Use the microscope to come up with a simple test to classify local extrema.
Discover how functions can achieve extreme values on exotic shapes.
Develop a simple means for finding constrained extrema using gradient geometry.
Apply Lagrange's Method to a fun real-world example.
Practice all of your extrema-hunting strategies here.
Extremize functions of more than two variables with linear algebra.
Become a master of multivariable integration.
Gain double integral intuition through Riemann sums.
Evaluate simple double integrals with geometric reasoning.
Break difficult double integrals down into bite-sized pieces.
Master the art of integral domain slicing.
What does it mean to integrate a function with more than two variables?
Discover why multiple integrals are so useful.
Reshape a multiple integral into something easier through coordinate transformations.
Practice on real-world applications with symmetry.