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Applications_of_multivariable_derivatives

The tools of partial derivatives, the gradient, etc. can be used to optimize and approximate multivariable functions. These are very useful in practice, and to a large extent this is why people study multivariable calculus.

Green_s_Stokes_and_the_divergence_theorems

Here we cover four different ways to extend the fundamental theorem of calculus to multiple dimensions. Green's theorem and the 2D divergence theorem do this for two dimensions, then we crank it up to three dimensions with Stokes' theorem and the (3D) divergence theorem.

Integrating_multivariable_functions

There are many ways to extend the idea of integration to multiple dimensions: Line integrals, double integrals, triple integrals, surface integrals, etc. Each one lets you add infinitely many infinitely small values, where those values might come from points on a curve, points in an area, points on a surface, etc. These are all very powerful tools, relevant to almost all real-world applications of calculus. In particular, they are an invaluable tool in physics.

Thinking_about_multivariable_functions

The only thing separating multivariable calculus from ordinary calculus is this newfangled word "multivariable". It means we will deal with functions whose inputs or outputs live in two or more dimensions. Here we lay the foundations for thinking about and visualizing multivariable functions.

Derivatives_of_multivariable_functions

What does it mean to take the derivative of a function whose input lives in multiple dimensions? What about when its output is a vector? Here we go over many different ways to extend the idea of a derivative to higher dimensions, including partial derivatives​, directional derivatives, the gradient, vector derivatives, divergence, curl, etc.