Multivariable calculus

Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.
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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 that can have either multiple inputs or multiple outputs, like f(x, y) = (xy, x-3y). Before diving into the many new topics in calculus this seemingly small shift brings about, we take a moment in this tutorial to go through the different ways one can think about and visualize multivariable functions.

Double and triple integrals

Volume under a surface with double integrals. Triple integrals as well.

Partial derivatives, gradient, divergence, curl

Thinking about forms of derivatives in multi-dimensions and for vector-valued functions: partial derivatives, gradient, divergence and curl.

Line integrals and Green's theorem

Line integral of scalar and vector-valued functions. Green's theorem and 2-D divergence theorem.

Surface integrals and Stokes' theorem

Parameterizing a surface. Surface integrals. Stokes' theorem.

Divergence theorem

Divergence theorem intuition. Divergence theorem examples and proofs. Types of regions in 3D.

Line integrals and Green's theorem

Line integral of scalar and vector-valued functions. Green's theorem and 2-D divergence theorem.
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All content in “Line integrals and Green's theorem”

Line integrals for scalar functions

With traditional integrals, our "path" was straight and linear (most of the time, we traversed the x-axis). Now we can explore taking integrals over any line or curve (called line integrals).

Position vector functions and derivatives

In this tutorial, we will explore position vector functions and think about what it means to take a derivative of one. Very valuable for thinking about what it means to take a line integral along a path in a vector field (next tutorial).

Green's theorem

It is sometimes easier to take a double integral (a particular double integral as we'll see) over a region and sometimes easier to take a line integral around the boundary. Green's theorem draws the connection between the two so we can go back and forth. This tutorial proves Green's theorem and then gives a few examples of using it. If you can take line integrals through vector fields, you're ready for Mr. Green.

2D divergence theorem

Using Green's theorem (which you should already be familiar with) to establish that the "flux" through the boundary of a region is equal to the double integral of the divergence over the region. We'll also talk about why this makes conceptual sense.