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.

Divergence theorem

Divergence theorem intuition. Divergence theorem examples and proofs. Types of regions in 3D.
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All content in “Divergence theorem”

Divergence theorem (3D)

An earlier tutorial used Green's theorem to prove the divergence theorem in 2-D, this tutorial gives us the 3-D version (what most people are talking about when they refer to the "divergence theorem"). We will get an intuition for it (that the flux through a close surface--like a balloon--should be equal to the divergence across it's volume). We will use it in examples. We will prove it in another tutorial.

Types of regions in three dimensions

This tutorial classifies regions in three dimensions. Comes in useful for some types of double integrals and we use these ideas to prove the divergence theorem.

Divergence theorem proof

You know what the divergence theorem is, you can apply it and you conceptually understand it. This tutorial will actually prove it to you (references types of regions which are covered in the "types of regions in 3d" tutorial.