18.02SC | Fall 2010 | Undergraduate

# Multivariable Calculus

## 4. Triple Integrals and Surface Integrals in 3-Space

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### Unit 4 Introduction

In our last unit we move up from two to three dimensions. Now we will have three main objects of study:

1. Triple integrals over solid regions of space.
2. Surface integrals over a 2D surface in space.
3. Line integrals over a curve in space.

As before, the integrals can be thought of as sums and we will use this idea in applications and proofs.

We’ll see that there are analogs for both forms of Green’s theorem. The work form will become Stokes’ theorem and the flux form will become the divergence theorem (also known as Gauss’ theorem). To state these theorems we will need to learn the 3D versions of div and curl.

Part A: Triple Integrals

Part B: Flux and the Divergence Theorem

Part C: Line Integrals and Stokes’ Theorem

Exam 4

Physics Applications

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In this part we will learn to compute triple integrals over regions in space. We will learn to do this in three natural coordinate systems: rectangular, cylindrical and spherical.

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Here we will extend Green’s theorem in flux form to the divergence (or Gauss’) theorem relating the flux of a vector field through a closed surface to a triple integral over the region it encloses. Before learning this theorem we will have to discuss the surface integrals, flux through a surface and the divergence of a vector field.

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In this part we will extend Green’s theorem in work form to Stokes’ theorem. For a given vector field, this relates the field’s work integral over a closed space curve with the flux integral of the field’s curl over any surface that has that curve as its boundary.

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We finish the course by showing the use of Gauss’ and Stokes’ theorems in Maxwell’s equations. In particular we use the theorems to show the equivalence of the integral and differential forms of the equations.

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## Course Info

Fall 2010
##### Learning Resource Types
Simulations
Exams with Solutions
Problem Sets with Solutions
Lecture Videos
Lecture Notes