4. Triple Integrals and Surface Integrals in 3-Space

Unit 4 Introduction

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

Triple integrals over solid regions of space.
Surface integrals over a 2D surface in space.
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

» Session 74: Triple Integrals: Rectangular and Cylindrical Coordinates
» Session 75: Applications and Examples
» Session 76: Spherical Coordinates
» Session 77: Triple Integrals in Spherical Coordinates
» Session 78: Applications: Gravitational Attraction
» Problem Set 10

Part B: Flux and the Divergence Theorem

» Session 79: Vector Fields in Space
» Session 80: Flux Through a Surface
» Session 81: Calculating Flux; Finding ndS
» Session 82: ndS for a Surface z = f(x, y)
» Session 83: Other Ways to Find ndS
» Session 84: Divergence Theorem
» Session 85: Physical Meaning of Flux; Del Notation
» Session 86: Proof of the Divergence Theorem
» Session 87: Diffusion Equation
» Problem Set 11

Part C: Line Integrals and Stokes’ Theorem

» Session 88: Line Integrals in Space
» Session 89: Gradient Fields and Potential Functions
» Session 90: Curl in 3D
» Session 91: Stokes’ Theorem
» Session 92: Proof of Stokes’ Theorem
» Session 93: Example
» Session 94: Simply Connected Regions; Topology
» Session 95: Stokes’ Theorem and Surface Independence
» Session 96: Summary of Multiple Integration
» Problem Set 12