module seventeen: Vector fields and line integrals

Objectives

By the end of this module, you should be able to

  • Evaluate vector fields at a given point and use software to sketch in the plane and in 3D
  • Compute the gradient of a scalar field
  • Compute divergence and curl for a vector field
  • Understand what makes a vector field conservative, and the implications of that
  • Compute line integrals of scalar functions
  • Computer line integrals of vector fields in various forms

Prerequisites

  • This material puts the previous 2/3 together - we now have "vector in, vector out", so you have all the material with vectors and vector valued functions joined with all the material on multivariable Calculus and surfaces.

Activities

Vector fields

  1. Printable lecture notes
  2. Flash lecture
  3. Plotting vector fields in Maple
  4. Plotting vector fields in MVT
  5. Physical examples of vector fields
  6. Suggested problems
  7. Solutions to suggested problems

grad, div, and curl

  1. Printable lecture notes
  2. Flash lecture
  3. grad curl and div in Maple
  4. Suggested problems
  5. Solutions to suggested problems

What is all this good for anyway? (Supplemental reading/video)

It's traditional at this point in the semester to throw the last chunk of stuff out there and say "uh, here you go, just do this...it's good for something in Physics later on. Really." The time crunch at this point is pretty bad, and so neither I (nor a standard Calc text) spend a whole lot of time on what these things are. There's the added issue that coming at them in a Math program, and not as a Physics grad, they were presented to *me* even at the grad school level in more of a purely math-y perspective (I see them in terms of operators and the theory thereof), so I'm weak on the physical interpretations and probably wouldn't do a great job of explaining them even if we had the time.

Internet to the rescue. There are two nice sources I've found for this stuff - the Khan video guy does a great intro to the topics (explaining just what is curling in curl!), and there's another nifty website called Better Explained that explains, oh all sorts of things...inculding some pretty decent math! So, for your enjoyment and further reading...

  1. Better Explained: Understanding the gradient
  2. Better Explained: Understanding flux (we don't do much with flux, but you need to read up on it to understand a physical interpretation of divergence)
  3. Better Explained: Understanding divergence
  4. Better Explained: Understanding circulation and curl
  5. Khan Academy: Divergence explained - articles plus videos. This link should take you to a page that shows a chain of four articles on a left menu, starting with "Divergence" and ending on "Curl, fluid rotation in three dimensions." When you get a chance, go through it - it's a great presentation on the topics.

Conservative vector fields

  1. Printable lecture notes
  2. Flash lecture
  3. Live example notes
  4. Live example one
  5. Live example two
  6. Suggested problems
  7. Solutions to suggested problems

Line integrals

  1. Printable lecture notes
  2. Flash lecture
  3. Live example notes
  4. Live example one
  5. Live example two
  6. Not-live example three (I wrote this one out - it's got so many little fiddly bits and trips back to Maple it's a misery to do live)
  7. Line integrals in Maple
  8. Suggested problems
  9. Solutions to suggested problems

Line integrals of vector fields

  1. Printable lecture notes
  2. Flash lecture
  3. Live example notes
  4. Live example one
  5. Live example two
  6. Suggested problems
  7. Solutions to suggested problems

Line integrals of vector fields - differential form

  1. Printable lecture notes
  2. *No slideshow on this one
  3. Live example one
  4. Live example two
  5. Suggested problems
  6. Solutions to suggested problems