Vectors and Space Curves

Copyright 2007 by Evans M. Harrell II.  All rights reserved.

You are welcome to copy this Mathematica  notebook without restrictions for educational purposes at Georgia Tech.  It is, however, copyrighted material, so you should get permission to use it independently of Georgia Tech classes taught by Carlen or Harrell.

Off campus users should treat the notebook as shareware - try it out for a limited time before contacting the author: harrell@math.gatech.edu or via the Web at http://www.math.gatech.edu/~harrell/  .

Notes for the instructor.

This contains some standard material from Math 2507, implemented in Mathematica.

Instructions for using Mathematica  notebooks in Carlen's and Harrell's Calculus 2507.

(Double-click on the arrow-shaped thing at the right side of a cell like this one to see its contents or to close it when finished.)

Mathematica  is a powerul software package, which can be of great use to you with most of the mathematics you will encounter in engineering.  Some of the calculus classes at Georgia try to teach you to use Mathematica  by requiring term papers, which those students who get seriously involved with Mathematica  find valuable.  In this class, however, Mathematica  will be a strictly optional element, which you are encouraged to use with your homework.  

In order to make the use of software as painless and useful as possible, we will provide notebooks like this one with a substantial amount of code, which you are welcome to copy without restrictions for educational purposes at Georgia Tech.  It is, however, copyrighted material, so you should get permission to use it independently of Georgia Tech classes taught by Carlen or Harrell.

There are three ways you can use Mathematica  in the vector calculus class.  

1.  You can work through notebooks like this one in order to learn the subject matter of the notebook.  This should be a hands-on effort; modify the code by changing the functions, seeking alternatives, etc. to get a feel for the concepts of vector calculus

2.  You can use Mathematica  to perform calculations for your homework, thereby avoiding tedious integrals and the like.  If you do this, you will not get credit for handing in homework with the statement that "Mathematica  calculated this for me."  Instead, you need to include the code in such a way that it will actually run.  Also, you should think about the answer.  For instance, is an integral turns out to be zero, try to explain why.

Ideally, you would submit such homework both in printed form and with a diskette.

3.  Mathematica  makes a great study aid.  Do calculations by hand and then see if Mathematica  gets the same answer.  Then you have some instantaneous feedback telling you whether you have the right answer.  (Or perhaps you are lucky, of course, but that won't happen consistently unless you are doing something right.)  Mathematica  is particularly useful for feedback in first year calculus, where many of the calculations are routine, but it is still sometimes useful in vector calculus.

Reminders about common Mathematica  notation:

	Parentheses () are used to group items as in algebra.
	Brackets [] are used for inputs to functions and operations.
	Braces {} are used for lists.  In particular, vectors are written as   {x,y,z}.
	The backslash indicates that the command continues onto   \
	           the next line without a break.

For this notebook, it is not necessary to load the package Calculus`VectorAnalysis` .

Displaying Vectors

<< Graphics`Arrow`

Show[
      Graphics[{Arrow[{0, 0}, {1, 1}],  
          Arrow[{0, 0}, {-3, -1}]}]]

Space Curves

ParametricPlot3D[{s,Sin[4 Pi s],2 s}, {s, 0, Pi}, ViewPoint -> {5,5,1}]

ParametricPlot3D[{Cos[Pi s],Sin[2 Pi s],Cos[3 Pi s]}, {s, 0, 4}]

ParametricPlot3D[{Cos[Pi s], Sin[2 Pi s], Cos[3 Pi s]}, {s, 0, 4}, 
      ViewPoint -> {5, 5, 4}]