Introduction to Mathematica

1. Introduction to Mathematica AE6381 Special Lecture Prof. J. Craig Fall 2008

2. What is Mathematica? • “A system for doing mathematics by computer” • Also for modeling, simulation, visualization, development, documentation, and deployment. • Available on all public computer clusters. • Can be installed on any GT-owned computer • Current version: 6.0.3 • Versions for: • Windows XP, Vista • Mac OS-X • Unix/Linux

3. Why consider Mathematica? • Powerful system for symbolic mathematical but also handles numerical mathematics, graphics, data visualization, simulation. • Larger community of users than Maple. • Versatile Notebook and Player for flexible documentation & distribution. • CONS: • VERY steep learning curve • Proprietary

4. What is this introduction? • I’ve been using Mathematica in AE3125 Aerospace Structural Analysis in order to simplify tedious calculations and solve problems not readily doable by hand. • All homework solutions are in Mathematica. • I’ll try to provide a simple and very basic introduction using this as my reference. • I’ll suggest other uses and point to the Wolfram web pages and ScreenCasts.

5. References • Wolfram, S., “Mathematica,” 2nd Edition, Addison-Wesley, 1991. • http://www.wolfram.com is main web site. • Built-in Mathematica documentation (all written in Mathematica Notebook code) • Tutorial screencastsare an excellent source of helpful information and can be found at: http://www.wolfram.com/broadcast/screencasts/ • Demonstration Project has great user demos at: http://demonstrations.wolfram.com/ (create your own using File>New>Demonstration) • Mathematica Journal at http://www.mathematica-journal.com

6. Launching Mathematica • User sees the Notebook interface. • An extensive online documentation system is available. • Palettes • A separate kernel is used for all mathematical processing

7. Basic Notebook Operation • Type an expression; default style is Input • Press Enter (in numeric pad) or Shift-Return which will send Input to kernel • Result will appear on line below • Sequential line numbers are prepended by kernel • Style determines appearance and what happens to line(s). Input is always sent to kernel. • Styles define appearance of blocks of cells shown by brackets at right margin. • Click cell bracket to select (to change, cut, copy…) • Double-click cell bracket to collapse that cell • Can collapse an entire bracket to the first cell

8. Notebooks • Provide a very readable display of code along with comments to fully document • Can collapse/hide portions (styles are useful here) • Layout is controlled by styles that can be modified by the user to change appearance • Can insert graphics, logos, etc. • Previous output can be saved in Notebook, including plots. • Use Evaluation>Evaluate Notebook to compute Notebook (send code to kernel). • Can also click on style bracket(s) and hit Enter (or Shift-Return) to compute that selected portion of Notebook.

9. Notebook and Kernel Issues • Kernel retains state of computations and sequentially numbers all input and output lines. • Must manually clear any previous variables • Clear[variables] (* clears values of variables *) • x=. (* same as Clear *) • Remove[“Global`*”] (* removes all variables *) • May need to “quit” local kernel to reset counter. See Evaluation>Quit Kernel<Local • May also connect to remote kernel(s) for greater power. • Line numbers show calculation sequence…

10. Using Mathematica • Startup Palette opens in a window by default when Mathematica is started. Go here for all documentation and help. • All documentation is in “live” Notebook format and you can cut & paste into your own code. • Good tutorials available.

11. Useful details… • Use of brackets: • [ ] =function arguments • { } =defines a list (can be nested) • [[n ]] =indexes a list (equiv to Part[ ] ) • ( ) =groups mathematical operations • N[expr]=numeric value • expr//=operator postfix notation • expr/.x→2=transformation rule (substitution) • % =previous result (%%=next to previous) • %n =result on Output line n

12. Basic operations • Simple numerical computations • Simple symbolic computations • Using built-in Functions • …

13. Useful Algebraic Manipulation • Can use Algebraic palette: • Click to apply directly to current expression • Type function and execute • Some functions have optional arguments Simplify[ expr,assum] = Simplify to expression with smallest number of terms. Optional: assumptions, i.e, x>0. FullSimplify[ ]= same but with more effort… Expand[ expr]= Expand numerators, keeping denominator ExpandAll[expr]= Expand numerators and denominators ExpandNumerator[expr]= Expand only numerator (like Expand[ ]) ExpandDenominator[ ]= Expand only denominator Factor[expr] = Factor a polynomial Together[expr] = Puts terms in a sum over a common denominator and cancels factors in the result. Apart[expr] = Rewrites a rational expression as a sum of terms with minimal denominators. Cancel[expr] = Cancels out common factors in the numerator and denominator of expr. Collect[expr,[x1,..}]= Collect together terms with same powers of objects matching x1 …

14. Examples from AE3125 • Shear flow in a thin-wall cross-section beam. • Function definitions • Symbolic integration • Plotting • Bending-shear-torsion in a complex thin-wall cross-section beam. • Solve ODE with boundary conditions • Rotational transformation using matrices

15. Another example… • Needed to explain to undergrad what happens to structural dynamics when you add a tip mass to a cantilever beam: • Set up PDE and use separation of variables • Solve spatial ODE • Compute eigenvalues and modeshapes • Animate?

16. Graphics • Large and sophisticated collection of graphics functions and objects. • Easy functional evaluation as well as basic numeric data plotting. • Many interesting examples available in Document Center and in “Demonstration Project.” • Screencasts: http://www.wolfram.com/broadcast/screencasts/

17. Data Manipulation • Easy to import data in many different file formats. • Many data sources (lots of new ones in 6.0). • Computation as well as plotting. • Screencast: http://www.wolfram.com/broadcast/screencasts/integrateddata/

18. Other topics… • Programming in Mathematica • Developing attractive Notebooks • Using Packages • Developing Packages • Creating Notebook Player files (.nbp)