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

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: – Versions for: Windows XP, Vista Mac OS-X Unix/Linux

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

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.

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

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

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

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.

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…

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.

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

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

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,[x 1,..}]= Collect together terms with same powers of objects matching x 1 …

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

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?

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:

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: egrateddata/ egrateddata/

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