Mathematica Laboratory

Availability The Mathematica program can be obtained here. The current version is Mathematica 9. During installation you will be prompted for an Activation Key. Students Faculty/Staff. Creation of a Wolfram account ID is optional. The contact information you provide must include your Harvard email address. Send me an email if you plan to use Mathematica on a linux system.
Getting the notebook The project is here. If you need more information, you find more examples here.
  • The mathematica file written live on July 18 during the mini workshop.
  • Running Mathematica Mathematica starts like any other application on OS X or Windows. On Linux, type "mathematica" in a terminal to start the notebook version, or "math" if you want to use the terminal version.
    Some basic commands:
    Plot[ x Sin[x],{x,-10,10}] Graph function of one variable
    Plot3D[ Sin[x y],{x,-2,2},{y,-2,2}] Graph function of two variables
    ParametricPlot[ {Cos[3 t],Sin[5 t]} ,{t,0,2Pi}] Plot planar curve
    ParametricPlot3D[{Cos[t],Sin[t],t} ,{t,0,4Pi},AspectRatio->1] Plot space curve
    ParametricPlot3D[{Cos[t] Sin[s],Sin[t] Sin[s],Cos[s]},{t,0,2Pi},{s,0,Pi}] Parametric Surface
    SphericalPlot3D[(2+Sin[2 t] Sin[3 s]),{t,0,Pi},{s,0,2 Pi}] Spherical Plot
    RevolutionPlot3D[{2 + Cos[t], t}, {t,0,2 Pi}] Revolution Plot
    ContourPlot[Sin[x y],{x,-2,2},{y,-2,2} ] Contour lines (traces)
    ContourPlot3D[x^2+2y^2-z^2,{x,-2,2},{y,-2,2},{z,-2,2}] Implicit surface
    VectorPlot[{x-y,x+y},{x,-3,3},{y,-3,3}] Vectorfield plot
    VectorPlot3D[{x-y,x+y,z},{x,-3,3},{y,-3,3},{z,0,1}] Vectorfield plot 3D
    Integrate[x Sin[x], x] Integrate symbolically
    Integrate[x y^2-z,{x,0,2},{y,0,x},{z,0,y}] 3D Integral
    NIntegrate[Exp[-x^2],{x,0,10}] Integrate numerically
    D[ Cos^5[x],x ] Differentiate symbolically
    Series[Exp[x],{x,0,3} ] Taylor series
    DSolve[ x''[t]==-x[t],x,t ] Solution to ODE
    DSolve[{D[u[x,t],t]==D[u[x,t],x],u[x,0]==Sin[x]},u[x,t],{x,t}] Solution to PDE
    Classify extrema:
    ClassifyCriticalPoints[f_,{x_,y_}]:=Module[{X,P,H,g,d,S}, X={x,y}; 
    P=Sort[Solve[Thread[D[f,#] & /@ X==0],X]]; H=Outer[D[f,#1,#2]&,X,X];g=H[[1,1]];d=Det[H];
    S[d_,g_]:=If[d<0,"saddle",If[g>0,"minimum","maximum"]];
    TableForm[{x,y,d,g,S[d,g],f} /.P,TableHeadings->{None,{x,y,"D","f_xx","Type","f"}}]]
    ClassifyCriticalPoints[4 x y - x^3 y - x y^3,{x,y}]
    
    Solve a Lagrange problem:
    F[x_,y_]:=2x^2+4 x y;     G[x_,y_]:=x^2 y;
    Solve[{D[F[x,y],x]==L*D[G[x,y],x],D[F[x,y],y]==L*D[G[x,y],y],G[x,y]==1},{x,y,L}]
    
    Check that a function solves a PDE:
    f[t_,x_]:=(x/t)*Sqrt[1/t]*Exp[-x^2/(4 t)]/(1+ Sqrt[1/t] Exp[-x^2/(4 t)]);
    D[f[t,x],t]+f[t,x]*D[f[t,x],x]-D[f[t,x],{x,2}]
    Simplify[%] Chop[%]