Mathematica Laboratory

The mathematica project is online.
Availability The Mathematica program can be obtained here. The current version is Mathematica 10. 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 More examples here.
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[%]