Mathematica Laboratory

The Mathematica Assignment Fall 2016

The Mathematica project of Fall 2016 is

here

We ad a workshop on Wednesday November 16 5:30-6:30 in Science center Hall C. A few important points:
  • There is a strict deadline, December 3, midnight
  • Submission is by email and will be acknowledged.
  • Be creative. Its fine to consult with others or ask around or consult the web. But each submits his or her own project. Its not a group project.
  • If using external sources, acknowledge it, as custom in any academic setting. If you like to add in part 6 a title or context pointer (if not obvious), please do so.
The lab is of creative nature as in previous years and be posted soon. Here are galleries from previous years:
Example Summer 2016, Example Fall 2015 Example Fall 2014 Example Fall 2013
  • The Mathematica program can be obtained here. During installation you will be prompted for an Activation Key. Students Faculty/Staff. Make sure to use your Harvard email address when registering. Contact me (knill@math.harvard.edu) if you plan to use Mathematica on a linux system.
  • Mathematica is started like any other application on Macintoshs or PC's. On Linux, just type "mathematica" in a terminal to start the notebook version, or "math" to start the terminal version.
  • Once Mathematica is running, copy paste any of the following lines into a cell, click with the mouse somewhere into the cell, then hold "Shift" and hit "Enter".
Some frequently used 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 with 2 variables 
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}]
With 3 variables 
F[x_,y_,z_]:=2x^2+4 x y+z;     G[x_,y_,z_]:=x^2 y + z;   c=1; 
Solve[{D[F[x,y,z],x]==L*D[G[x,y,z],x],
       D[F[x,y,z],y]==L*D[G[x,y,z],y],
       D[F[x,y,z],z]==L*D[G[x,y,z],z],
       G[x,y,z]==c},{x,y,z,L}]
With 3 variables and two constraints 
F[x_,y_,z_]:=z;     G[x_,y_,z_]:=z^2-x^2-y^2;  H[x_,y_,z_]:=4x-3y+8z; c=0; d=5; 
Solve[{D[F[x,y,z],x]==L*D[G[x,y,z],x] + M D[H[x,y,z],x],
       D[F[x,y,z],y]==L*D[G[x,y,z],y] + M D[H[x,y,z],y],
       D[F[x,y,z],z]==L*D[G[x,y,z],z] + M D[H[x,y,z],z],
       G[x,y,z]==c,
       H[x,y,z]==d},
{x,y,z,L,M}]
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[%]
Solving a partial differential equation numerically Please use code from here. A partial differential equation (wave equation) with three variables: 
A=Rectangle[{0,0},{1,1}]; Clear[t,x,y];
f[x_,y_]:=Sin[2 Pi x] Abs[Sin[3 Pi y]];
g[x_,y_]:=3 Sin[Pi x] Sin[Pi y];
U=NDSolveValue[{D[u[t,x,y],{t,2}]
 -Inactive[Laplacian][u[t,x,y],{x,y}]==0,
  u[0,x,y] == f[x,y],
  Derivative[1,0,0][u][0,x,y]==g[x,y],
  DirichletCondition[u[t,x,y] ==0,True]},
  u,{t, 0, 2 Pi}, {x,y} \[Element] A];
Plot3D[U[4,x,y],{x,0,1},{y,0,1}]