Mathematica

  • The Mathematica lab is here. The assignment is at the end of the lab.
  • Here is an example on how to check a PDE with mathematica. 
    f[t_,x_]:=(1/Sqrt[t])*Exp[-x^2/(4t)];
FullSimplify[D[f[t,x],t] == D[f[t,x],{x,2}]]


    Graphs and contour plots of functions of two variables
    Please distinguish "graph", "contour curve", "contour surface", "contour map". Mathematica does this also nicely: If you do not specify a value in Contour Plot, it produces a contour map (several surfaces). 
    f[x_, y_] := x^2 + y^2; 
ContourPlot[f[x, y], {x, -2, 2}, {y, -2, 2}]
Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}]
ContourPlot3D[ z - f[x, y], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]
ContourPlot3D[f[x, y] == 1, {x, -2, 2}, {y, -2, 2}]
ContourPlot3D[z - f[x, y] == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]

    Lagrange problems

    Solve a Lagrange problem with 2 variables 
    f=2x^2+4 x y;  g=x^2 y;
Solve[{D[f,x]==L*D[g,x],D[f,y]==L*D[g,y],g==1},{x,y,L}]
    With 3 variables: 
    f=2x^2+4 x*y+z;     g=x^2 y + z;   c=1;
Solve[{D[f,x]==L*D[g,x], D[f,y]==L*D[g,y], D[f,z]==L*D[g,z], g==c},{x,y,z,L}]
    With 3 variables and two constraints 
    f=x y z;  g=x*y+2 y z+2 x z;  h=2x+2y+4z; d=4; c=1
Solve[{D[f,x]==L*D[g,x] + M*D[h,x],
       D[f,y]==L*D[g,y] + M*D[h,y],
       D[f,z]==L*D[g,z] + M*D[h,z],
       g==c, h==d},{x,y,z,L,M}]

    Classifying critical points

    Here is example code on how to compute the gradient and the discriminant 
    f=x^3 y^3- x y^2;
Grad[f,{x,y}]
D[f,{x,2}]*D[f,{y,2}]-D[f,{x,1},{y,1}]^2

    Classifying critical points

    This produces a nice table. 
    f=4 x y - x^3 y - x y^3;
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[f,{x,y}]

    Installation

    We will have a Mathematica lab at the end of this course. Please help each other installing mathematica. You should have done so until Tuesday 6/30/2020, when we start graphing.
    • Make an gmail account from harvard
    • You can get Mathematica from here. There are instructions there to download and install Mathematica.
    • You need to make a wolfram account. Start here to make an account (use your Harvard summer school email address), Use the latest Mathematica version.