## Mathematica Laboratory

 Download the project. (option click the link to save it). We had a Mathematica workshop on Wednesday, November 19, 2014, from 7 to 8:30 PM in Science center C. Video. The demofile we produced during that workshop. There are three problems: make a region plot. Create and render a surface. Create something else. Have in mind that it could be printed, used in a game or in google earth. The project is due Saturday, December 6, 2014.
• The Mathematica program can be obtained here. The current version is Mathematica 10.0.1. 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 ```f[x_]:=Sin[Pi 7x]; g[x_]:=5 Sin[5 Pi x]; U = NDSolveValue[ {D[u[t,x],{t,2}]-D[u[t,x],{x,2}]==0, u[0,x]==f[x], Derivative[1,0][u][0,x] == g[x], DirichletCondition[u[t,x]==f,x==0], DirichletCondition[u[t,x]==f,x==1]}, u,{t,0,1},{x,0,1}]; Plot[U[t,0.5], {t, 0, 1}] ``` 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}] ```