Fall 2006

# Mathematics Math21a Fall 2006

## Multivariable Calculus

Office: SciCtr 434
Email: knill@math.harvard.edu
Harvard Mathematics

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## Mathematica Laboratory

 Gallery: there were many wonderful submissions. There are still a few dozen projects which need to be processed. Enter the gallery.

 Workshop: Thursday November 30 at 7-8 PM in lecture hall E,

 Deadline The Mathematica project is due December 18/19 in class. You hand in the Mathematica project printout with the last homework. We need a physical printout which contains the graphics and answers to the questions at the end of the notebook. Show your work. Also for the graphics plots, we need to see the command, which produced the plot.
Availability Mathematica is installed on some computers in the basement of the Science center. It might also be installed on some computers in the houses. If you want to try an installation of Mathematica on your own computer, get it here. Note that you have to be on a Harvard network and have your PIN ready to download the software and that requesting the Mathematica Password requires you to send the request from a Harvard computer.
Installation After you downloaded the program to your computer, start the application and follow the instructions. During the installation progress, you have to enter the Harvard Licence number L2482-2405. The number which you will get in return has to be entered in the Mathematica Registration page. You will then be sent a password by email. This is what you see during installation in Send email to math21a@fas, if you plan to use Mathematica on a linux system. We can provide you with a CD.
Getting the notebook
Running mathematica Mathematica is started like any other application on Macintoshs or PC's. On Linux, just type "mathematica" to start the notebook version, or "math" to start the terminal version.
Some frequently used commands: ( See more what you can do with Mathematica)
 Example how to check homework: change the function and get an analysis about its critical points: ``` (* Mathematica code to classify critical points, O. knill, 2000 *) f[x_,y_]:=4 x y - x^3 y - x y^3; a[x_,y_]:=D[f[u,v],u] /. {u->x,v->y}; b[x_,y_]:=D[f[u,v],v] /. {u->x,v->y}; A=Solve[{a[x,y]==0,b[x,y]==0},{x,y}]; CriticalPoints=Table[{A[[i,1,2]],A[[i,2,2]]},{i,Length[A]}]; H[{x_,y_}]:={{D[f[u,v],{u,2}],D[D[f[u,v],v],u]},{D[D[f[u,v],u],v],D[f[u,v],{v,2}]}} /. {u->x,v->y}; F[A_]:=A[[1,1]]; Discriminant=Map[Det,Map[H,CriticalPoints]] FirstEntry=Map[F,Map[H,CriticalPoints]] Decide[B_]:=If[Det[B]<0,"saddle",If[B[[1,1]]<0,"max","min"]]; Analysis=Map[Decide, Map[H,CriticalPoints]]; Table[{CriticalPoints[[i]],Analysis[[i]]},{i,Length[CriticalPoints]}] ``` Here is a slicker code, doing the same and even presenting it as a nice table (credit: Matt Leingang, 2006) ```ClassifyCriticalPoints[f_,{x_,y_}] := Module[{X,P,H,g,d,S}, X={x,y}; P=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} /. Sort[P],TableHeadings->{None,{x,y,"D","f_xx","Type","f"}}]] ClassifyCriticalPoints[4 x y - x^3 y - x y^3,{x,y}] ``` Here is an example how to 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[%] ```

 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 ContourPlot[ Sin[x y],{x,-2,2},{y,-2,2} ] Contour lines (traces) Integrate[ x Sin[x], x] Integrate symbolically 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 Get["Graphics`ContourPlot3D`"]; ContourPlot3D[x^2+2y^2-z^2-1,{x,-2,2},{y,-2,2},{z,-2,2}] Implicit surface Get["Calculus`VectorAnalysis`"] SetCoordinates[Cartesian[x, y, z]] Curl[{x y,x z, y z}] Div[{x y,x z,y z}] Grad[x y z]