Strict deadline: Wednesday, December 4. Midnight.
(The email has to have a December 4 time stamp).
Submit by email to knill@math.harvard.edu
Please remove all graphics first from the notebook:
(Menu: Cell -> Delete all Output) and attach the .nb file to your email.
Do not use canvas nor any dropbox/google drive/cloud type service
to submit. Just attach the .nb file to a regular email.
A short Mathematica workshop gave more details about the project
It happened on Thursday, November 21 at 3 PM in Hall B: a selection of
footage from the Mathematica workshop:
f=z; g=z^2-x^2-y^2; h=4x-3y+8z; c=0; d=5;
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}]
If the numerical solution does not work, replace Solve with NSolve as follows:
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[NSolve[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}]
Here is a check that a function f satisfies the heat equation.
In HW 12, you have to verify such a thing using technology but
for a bit more complicated function and also for a bit more
complicated PDE:
In one Data project, we have used
this notebook about roller coaster to learn how to run a notebook.
At the end of the course, we will have a
Mathematica project. There is no need to buy the software.
Harvard has a site license.