Higher Dimensional Lagrange Multipliers

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This Mathematica notebook lists a few of the commands you might find useful while exploring the higher dimensional Lagrange multipliers lab.

Graphing Contour Surfaces

The following command will find the contour surface of the function written in red for the level written in blue. The ranges of the variables x, y, and z are colored green. The rest of the command merely dresses up the way Mathematica displays the graph. You can rotate the surface by dragging it, although more complicated surfaces may require patience.
In[]:=
ContourPlot3D[Cos[x]Sin[y]+Cos[y]Sin[z]+Cos[z]Sin[x]==0,{x,-2π,2π},{y,-2π,2π},{z,-2π,2π},ContourStyleDirective[FaceForm[Orange,Red],Specularity[White,30]],MeshNone]
Out[]=

Graphing Multiple Contour Surfaces

The following command will find the contour surface of the functions written in red and orange for the levels written in blue. The ranges of the variables x, y, and z are colored green. The rest of the command merely dresses up the way Mathematica displays the graph. You can rotate the surface by dragging it, although more complicated surfaces may require patience.
In[]:=
ContourPlot3D[{x^3+y^2-z^2==0,x^2+y^2+z^26},{x,-2π,2π},{y,-2π,2π},{z,-2π,2π},ContourStyleDirective[FaceForm[Orange,Red],Specularity[White,30]],MeshNone]
Out[]=

Solving Simultaneous Equations

The following equation will solve the system of equations colored red for the variables x and y.
In[]:=
Solve[{2x+y==7,5x-y==1},{x,y}]
Out[]=
x
8
7
,y
33
7


Partial Derivatives

You can have Mathematica to compute partial derivatives for you:
In[]:=
∂
x
(x^(1/3)y^(1/2)z^(1/6))
Out[]=
y
1/6
z
3
2/3
x