
MTH243 (Calculus
for Functions of Several Variables)
SAGE. Chapter 12:
Functions of Several Variables
Vladimir A. Dobrushkin,Lippitt
Hall 202C, 8745095,dobrush@uri.edu
In this course we will use Sage computer algebra system (CAS), which is a free software.
The Sage projects are created to help you learn new concepts. Sage is very useful in visualizing graphs
and surfaces in three dimensions. Matlab (commercial software) is also available at engineeering labs. Its free version
is called Octave. The university has a license for computer algebra system Mathematica, so it is free to use for its students. A student can also use free CASs: SymPy (based on Python), or Maxima.

Section 12.3. Contour Diagrams
Example 1.
Example 2.
Example 3.
Example 4.
Draw a contour diagram for \( f(x,y) = \sqrt{x^2 + y^2} \) and relate it to the graph of f.
First, we plot cones:
figure
[X,Y,Z] = meshgrid(10:0.5:10,10:0.5:10,10:0.5:10);
a=1;
b=1;
c=1;
V = X.^2/a^2 + Y.^2/b^2  Z.^2/c^2;
p=patch(isosurface(X,Y,Z,V,0));
set(p,'FaceColor','GREEN','EdgeColor','none');
daspect([1 1 1])
view(3);
camlight
title('12.77 Cone')
a = 1.5; b = 1.5; c = 1;
ContourPlot3D[
x^2/a^2 + y^2/b^2  z^2/c^2 == 0, {x, 3, 3}, {y, 3, 3}, {z, 2,
2}, ColorFunction > Hue]
ContourPlot3D[
x^2 + y^2  z^2 == 0, {x, 3, 3}, {y, 3, 3}, {z, 3, 3},
ContourStyle > LightGray]
ContourPlot3D[
x^2 + y^2  z^2 == 0, {x, 4., 4.}, {y, 4., 4.}, {z, 4., 4.},
Axes > True, BoxRatios > {1., 1., 1.},
ViewPoint > {1.5, 2.5, 0.5}, PlotRange > All,
AxesLabel > {"x", "y", "z"},
ContourStyle >
Directive[RGBColor[1, 0.8, 0.3],
Specularity[RGBColor[0.2, 0.4, 0.9], 20]], Lighting > "Neutral",
ColorFunction > None, BoxStyle > GrayLevel[0.4, 0.35]]
ContourPlot3D[
x^2 + y^2  z^2 == 0, {x, 3, 3}, {y, 3, 3}, {z, 3, 3},
Mesh > 4, MeshFunctions > {#1 &, #2 &, #3 &}]
ContourPlot3D[
x^2 + y^2  z^2 == 0, {x, 3, 3}, {y, 3, 3}, {z, 3, 3},
MeshShading > {Blue, Orange}, MeshFunctions > {#3 &}]
Plot3D[Sqrt[x^2 + y^2], {x, 5, 5}, {y, 5, 5}, BoxRatios > {1, 1, 1}]
Example 5.
Example 6.
Example 7.
Example 8.
 