Prof. Vladimir Dobrushkin
Department of Mathematics
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MTH243 (Calculus for Functions of Several Variables)
MATHEMATICA. Chapter 13: Vectors
, Section 13.4

Vladimir A. Dobrushkin,Lippitt Hall 202C, 874-5095,dobrush@uri.edu

In this course we will use Mathematica computer algebra system (CAS), which is available in computer labs at URI. The Mathematica projects are created to help you learn new concepts. Mathematica 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. A student can also use free CASs: SymPy (based on Python), Maxima, or Sage.

Section 13.4. The Cross Product

The first traceable work on cross product is found in the book "Vector Analysis," founded upon the lectures of J. Willard Gibbs, second edition, by Edwin Bidwell Wilson (1879--1964), published by Charles Scribner's Sons in 1909.

Area of a parallelogram :

Graphics[Parallelogram[]]
or
Graphics[Parallelepiped[{0, 0}, {{1, 0}, {1, 1}}]]
With blue color:
Graphics[{Blue, Parallelogram[]}]
or
R = Parallelogram[{0, 0}, {{1, 0}, {1, 1}}];
Graphics[{Blue, R}]
   
Another approach:
S = Parallelogram[{0, 0}, {{1, 0}, {1, 1}}]; {Graphics[{EdgeForm[Dotted], Green, S}]}
or
S = Parallelogram[{0, 0}, {{1, 0}, {1, 1}}]; {Graphics[{EdgeForm[Thick], Yellow, S}]}
   
With dashed boundary:
Show[ListLinePlot[{{0, 0}, {.5, .5}, {1, 1}} -> {"", "w"}, Filling -> Axis],
ListLinePlot[{{0, 0}, {.5, 0}, {1, 0}} -> {"", "v"}],
ListLinePlot[{{1, 0}, {2, 1}}, Filling -> Top, PlotStyle -> Dashed],
ListLinePlot[{{1, 1}, {2, 1}}, PlotStyle -> Dashed], PlotRange -> {{0, 2}, {0, 1}}]
or
\[ScriptCapitalR] = Parallelogram[{0, 0}, {{1, 0}, {1, 1}}];
{Graphics[{EdgeForm[Dashed], Blue, \[ScriptCapitalR]}]}
   

Example 1. Find \( {\bf i} \times {\bf j} \)

Now we create a new graph: \( g (x, y) = x^2 + y^2 + 3 \)
Plot3D[(x^2 + y^2 + 3), {x, -3, 3}, {y, -3, 3}, Axes -> True]
Another graph of \( h(x, y) = 5 - x^2 - y^2 \)
Plot3D[(5 - x^2 - y^2), {x, -3, 3}, {y, -3, 3}, Axes -> True, PlotStyle -> Orange]
One more: \( k (x, y) = x^2 + (y - 1)^2 \)
Plot3D[(x^2 + (y - 1)^2), {x, -3, 3}, {y, -3, 3}, PlotStyle -> None]
Example 2. Plotting the Graph of the Function \( G(x,y)=e^{-(x^2+y^2)} \)
Plot3D[(E^-(x^2 + y^2)), {x, -5, 5}, {y, -5, 5}, PlotStyle -> Opacity[.8]]
Cross Sections and the Graph of a Function where x=2
Plot3D[{(x^2 + y^2), (4 + y^2)}, {x, -3, 3}, {y, -3, 3}]