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$Some fractional functional inequalities - UPC$

Figure 11. Constructing a spiral helix, with plane projectiononto an Archimedes spiral, and then putting it in a precise positiononto the plane x + y + z =3.example specular symmetry (see gure 12).Figure 12. Toothed wheels, prisms and specular symmetry.Another interesting exercise is the generation of polyhedra of revolutionfrom a plane polygonal prole, as a standard tool for volume generationin solid modelling and CAD one of the most popular objects is the chessqueen, that we also integrate in more complex geometric compositions, suchas in gure 13.14
Figure 13. Compositionwith chess piecesof revolution.ConclusionsOur geometry lab is not merely an experimental complement in our course,but it is an integral part of it, and one of the most important ones. Withoutthe possibilityofusingMathematica wewould not have been able to organizethe course as we have done, and without it our course would have been verydierent.Our impression is that students learn much better geometry with such kindof lab exercices, they become enthousiastic about them and they realize theimportance of the relationship between theory and applications.With our experience, we hope wehavecontributed to makemany people morefamiliar with mathematics laboratories, specially geometry ones, and to behelping that way to the incorporation of the new developing technologiesin mathematical education, to put the teaching of mathematics in a newhigh-tech environment, and consequently, on a new and modern basis.As a byproduct of our preparation of materials for the course we have atour disposal an extensive set of Mathematica functions that constitute apowerful tool for mathematical calculation and illustration (specically, inthe dicult eld of 3D geometrical illustration) once it will be organized asa Mathematica package, it will become available. By now, our contributionto this use of Mathematica in preparing graphic material for teaching canbe found in [2]. Some other authors in our Department have developed15
Page 1 and 2: Geometry Lab with MathematicaMathem
Page 3 and 4: Figure 2. Rhodonea, epitrochoids an
Page 5 and 6: X and X 0 are also column vectors.
Page 7 and 8: Mathematica does the rest, speciall
Page 9 and 10: environment. That is the point that
Page 11 and 12: Figure 8A more complete exampleLetu
Page 13: Show[Graphics3D[{{GrayLevel[.7],cyl
Page 17: Both are enthousiastic about the us

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