Mathematics 153
Notes from May 22, 2020, lecture session

What was covered:
The remaining material from Unit 11.  One part treats the development of coordinate geometry in the work of R. Descartes, P. de Fermat, and several other researchers who built upon their work.   This includes a discussion of the similarities and differences in the contributions by Descartes, Fermat and several other individuals.  There were also remarks about other mathematical results due to Descartes and Fermat (but none of these were due to both).  In Descartes' case one algebraic result is cited (however, he also has priority for the previously cited result on the vertices, edges and faces of a polyhedron which is usually called Euler's Formula) and in Fermat's case the legacy is an extensive list of highly influential observations in number theory.  His writings did not contain proofs for all of them, but he was wrong only a few times and most of his observations were proved by the end of the 18th century.   One famous exception (Fermat's Last Theorem) was not proved until the end of the 20th century;  work on this and related problems led to the creation of large bodies of mathematics that did not exist in Femat's time.   Much of this mathematics has turned out to have numerous important applications (for example, in subjects as varied as cryptography and modern physics).
Relevant directory files for Unit 11:
Here is a listing of files from Unit 11 which are related to the May 22 lecture, with comments on their significance.  These links are all clickable. 
http://math.ucr.edu/~res/math153-2020/week8/unit11/history11.pdf  This is the main file for the unit and should ultimately be read from beginning to end.  After doing so one should read the cited sections of Burton.
http://math.ucr.edu/~res/math153-2020/week8/unit11/zzz-Projectile%20motion%20with%20simulator.mp4   This is an animated file showing the parabolic motion of a projectile fired upward (with air resistance not taken into account).  Galileo discovered this phenomenon.
https://mathworld.wolfram.com/Cycloid.html   An animated description of the cylcoid curve, which was first mentioned in the 15th century and was analyzed in detail during the next two centuries, mainly before the development of calculus.
http://math.ucr.edu/~res/math153-2020/week8/unit11/catenary+parabola.png   Galileo also conjectured that the curve defined by a hanging chain is a parabola, but it turns out that this curve is a slightly different one called a catenary (the graph of a curve whose graph has the shape of the hyperbolic cosine function).  One significant point for our purposes is that Renaissance physics was leading to the discovery of previously unknown curves, and these discoveries indicated a need for more powerful techniques for analyzing geometrical figures.  The file depicts a catenary and a comparable parabola, both of which pass through two points at the same distance from the ground..
http://math.ucr.edu/~res/math153-2020/week8/unit11/history11c.pdf   The basic result on choosing coordinate axes in the plane which often yield simplifications in proofs based upon coordinate geometry.  For example, one can always choose coordinates so that a given line is represented by the x-axis.
http://math.ucr.edu/~res/math153-2020/week8/unit11/aristotle.pdf   Some of the most difficult problems in classical Greek geometry were locus problems, whose goals are to give familiar descriptions for sets (or loci) of points satisfying certain conditions.   Coordinate geometry provides effective methods for successfully answering such questions in a relatively straifhtforward manner.   This is illustrated by a locus problem which appears in the writings of Aristotle.
http://math.ucr.edu/~res/math153-2020/week8/unit11/locus-problems.pdf      AND
http://math.ucr.edu/~res/math153-2020/week8/unit11/locus-problems1.pdf    AND
http://math.ucr.edu/~res/math153-2020/week8/unit11/locus-problems2.pdf   AND
http://math.ucr.edu/~res/math153-2020/week8/unit11/locus-problems3.pdf   Four files with many examples of solved locus problems.  There is a very good chance that a problem of this type will appear on the final examination.
http://math.ucr.edu/~res/math153-2020/week8/unit11/history11b.pdf  The last page of this file contains a worked out example involving coordinate geometry.
http://math.ucr.edu/~res/math153-2020/week8/unit11/transcendentals.pdf    AND
http://math.ucr.edu/~res/math153-2020/week8/unit11/transcurves.pdf           AND
http://math.ucr.edu/~res/math153-2020/week8/unit11/transcurves2.pdf        AND  
http://math.ucr.edu/~res/math153-2020/week8/unit11/transcurves3.pdf   These four files discuss the concepts of algebraic and nonalgebraic (transendental) curves.  The goal is to prove that curves which one might expect to be transcendental are in fact transcendental in a precise mathematical sense.  These also provide a justification for the term "transcendental functions" which is frequently seen in precalculus and first year calculus texts.   This is included only for reference purposes and will not be tested.
https://www.youtube.com/watch?v=xlZHY0srIew   A video describing a bizarre curve called the Koch snowflake, which is much more considerably bizarre than the curves than those which Descartes and some of his contemporaries (and forerunners) often dismissed as "mechanical."  This curve is an example of an object called a fractal.   None of this will be tested.
https://math.hmc.edu/funfacts/space-filling-curves/  An even more bizarre example of a continuously parametrized curve in the plane whose image contains every point of a solid square.
http://math.ucr.edu/~res/math153-2020/week8/unit11/history11a.pdf           AND
http://math.ucr.edu/~res/math153-2020/week8/unit11/morefermat.pdf     These files contain proofs for some of Fermat's results on number theory which are mentioned in the main notes for Unit 11.  Some of the proofs use graduate level topics in algebra, and accordingly none of the material in these files will be tested.  
https://mathpages.com/home/kmath195/kmath195.htm  Although Fermat often did not include proofs for his assertions, there are only a few incorrect or questionable assertions in his writings.  This article discusses the problematic statements.