| On means, polynomials and special functions | 1$-$20 |
Abstract
We discuss how derivatives can be considered as a game of cubes and beams, and of geometric means. The same principles underlie wide classes of polynomials. This results in an unconventional view on the history of the differentiation and differentials.
Keywords: Derivative; polynomial; special function; arithmetic, geometric and harmonic means.
MSC Subject Classification: 97E55, 97H25, 51M15
MathEduc Subject Classification: F50, H20
| Euler-Poincaré characteristic---a case of topological self-convincing | 21$-$33 |
Abstract
In this paper we establish a topological property of geometric objects (lines, surfaces and solids) called Euler-Poincaré characteristic. Since the paper is intended for a large profile of mathematics teachers, our approach is entirely intuitive and majority of readers can omit two addenda whose understanding requires a solid knowledge of topology. E-P characteristic is an integer which we calculate here decomposing lines into fibers being finite sets of points, surfaces into fibers being lines and solids into fibers being surfaces. When an object is subjected to a ``plastic'' deformation its shape and size changes as well as the alternating sums resulting from the method of calculation, but the value of these sums stays unchanged. This fact serves to convince the reader that E-P characteristic is a stable topological property. The same fact gives this approach an advantage over usually practiced ones which require a triangulation of geometric objects.
Keywords: Euler-Poincaré characteristic; decomposition of lines into finite sets of points, surfaces into lines, solids into surfaces.
MSC Subject Classification: 97G99
MathEduc Subject Classification: G95
| A proof of method of cylindrical shells based on a generalized integral representation of additive interval function | 34$-$38 |
Abstract
In this paper we provide a generalized integral representation of additive interval function based on a fundamental integral representation of additive interval function given in Zorich's textbook, Mathematical Analysis, Vol I. Then we use it to give a rigorous proof of the method of cylindrical shells for the evaluation of volume of solid of revolution about vertical line.
Keywords: Additive interval function; method of cylindrical shells; Riemann integrable function.
MSC Subject Classification: 97I50
MathEduc Subject Classification: I55
| Cauchy-type inclusion and exclusion regions for polynomial zeros | 39$-$50 |
Abstract
A classical result by Cauchy defines a disk containg all the zeros of a polynomial. We derive several related results by using similarity transformations of a polynomial's companion matrix, together with Gershgorin's theorem. We thus show that Cauchy's original result can be seen as but one member of a family of related results.
Keywords: Cauchy; Gershgorin; zero location; root location; polynomial.
MSC Subject Classification: 97H30, 12D10, 15A18
MathEduc Subject Classification: H35