摘要:

† An invited address given at the 21st annual conference of the Australian Mathematical Society on 19 May 1977 at La Trobe University, Melbourne, Australia.

ISSN:0020-739X    

DOI:10.1080/0020739790100101

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

A course of mathematical methods for students of different degree programmes is outlined. An approach to the teaching of the course is described which allows continuity between sections of apparently different material, thus avoiding the criticism that courses in mathematical methods are disjoint.

ISSN:0020-739X    

DOI:10.1080/0020739790100102

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

Three approaches to the same geometrical problem are presented to demonstrate the role of insight or intuition in mathematics. Insight is the ability to directly and spontaneously perceive a proof, as a whole rather than line‐by‐line.

ISSN:0020-739X    

DOI:10.1080/0020739790100103

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

A new approach to the calculus is introduced using the calculus of finite differences and finite sums, both as motivation and as useful tools. The main reason for this development is to create an ideal context in which the computer can be naturally incorporated into calculus courses. Most topics are first developed in the discrete setting and then extended to the continuous case by applying the limit. This provides a constant review of all topics, a definite aid to most students. Furthermore, the numerical techniques used by computers are all within the scope of the finite calculus. Moreover, by starting with the finite calculus, a wider range of topics can be naturally included in the calculus course, including discrete models based on difference equations (especially useful for non‐science majors) and various approximation and interpolation techniques which make calculus usable in practical situations.

ISSN:0020-739X    

DOI:10.1080/0020739790100104

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

The rate of convergence of a Fourier‐series representation of a given function depends on the nature of the function and of its derivatives. This is shown by using the graphical outputs of a desk computer for different cases. For full‐range series, the effects of continuity and discontinuity of the function and its first derivatives are shown first. The advantage of half‐range formulae due to the free choice of function in the second half‐range are demonstrated next, along with the importance of choice of sine and cosine series according to the function being represented. Finally, a method is given of modifying the given function in a simple manner whereby a dramatic increase in the rate of convergence of the Fourier series is obtained. Examples of the Gibbs overshoot and its elimination are included.

ISSN:0020-739X    

DOI:10.1080/0020739790100105

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

In lecture courses on stochastic processes, simple, continuous‐time Markov processes with discrete states are usually solved by deriving Kolmo‐gorov forward equations, summarizing them by a single Lagrange partial‐differential equation for a generating function, and then solving the resulting equation by means of the auxiliary equations. Students frequently find this approach confusing. The aim of this paper is to present three alternative approaches which may be employed in order to assist in the teaching of the subject, and which are frequently omitted from text books; the approach described in § 3, in particular, is not readily available. At the same time, it is hoped that students of mathematics in general will find in these processes interesting applications of mathematical methods.

ISSN:0020-739X    

DOI:10.1080/0020739790100106

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

In this paper we shall consider sophisticated computer systems designed to help students in learning mathematics. In the first part of the paper (§§ 1‐3) the possibilities of such systems are explored. In the second part (§§ 4‐9) we discuss some of the major problems which have to be solved before these possibilities can be realized.

ISSN:0020-739X    

DOI:10.1080/0020739790100107

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

A method of generating Pythagorean triples is described in this paper.

ISSN:0020-739X    

DOI:10.1080/0020739790100108

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

This paper describes a self‐study programme in mathematics for undergraduates. A series of units on topics not normally covered in traditional curricula are written up for the students by the joint efforts of the interested members of the faculty. A guideline is provided for the students, who may elect to study some of the materials for academic credits. A summary of the units which the author used is included.

ISSN:0020-739X    

DOI:10.1080/0020739790100109

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

A notation is presented to assist in the discussion of units and the consistency of equations. It is extended to cover vectors and some mathematical functions. It highlights various points which give rise to confusion in teaching.

ISSN:0020-739X    

DOI:10.1080/0020739790100110

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor