European Mathematical Society - 97I99
https://euro-math-soc.eu/msc-full/97i99
enCalculus Reordered
https://euro-math-soc.eu/review/calculus-reordered
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
For reasons of sequentiality, an elementary calculus course is usually organising the topics that are discussed in the order where first limit is defined, which is needed to define continuity, then the derivative as the limit of a ratio, and only then the integral as the limit of a Riemann sum. The same order can be repeated for functions of a discrete variable (sequences) where the variable is usually denoted as an index. The limit can only be considered when the index goes to infinity (the only accumulation point), continuity and derivative don't make much sense (unless one wants to discuss finite differences) and (improper) integrals correspond to (infinite) sums, and the convergence of series. At the beginning some functions are assumed known: algebraic functions are a minimum. Sometimes also goniometric functions, and possibly the exponential and logarithm, but in principle all of these and the more "advanced" ones can be defined later. Functions are "defined" when no name exists for a converging series or a primitive function of an integral. The logarithm for example is the integral of 1/x, and the exponential function is its inverse. At least this is how I organized my lecture notes, but this is not always the order used, and it is also not in this order that all these concepts were developed historically.</p>
<p>
What Bressoud does in this book is looking at the contents of a calculus course from an historical perspective. In what order were all these concepts developed, and where do all the well known theorems come from? In fact analysis came relatively late. The Greek were mainly doing geometry using rational numbers, and yet they computed the volume of a sphere and the area of a circle. Then the algebra came to Western Europe through the Arabs, and only then analysis took off with Newton and Leibniz who were already manipulating series or at least truncated series as interpolating approximations. It is only when analytic geometry bridged the gap between algebra and geometry, that analysis took over as the dominant tool for solving the practical analytic problems. The limit and its geometric interpretation came only very late.</p>
<p>
The first chapter of this book covers the period up to and including Newton and Leibniz. The second is about the further evolution of calculus, analytic geometry, the logarithm, differential equations, waves and field theory, culminating in the Maxwell equations. The emergence of Taylor and Fourier series are covered in chapter three, but only in chapter four, we find convergence criteria for series. This convergence, just like the proper definition of a derivative, and other concepts that are defined as the result of a limit, can only come to a conclusion by including the limiting value by upper and lower bounds approaching each other. Only using these bounding inequalities, will eventually lead to the proper concept of a limit. This new concept allowed a previously unseen expansion of analysis, requiring to rethink the concept of a function, since it was realized how exotic some functions can be, like everywhere continuous and nowhere differentiable. Bressoud's historical survey illustrates that the order of our calculus course is inverting history. First came accumulation (integral), then ratios of change (derivative), then sequences of partial sums (series) and only in the end the algebra of inequalities (limit).</p>
<p>
In an appendix, Bressoud adds his thoughts about how calculus should be taught. It is often the case that because of time restrictions that the integral is introduced as an anti-derivative, leading to cookbook recipes to compute integrals. Given the current technology, this is indeed a waste of time when there is no insight of the integral as a summation brought to a limit. Similarly teaching formulas to compute derivatives without seeing the derivative as a limiting process of ratios of change doesn't make much sense either, and series should be introduced as the limit of the sequence of its partial sums. And finally the limit is the result of a sandwich principle where upper and lower bounds approach each other. All of these insights are essential when looking at numerical analysis where exactly these insights are the elements that compute integrals, derivatives, and evaluate transcendental functions.</p>
<p>
As I have been teaching elementary courses in algebra, analysis, and numerical analysis, I can fully appreciate Bressoud's conclusions that I described in the previous paragraph and I fully realize the importance of algebra as an essential element in the development of calculus, certainly when one moves to functions of several variables, (linear) algebra, matrices and vector spaces become essential. However, it is somewhat unclear if Bressoud is promoting to keep also the historical order as he describes it in this book: integration, differentiation, series, limit. In my opinion the limit is the missing link that lets the whole building of calculus fall into a logical sequence and hence it should come first. It would be a waste of time to re-live the trial and errors of history in a calculus course. Apart from the pedagogical conclusions, the book gives a nice survey of how the main achievements and theorems that any student meets in a calculus course came about. Moreover, it is shown that the historical approach is sometimes quite different from what is written in the lecture notes, because the mathematical tools and the objectives of the person developing them were in those days quite different from what is available to a student in the 21st century.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The book is a survey of how the main ideas that underpin a modern calculus course were developed in their historical context. Based on this, Bressoud draws some conclusions about how we should teach a calculus course. An approach following the historical origin will be much closer to intuition and have didactic advantages.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-m-bressoud" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David M. Bressoud</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691181318 (hbk), 9780691189161 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 24.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">242</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/13397.html" title="Link to web page">https://press.princeton.edu/titles/13397.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97d40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97D40</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97i99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I99</a></li></ul></span>Mon, 05 Aug 2019 09:54:15 +0000Adhemar Bultheel49603 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-reordered#commentsThe Calculus Story
https://euro-math-soc.eu/review/calculus-story
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
David Acheson's popular math book <em>1089 and all that</em> (Oxford U. Press, 2002) was rather successful, and it got with this one a worthy successor. Small size, short chapters, amply illustrated, large font, airy layout, all properties that turn it into a storybook indeed. By storybook, I mean the kind of books you want to read to your children or grandchildren, that keeps their attention and makes them impatient and eager for the adventure to come in the next episode. In this sense, the title is very well chosen. It is indeed a story book, the story of calculus and how it came about, once upon a time...</p>
<p>
Since it is still about mathematics, you shouldn't try this on a toddler, but you might catch the attention of any novice to calculus from the age of about twelve or thirteen on. What exactly should be familiar before you can start telling the calculus story? You need some algebra to be able to manipulate equations. Also the concept of a proof, and curves as representations of functions are assumed, but that's about all. The mathematical content of the story is then the same as what you find in a classic textbook: limits, differentiation, integration, and infinite series. In the early chapters of the book Acheson sketches the preliminaries and sheds some light on what one wants to achieve with the next steps about to be taken. And later in the second half of the book, while he got the attention of the reader, Acheson moves on to differential equations, optimisation, complex numbers and chaos.</p>
<p>
For reasons of efficiency, a classic textbook will introduce a new concept by following a certain paradigm that consists of giving some motivating examples, followed by a precise and polished definition, or it is done the other way around: first the definition, and then some examples. However, this is not how the concept caught by the definition was developed historically. It is a much more natural approach to follow the historical path. In retrospect, the side tracks that turned out to be dead ends, can be pruned away, but still, letting the concept grow organically usually is a better choice. Decoupling mathematics from its history makes it abstract and dull. Including the history makes it less of a top-down rigid dogmatic doctrine that is forced upon the pupil, but something developed by real people, hence more "human".</p>
<p>
These historical elements make Acheson's book a mathematical (his)story(book). For example it is interesting to learn how people struggled with $\infty\times0=?$. Summing infinitely many infinitely small elements was used, and sometimes misused, by the founding fathers such as Kepler, Cavalieri, Wallis, Torricelli and others to compute and area or a volume, hence indirectly summing infinite series. Many of these computations were inspired by physics. The speed and acceleration of falling objects subject to gravity had been investigated but it was Newton who formulated the more general fundamental laws of motion. When applied to the force of gravity, these eventually explained the orbits in our planetary system. Kepler had described the "how" of the orbits, and Newton provided the "why". This has influenced Newton strongly in the way he developed his calculus. He made use of fluxions as it was based on the dynamics of coordinates like an object exposed to some action will move along a path describing its position as a function of time. Independently also Leibniz developed calculus. He used these infinitely small increments. The ratio $\delta y/\delta x$ of small changes gave rise to the notation $dy/dx$ for the derivative that mathematicians still use today, while Newton used $\dot{x}$ for the derivative of $x$, which is more commonly used in physics. With this climax in the calculus story, the birth of calculus, Acheson is about half way in his book.</p>
<p>
In the second half, the sine and cosine functions are used to connect an angle to periodic motion and for example to show the Leibniz formula $1−1/3+1/5−1/\cdots=\pi/4$ and other ways to compute <em>π</em>. The Leibniz formula had been hinted to in previous chapters, building up some tension. So, it feels like yet another success of calculus that it can explain why this formula holds. But periodic motion also means differential equations describing the pendulum or a vibrating string. The towering historical mathematician is now Euler. With calculus at our disposal, now topics such as calculus of variations, optimization, logarithms (including e and $i=\sqrt{-1}$), Taylor series expansions, Fourier series and other topics now come in rapid succession. It also requires to reconsider the definition of a limit to the more rigorous $(\epsilon,\delta)$ definition as developed by Weierstraß. Thus it is illustrated how this precise definition of the limit is the endpoint of a whole evolution and perhaps should not be the first definition to which a novice should be exposed. Acheson ends his story with a glimpse on Maxwell's equation of electromagnetism, Schrödinger's equation, and chaos theory.</p>
<p>
Thus Acheson introduces the reader with elementary steps to the concepts that matter, creating insight, and answering the why's and how's by calling the historical mathematicians to the stage and by citing from their papers. Thus it is not a bedtime story after all, but it should awaken the interest of the youngsters for the fascinating mathematics that in the end is describing the physics of the world we live in. Even for those students who had to assimilate calculus from a dull textbook, this story underlying all these definitions, theorems and computational rules, may soften their aversive attitude towards the subject. To pull youngsters away from the dark side of mathphobia, this booklet acts as a medicine to be applied to your wishes: preventively, remedially, or supplementary. This calculus story can be applied at all times to create a mathematical success story. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Acheson uses an historical approach to introduce calculus to beginners. Starting from the most elementary concepts, the reader is exposed to a world of limits, differentiation, integration, and series and their entanglement with physics. Describing our planetary system was an incentive for the development of calculus. A flavour is given of differential equations describing a pendulum or a vibrating string, and there is a preview of Maxwell's and Schrödinger's equations and chaos theory. While previously physics pushed the boundaries of mathematics, it is now mathematics that pushes the limitations of physics.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-acheson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Acheson</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780198804543 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 11.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">208</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/the-calculus-story-9780198804543" title="Link to web page">https://global.oup.com/academic/product/the-calculus-story-9780198804543</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97i99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I99</a></li></ul></span>Tue, 31 Oct 2017 06:17:37 +0000Adhemar Bultheel47981 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-story#comments