vol 4: Glossary
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Calculus
[Hazewinkel, Calculus] [Borowski, Calculus] [
Calculus was invented independently by Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716).
The world of calculus is populated by functions and operators. Operators transform functions into other functions. Differentiation reveals the local behaviour of a function, telling us how fast it is changing. Integration is concerned with the global behaviour of a function, providing us with a measure of the overall effect of the function. Differentiation and integration are inverses of one another, so that (under certain conditions) differentiation followed by integration returns the original function.
The functions of interest to calculus are all those functions which are either differentiable, integrable, or both. We may see integration as an extension of the idea of addition and differentiation as an extension of subtraction.
The simplest and clearest application is in Newtonian mechanics in ordinary three dimensional space. Calculus also finds application in more complex spaces, and its use is ubiquitous throughout mathematics, physics and engineering.
Further reading
Books
| Borowski, Ephraim J, & Johnathan M Borwein, Collins Dictionary of Mathematics, Harper Collins 1089 'It is the immodest hope of the authors that this dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary schools to a master's degree, but also offer much to interest a more general readership. Amazon back |
| Hazewinkel, Michiel, Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' Amazon back |
| Misner, Charles W, and Kip S Thorne, John Archibald Wheeler, Gravitation, Freeman 1973 Jacket: 'Einstein's description of gravitation as curvature of spacetime led directly to that greatest of all predictions of his theory, that the universe itself is dynamic. Physics still has far to go to come to terms with this amazing fact and what it means for man and his relation to the universe. John Archibald Wheeler. ... this is a book on Einstein's theory of gravity (general relativity).' Amazon back |
| Yourgrau, Wolfgang, and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory, Dover 1979 Variational principles serve as filters for parititioning the set of dynamic possibilities of a system into a high probability and a low probability set. The method derives from De Maupertuis (1698-1759) who formulated the principle of least action, which states that physical laws include a rule of economy, the principle of least action. This principle states that in a mathematically dsecribed dynamic system will move so as to minimise action. Mandelstam explains the application of this principle to a variety of physical systems. Amazon back |
Links
| Bert G Wachsmuth Interactive Real Analysis Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, and more. The text is changing constantly, and your comments are very welcome: please sign our guest book. The project was supported by a grant from Seton Hall University. back |
| Cambridge University Isaac Newton Institute for Mathematical Sciences 'The Isaac Newton Institute is an international research institute for the mathematical sciences. Seminars are open to all. Registration or invitation is usually required for longer-term participation in programmes or workshops. We are particularly eager to encourage participation by young researchers in the UK.' back |
| Cambridge University Isaac Newton Links 'Here at the Isaac Newton Institute for Mathematical Sciences, we are often asked about Newton's life and works. There are already many excellent and informative Web sites and books about Newton, so rather than duplicate those, we have put together a guide to some of the places, both real and virtual, where you can find out more.' back |
| Mathematical Archives Calculus Resources on Line 'Welcome to the Calculus Resources On-line area of the Mathematics Archives. This area contains information and links to numerous Internet resources, which could be used for teaching and learning of calculus.' back |
| Newton.org Sir Isaac Newton 'Welcome to newton.org.uk - the virtual museum of Sir Isaac Newton and the history of science.' back |
| Phill Schultz Leibniz' Calculus 'Lecture 19 Leibniz' Invention of calculus.' back |
| School of Mathematics and Statistics University of St Andrews, Scotland Leibniz School of Mathematics and Statistics, University of Saint Andrews, Scotland. back |
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