Calculus a Very Short

This book addresses the issue of uniqueness of a solution to a problem – a very important topic in science and technology, particularly in the field of partial differential equations, where uniqueness guarantees that certain partial differential equations are sufficient to model a given phenomenon.

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Addressed to both pure and applied probabilitists, including graduate students, this text is a pedagogically-oriented introduction to the Schwartz-Meyer second-order geometry and its use in stochastic calculus. P.A. Meyer has contributed an appendix: "A short presentation of stochastic calculus" presenting the basis of stochastic calculus and thus making the book better accessible to non-probabilitists also. No prior knowledge of differential geometry is assumed of the reader: this is covered within the text to the extent.

The origin of this book can be traced to courses on financial mathemat­ ics taught by us at the University of New South Wales in Sydney, Warsaw University of Technology (Politechnika Warszawska) and Institut National Polytechnique de Grenoble. Our initial aim was to write a short text around the material used in two one-semester graduate courses attended by students with diverse disciplinary backgrounds (mathematics, physics, computer sci­ ence, engineering, economics and commerce).

This study guide is intended to aid your understanding of multivariable calculus. We have organized it into chapters and sections corresponding to Basic Multivariable Calculus by Jerrold E. Marsden, Anthony ]. Tromba, and Alan Weinstein (Springer-Verlag/W.H. Freeman, 1993). Each section contains Goals, Study Hints, and (most important) Solutions to Every Other Odd Exercise. In addition, we have written chapter tests at the end of each review section.

The link between Calculus of Variations and Partial Differential Equations has always been strong, because variational problems produce, via their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can often be studied by variational methods. At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on a classical topic (the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to pde's resp.).

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