Applied Mathematics: Body and Soul: Volume 2: Integrals and by Kenneth Eriksson, Donald Estep, Claes Johnson PDF

By Kenneth Eriksson, Donald Estep, Claes Johnson

ISBN-10: 364205658X

ISBN-13: 9783642056581

ISBN-10: 3662057980

ISBN-13: 9783662057988

Applied arithmetic: physique & Soul is a arithmetic schooling reform venture built at Chalmers collage of expertise and contains a sequence of volumes and software program. this system is stimulated through the pc revolution starting new chances of computational mathematical modeling in arithmetic, technological know-how and engineering. It involves a synthesis of Mathematical research (Soul), Numerical Computation (Body) and alertness. Volumes I-III current a latest model of Calculus and Linear Algebra, together with constructive/numerical recommendations and purposes meant for undergraduate courses in engineering and technology. additional volumes current subject matters reminiscent of Dynamical structures, Fluid Dynamics, sturdy Mechanics and Electro-Magnetics on a complicated undergraduate/graduate point.

The authors are best researchers in Computational arithmetic who've written a variety of winning books.

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Extra resources for Applied Mathematics: Body and Soul: Volume 2: Integrals and Geometry in IRn

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Cauchy's Representation Formula . Taylor's Formula: Second Shot .. Power Series Representation of Analytic Functions Laurent Series. . . . . . . Residue Calculus: Simple Poles . . Residue Calculus: Poles of Any Order The Residue Theorem . . . . . 33 Computation of J~oo ~ dx . . . 34 Applications to Potential Theory in IR2 . 1 Introduction............... 3 Warm Up II: Series . . . . . . 4 Complex Fourier Series. . . . . 6 Truncated Fourier Series and Best L 2-Approximation .

13 The Integral as a Limit of Riemann Sums The Fundamental Theorem of Calculus states that the integral of f(x) over the interval [a, b] is equal to a limit of Riemann sums: where xi = a + ihn' hn = 2- n(b - a), or more precisely, for n = 1,2, ... 14 An Analog Integrator 449 where L f is the Lipschitz constant of f. We can thus define the integral f (x) dx as a limit of Riemann sums without invoking the underlying differential equation u' (x) = f (x). This approach is useful in defining integrals of functions of several variables (so-called multiple integrals like double integrals and triple integrals), because in these generalizations there is no underlying differential equation.

We may prove these properties in two ways: (i) by using the connection between the integral and the derivative and using properties of the derivative, or (ii) using that the integral is the limit of Riemann sum approximations, that is, using the area interpretation of the integral. We indicate both types of proofs to help the reader getting familiar with different aspects of the integral, and leave some of the work to the problem section. K. , Applied Mathematics: Body and Soul © Springer-Verlag Berlin Heidelberg 2004 454 28.

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Applied Mathematics: Body and Soul: Volume 2: Integrals and Geometry in IRn by Kenneth Eriksson, Donald Estep, Claes Johnson


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