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3190 results in Mathematical Methods

Page 114 of 160

  • 4 - Interpolation

  • Andi Klein, Los Alamos National Laboratory, Alexander Godunov, Old Dominion University, Virginia
  • Book:
    Introductory Computational Physics
    Published online:
    04 August 2010
    Print publication:
    09 March 2006, pp 25-36

  • Summary

    An important part in a scientist's life is the interpretation of measured data or theoretical calculations. Usually when you do a measurement you will have a discrete set of points representing your experiment. For simplicity, we assume your experiment to be represented by pairs of values: an independent variable “x,” which you vary and a quantity “y,” which is the measured value at the point x. As an illustration, consider a radioactive source and a detector, which counts the number of decays. In order to determine the half-life of this source, you would count the number of decays N0, N1, N2, …, Nk at times t0, t1, t2, …, tk. In this case t would be your independent variable, which you hopefully would choose in such a way that it is suitable for your problem. However, what you measure is a discrete set of pairs of numbers (tk, Nk) in the range of (t0, tk). In order to extract information from such an experiment, we would like to be able to find an analytical function which would give us N for any arbitrary chosen point t. But, sometimes trying to find an analytical function is impossible, or even though the function might be known, it is too time consuming to calculate or we might be only interested in a small local region of the independent variable.

  • Preface

  • K. F. Riley, University of Cambridge, M. P. Hobson, University of Cambridge
  • Book:
    Student Solution Manual for Mathematical Methods for Physics and Engineering Third Edition
    Published online:
    05 June 2012
    Print publication:
    06 March 2006, pp ix-x

  • Summary

    The second edition of Mathematical Methods for Physics and Engineering carried more than twice as many exercises, based on its various chapters, as did the first. In the Preface we discussed the general question of how such exercises should be treated but, in the end, decided to provide hints and outline answers to all problems, as in the first edition. This decision was an uneasy one as, on the one hand, it did not allow the exercises to be set as totally unaided homework that could be used for assessment purposes, but, on the other, it did not give a full explanation of how to tackle a problem when a student needed explicit guidance or a model answer.

    In order to allow both of these educationally desirable goals to be achieved, we have, in the third edition, completely changed the way this matter is handled. All of the exercises from the second edition, plus a number of additional ones testing the newly added material, have been included in penultimate subsections of the appropriate, sometimes reorganised, chapters. Hints and outline answers are given, as previously, in the final subsections, but only to the odd-numbered exercises. This leaves all even-numbered exercises free to be set as unaided homework, as described below.

    For the four hundred plus odd-numbered exercises, complete solutions are available, to both students and their teachers, in the form of this manual; these are in addition to the hints and outline answers given in the main text.

Page 114 of 160