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An Elementary Treatise on Fourier's Series
and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical
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- 304 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android
eBook - ePub
An Elementary Treatise on Fourier's Series
and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical
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About This Book
Originally published over a century ago, this work remains among the most useful and practical expositions of Fourier's series, and spherical, cylindrical, and ellipsoidal harmonics. The subsequent growth of science into a diverse range of specialties has enhanced the value of this classic, whose thorough, basic treatment presents material that is assumed in many other studies but seldom available in such concise form. The development of functions, series, and their differential equations receives detailed explanations, and throughout the text, theory is applied to practical problems, with the solutions fully worked out. In addition, 190 problems, many with hints, are included. 1893 edition. Appendix of 6 tables.
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Yes, you can access An Elementary Treatise on Fourier's Series by William Elwood Byerly in PDF and/or ePUB format, as well as other popular books in Matematica & Analisi matematica. We have over one million books available in our catalogue for you to explore.
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Topic
MatematicaSubtopic
Analisi matematicaCHAPTER I.
INTRODUCTION.
1.In many important problems in mathematical physics we are obliged to deal with partial differential equations of a comparatively simple form.
For example, in the Analytical Theory of Heat we have for the change of temperature of any solid due to the flow of heat within the solid, the equation
where u represents the temperature at any point of the solid and t the time.
In the simplest case, that of a slab of infinite extent with parallel plane faces, where the temperature can be regarded as a function of one coƶrdinate, [I] reduces to
a form of considerable importance in the consideration of the problem of the cooling of the earthās crust.
In the problem of the permanent state of temperatures in a thin rectangular plate, the equation [I] becomes
In polar or spherical coƶrdinates [I] is less simple, it is
In the case where the solid in question is a sphere and the temperature at any point depends merely on the distance of the point from the centre [IV] reduces to
In cylindrical coƶrdinates [I] becomes
In considering the flow of heat in a cylinder when the temperature at any point depends merely on the distance r of the point from the axis [VI] becomes
In Acoustics in several problems we have the equation
for instance, in considering the transverse or the longitudinal vibrations of a stretched elastic string, or the transmission of plane sound waves through the air.
If in considering the transverse vibrations of a stretched string we take account of the resistance of the air [VIII] is replaced by
In dealing with the vibrations of a stretched elastic membrane, we have the equation
or in cylindrical coƶrdinates
In the theory of Potential we constantly meet Laplaceās Equation
which in spherical coƶrdinates becomes
and in cylindrical coƶrdinates
In curvilinear coƶrdinates it is
represent a set of surfaces which cut one another at right angles, no matter what values are given to Ļ1, Ļ2, and Ļ3; and where
and, of course, must be expressed in terms of Ļ1, Ļ2, and Ļ3.
If it happens that ā2Ļ1 = 0, ā2Ļ2 = 0, and ā2Ļ3 = 0, then Laplaceās Equation [XV] assumes the very simple form
...
Table of contents
- Cover Page
- Title Page
- Copyright Page
- Preface
- Contents
- Chapter I. Introduction
- Chapter II. Development In Trigonometric Series
- Chapter III. Convergence of Fourierās Series
- Chapter IV. Solution of Problems in Physics by the Aid of Fourierās Integrals and Fourierās Series
- Chapter V. Zonal Harmonics
- Chapter VI. Spherical Harmonics
- Chapter VII. Cylindrical Harmonics (Besselās Functions)
- Chapter VIII. Laplaceās Equation in Curvilinear Coordinates. Ellipsoidal Harmonics
- Chapter IX. Historical Summary
- Appendix