Mathematical Methods

by Hassani, Sadri

• ISBN13: 9780387989587
• ISBN10: 0387989587
• Format: Hardcover
• Copyright: 2000-06-01
• Publisher: Springer Verlag

Summary

Intended to follow the usual introductory physics courses, this book has the unique feature of addressing the mathematical needs of sophomores and juniors in physics, engineering and other related fields. Many original, lucid, and relevant examples from the physical sciences, problems at the ends of chapters, and boxes to emphasize important concepts help guide the student through the material.Beginning with reviews of vector algebra and differential and integral calculus, the book continues with infinite series, vector analysis, complex algebra and analysis, ordinary and partial differential equations. Discussions of numerical analysis, nonlinear dynamics and chaos, and the Dirac delta function provide an introduction to modern topics in mathematical physics.This new edition has been made more user-friendly through organization into convenient, shorter chapters. Also, it includes an entirely new section on Probability and plenty of new material on tensors and integral transforms.Some praise for the previous edition:"The book has many strengths. For example: Each chapter starts with a preamble that puts the chapters in context. Often, the author uses physical examples to motivate definitions, illustrate relationships, or culminate the development of particular mathematical strands. The use of Maxwell's equations to cap the presentation of vector calculus, a discussion that includes some tidbits about what led Maxwell to the displacement current, is a particularly enjoyable example. Historical touches like this are not isolated cases; the book includes a large number of notes on people and ideas, subtly reminding the student that science and mathematics are continuing and fascinating human activities."--Physics Today "Very well written (i.e., extremely readable), very well targeted (mainly to an average student of physics at a point of just leaving his/her sophomore level) and very well concentrated (to an author's apparently beloved subject of PDE's with applications and with all their necessary pedagogically-mathematical background)...The main merits of the text are its clarity (achieved via returns and innovations of the context), balance (building the subject step by step) and originality (recollect: the existence of the complex numbers is only admitted far in the second half of the text!). Last but not least, the student reader is impressed by the graphical quality of the text (figures first of all, but also boxes with the essentials, summarizing comments in the left column etc.)...Summarizing: Well done."--Zentralblatt MATH

Table of Contents

Preface	vii
Note to the Reader	xi
Coordinate Systems and Vectors	1	(44)
Vectors in a Plane and in Space	1	(9)
Coordinate Systems	10	(5)
Vectors in Different Coordinate Systems	15	(17)
Relations Among Unit Vectors	32	(6)
Problems	38	(7)
Differentiation	45	(32)
The Derivative	46	(3)
Partial Derivatives	49	(13)
Elements of Length, Area, and Volume	62	(9)
Problems	71	(6)
Integration	77	(62)
``f'' Means ``f um''	77	(4)
Properties of Integral	81	(11)
Applications: General Guidelines	92	(7)
Applications: Single Integrals	99	(16)
Applications: Double Integrals	115	(8)
Applications: Triple Integrals	123	(7)
Problems	130	(9)
Vectors	139	(66)
Vectors in a Plane Revisited	140	(20)
Vectors in Space	160	(11)
Determinant	171	(5)
The Jacobian	176	(4)
Finite-Dimensional Vector Spaces	180	(13)
Orthogonal Polynomials	193	(4)
Systems of Linear Equations	197	(4)
Problems	201	(4)
Infinite Series	205	(59)
Infinite Sequences	205	(3)
Summations	208	(5)
Infinite Series	213	(8)
Sequences and Series of Functions	221	(6)
Power Series	227	(16)
Fourier Series	243	(6)
Multivariable Taylor Series	249	(3)
Application to Differential Equations	252	(4)
Problems	256	(8)
Integrals and Series as Functions	264	(25)
Integrals as Functions	264	(11)
Power Series as Functions	275	(10)
Problems	285	(4)
Dirac Delta Function	289	(31)
One-Variable Case	289	(13)
Two-Variable Case	302	(6)
Three-Variable Case	308	(7)
Problems	315	(5)
Vector Analysis	320	(93)
Solid Angle	321	(7)
Time Derivative of Vectors	328	(6)
The Gradient	334	(5)
Flux of a Vector Field	339	(5)
Flux Density = Divergence	344	(13)
The Line Integral	357	(5)
Curl of a Vector Field and Strokes' Theorem	362	(8)
Conservative Vector Fields	370	(6)
Double Del Operations	376	(5)
Maxwell's Equations	381	(5)
Curvilinear Vector Analysis	386	(13)
Vectors and Indices	399	(6)
Problems	405	(8)
Complex Arithmetic	413	(19)
Cartesian Form of Complex Numbers	413	(5)
Polar Form of Complex Numbers	418	(7)
Fourier Series Revisited	425	(3)
Problems	428	(4)
Complex Analysis	432	(42)
Complex Functions	432	(16)
Complex Series	448	(4)
Taylor and Laurent Series	452	(4)
Calculus of Residues	456	(11)
Problems	467	(7)
Differntial Equations	474	(45)
From PDEs to ODEs	475	(8)
First-Order Differential Equations	483	(11)
Second-Order Linear Differential Equations	494	(8)
SOLDEs with Constant Coefficients	502	(12)
Problems	514	(5)
Laplace's Equation	519	(68)
Uniqueness of Solutions	520	(2)
Cartesian Coordinates	522	(10)
Spherical Coordinates	532	(30)
Cylindrical Coordinates	562	(18)
Problems	580	(7)
Other PDEs of Mathematical Physics	587	(31)
The Heat Equation	587	(5)
The Schrodinger Equation	592	(15)
The Wave Equation	607	(8)
Problems	615	(3)
Nonlinear Dynamics and Chaos	618	(29)
Systems Obeying Iterated Maps	619	(9)
Systems Obeying DEs	628	(11)
Universality of Chaos	639	(5)
Problems	644	(3)
Bibliography	647	(2)
Index	649