Basic Training in Mathematics

by Shankar, Ramamurti

• ISBN13: 9780306450358
• ISBN10: 0306450356
• Format: Hardcover
• Copyright: 1995-08-01
• Publisher: Plenum Pub Corp

Summary

Based on course material used by the author at Yale University, this practical text addresses the widening gap found between the mathematics required for upper-level courses in the physical sciences and the knowledge of incoming students. This superb book offers students an excellent opportunity to strengthen their mathematical skills by solving various problems in differential calculus. By covering material in its simplest form, students can look forward to a smooth entry into any course in the physical sciences.

Table of Contents

Differential Calculus of one Variable	1	(32)
Introduction	1	(1)
Differential Calculus	1	(4)
Exponential and Log Functions	5	(14)
Trigonometric Functions	19	(4)
Plotting Functions	23	(2)
Miscellaneous Problems on Differential Calculus	25	(4)
Differentials	29	(1)
Summary	30	(3)
Integral Calculus	33	(18)
Basics of Integration	33	(11)
Some Tricks of the Trade	44	(5)
Summary	49	(2)
Calculus of Many Variables	51	(24)
Differential Calculus of Many Variables	51	(10)
Lagrange multipliers	55	(6)
Integral Calculus of Many Variables	61	(11)
Solid angle	68	(4)
Summary	72	(3)
Infinite Series	75	(14)
Introduction	75	(2)
Tests for Convergence	77	(3)
Power Series in x	80	(7)
Summary	87	(2)
Complex Numbers	89	(18)
Introduction	89	(1)
Complex Numbers in Cartesian Form	90	(4)
Polar Form of Complex Numbers	94	(4)
An Application	98	(6)
Summary	104	(3)
Functions of a Complex Variable	107	(42)
Analytic Functions	107	(9)
Singularities of analytic functions	111	(3)
Derivatives of analytic functions	114	(2)
Analytic Functions Defined by Power Series	116	(10)
Calculus of Analytic Functions	126	(6)
The Residue Theorem	132	(7)
Taylor Series for Analytic Functions	139	(5)
Summary	144	(5)
Vector Calculus	149	(56)
Review of Vectors Analysis	149	(6)
Time Derivatives of Vectors	155	(3)
Scalar and Vector Fields	158	(1)
Line and Surface Integrals	159	(8)
Scalar Field and the Gradient	167	(5)
Gradient in other coordinates	171	(1)
Curl of a Vector Field	172	(10)
Curl in other coordinate systems	181	(1)
The Divergence of a Vector Field	182	(4)
Divergence in other coordinate systemes	186	(1)
Differential Operators	186	(2)
Summary of Integral Theorems	188	(1)
Higher Derivatives	188	(4)
Digression on vector identities	190	(2)
Applications from Electrodynamics	192	(10)
Summary	202	(3)
Matrices and Determinants	205	(24)
Introduction	205	(6)
Matrix Inverses	211	(4)
Determinants	215	(5)
Transformations on Matrices and Special Matrices	220	(7)
Action on row vectors	223	(4)
Summary	227	(2)
Linear Vector Spaces	229	(76)
Linear Vector Spaces: Basics	229	(8)
Inner Product Spaces	237	(10)
Linear Operators	247	(5)
Some Advanced Topics	252	(3)
The Eigenvalue Problem	255	(11)
Degeneracy	263	(3)
Applications of Eigenvalue Theory	266	(11)
Normal modes of vibration: two coupled masses	266	(8)
Quadratic forms	274	(3)
Function Spaces	277	(17)
Generation of orthonormal bases	279	(8)
Fourier integrals	287	(2)
Normal modes: vibrating string	289	(5)
Some Terminology	294	(1)
Tensors: An Introduction	294	(6)
Summary	300	(5)
Differential Equations	305	(46)
Introduction	305	(2)
ODEs with Constant Coefficients	307	(8)
ODEs with Variable Coefficients: First Order	315	(3)
ODEs with Variable Coefficients: Second Order and Homogeneous	318	(11)
Frobenius method for singular coefficients	325	(4)
Partial Differential Equations	329	(16)
The wave equation in one and two space dimensions	329	(8)
The heat equation in one and two dimensions	337	(8)
Green's Function Method	345	(2)
Summary	347	(4)
Answers	351	(8)
Index	359