Approximating Perfection

by Lebedev, L. P.

• ISBN13: 9780691117263
• ISBN10: 0691117268
• Format: Hardcover
• Copyright: 2004-04-12
• Publisher: Princeton Univ Pr

Summary

This is a book for those who enjoy thinking about how and why Nature can be described using mathematical tools.Approximating Perfectionconsiders the background behind mechanics as well as the mathematical ideas that play key roles in mechanical applications. Concentrating on the models of applied mechanics, the book engages the reader in the types of nuts-and-bolts considerations that are normally avoided in formal engineering courses: how and why models remain imperfect, and the factors that motivated their development. The opening chapter reviews and reconsiders the basics of calculus from a fully applied point of view; subsequent chapters explore selected topics from solid mechanics, hydrodynamics, and the natural sciences. Emphasis is placed on the logic that underlies modeling in mechanics and the many surprising parallels that exist between seemingly diverse areas. The mathematical demands on the reader are kept to a minimum, so the book will appeal to a wide technical audience.

Table of Contents

Preface	vii
The Tools of Calculus	1	(66)
Is Mathematical Proof Necessary?	2	(4)
Abstraction, Understanding, Infinity	6	(2)
Irrational Numbers	8	(3)
What Is a Limit?	11	(4)
Series	15	(4)
Function Continuity	19	(2)
How to Measure Length	21	(12)
Antiderivatives	33	(2)
Definite Integral	35	(7)
The Length of a Curve	42	(2)
Multidimensional Integrals	44	(3)
Approximate Integration	47	(5)
On the Notion of a Function	52	(1)
Differential Equations	53	(6)
Optimization	59	(2)
Petroleum Exploration and Recovery	61	(2)
Complex Variables	63	(2)
Moving On	65	(2)
The Mechanics of Continua	67	(84)
Why Do Ships Float?	67	(4)
The Main Notions of Classical Mechanics	71	(3)
Forces, Vectors, and Objectivity	74	(2)
More on Forces; Statics	76	(4)
Hooke's Law	80	(4)
Bending of a Beam	84	(10)
Stress Tensor	94	(6)
Principal Axes and Invariants of the Stress Tensor	100	(2)
On the Continuum Model and Limit Passages	102	(2)
Equilibrium Equations	104	(4)
The Strain Tensor	108	(5)
Generalized Hooke's Law	113	(1)
Constitutive Laws	114	(1)
Boundary Value Problems	115	(3)
Setup of Boundary Value Problems of Elasticity	118	(2)
Existence and Uniqueness of Solution	120	(6)
Energy; Minimal Principle for a Spring	126	(2)
Energy in Linear Elasticity	128	(4)
Dynamic Problems of Elasticity	132	(2)
Oscillations of a String	134	(3)
Lagrangian and Eulerian Descriptions of Continuum Media	137	(3)
The Equations of Hydrodynamics	140	(2)
D'Alembert---Euler Equation of Continuity	142	(2)
Some Other Models of Hydrodynamics	144	(1)
Equilibrium of an Ideal Incompressible Liquid	145	(3)
Force on an Obstacle	148	(3)
Elements of the Strength of Materials	151	(50)
What Are the Problems of the Strength of Materials?	151	(1)
Hooke's Law Revisited	152	(5)
Objectiveness of Quantities in Mechanics Revisited	157	(2)
Plane Elasticity	159	(2)
Saint-Venant's Principle	161	(2)
Stress Concentration	163	(2)
Linearity vs. Nonlinearity	165	(1)
Dislocations, Plasticity, Creep, and Fatigue	166	(6)
Heat Transfer	172	(3)
Thermoelasticity	175	(2)
Thermal Expansion	177	(1)
A Few Words on the History of Thermodynamics	178	(2)
Thermodynamics of an Ideal Gas	180	(2)
Thermodynamics of a Linearly Elastic Rod	182	(4)
Stability	186	(2)
Static Stability of a Straight Beam	188	(5)
Dynamical Tools for Studying Stability	193	(2)
Additional Remarks on Stability	195	(3)
Leak Prevention	198	(3)
Some Questions of Modeling in the Natural Sciences	201	(18)
Modeling and Simulation	201	(2)
Computerization and Modeling	203	(3)
Numerical Methods and Modeling in Mechanics	206	(2)
Complexity in the Real World	208	(1)
The Role of the Cosine in Everyday Measurements	209	(2)
Accuracy and Precision	211	(2)
How Trees Stand Up against the Wind	213	(3)
Why King Kong Cannot Be as Terrible as in the Movies	216	(3)
Afterword	219	(2)
Recommended Reading	221	(2)
Index	223