Second Year Calculus

From Celestial Mechanics to Special Relativity
1991
ISBN: 978-0-387-97606-8
Verlag: Springer

Buch, Englisch, 404 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 1250 g

Reihe: Undergraduate Texts in Mathematics

Second Year Calculus: From Celestial Mechanics to Special Relativity covers multi-variable and vector calculus, emphasizing the historical physical problems which gave rise to the concepts of calculus. The book carries us from the birth of the mechanized view of the world in Isaac Newton's Mathematical Principles of Natural Philosophy in which mathematics becomes the ultimate tool for modelling physical reality, to the dawn of a radically new and often counter-intuitive age in Albert Einstein's Special Theory of Relativity in which it is the mathematical model which suggests new aspects of that reality. The development of this process is discussed from the modern viewpoint of differential forms. Using this concept, the student learns to compute orbits and rocket trajectories, model flows and force fields, and derive the laws of electricity and magnetism. These exercises and observations of mathematical symmetry enable the student to better understand the interaction of physics and mathematics.

Zielgruppe

Lower undergraduate

Autoren/Hrsg.

Bressoud, David M.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Elementare Analysis und Allgemeine Begriffe

Inhaltsverzeichnis

1 F=ma.- 1.1 Prelude to Newton’s Principia.- 1.2 Equal Area in Equal Time.- 1.3 The Law of Gravity.- 1.4 Exercises.- 1.5 Reprise with Calculus.- 1.6 Exercises.- 2 Vector Algebra.- 2.1 Basic Notions.- 2.2 The Dot Product.- 2.3 The Cross Product.- 2.4 Using Vector Algebra.- 2.5 Exercises.- 3 Celestial Mechanics.- 3.1 The Calculus of Curves.- 3.2 Exercises.- 3.3 Orbital Mechanics.- 3.4 Exercises.- 4 Differential Forms.- 4.1 Some History.- 4.2 Differential 1-Forms.- 4.3 Exercises.- 4.4 Constant Differential 2-Forms.- 4.5 Exercises.- 4.6 Constant Differential k-Forms.- 4.7 Prospects.- 4.8 Exercises.- 5 Line Integrals, Multiple Integrals.- 5.1 The Riemann Integral.- 5.2 Line Integrals.- 5.3 Exercises.- 5.4 Multiple Integrals.- 5.5 Using Multiple Integrals.- 5.6 Exercises.- 6 Linear Transformations.- 6.1 Basic Notions.- 6.2 Determinants.- 6.3 History and Comments.- 6.4 Exercises.- 6.5 Invertibility.- 6.6 Exercises.- 7 Differential Calculus.- 7.1 Limits.- 7.2 Exercises.- 7.3 Directional Derivatives.- 7.4 The Derivative.- 7.5 Exercises.- 7.6 The Chain Rule.- 7.7 Using the Gradient.- 7.8 Exercises.- 8 Integration by Pullback.- 8.1 Change of Variables.- 8.2 Interlude with Lagrange.- 8.3 Exercises.- 8.4 The Surface Integral.- 8.5 Heat Flow.- 8.6 Exercises.- 9 Techniques of Differential Calculus.- 9.1 Implicit Differentiation.- 9.2 Invertibility.- 9.3 Exercises.- 9.4 Locating Extrema.- 9.5 Taylor’s Formula in Several Variables.- 9.6 Exercises.- 9.7 Lagrange Multipliers.- 9.8 Exercises.- 10 The Fundamental Theorem of Calculus.- 10.1 Overview.- 10.2 Independence of Path.- 10.3 Exercises.- 10.4 The Divergence Theorems.- 10.5 Exercises.- 10.6 Stokes’ Theorem.- 10.7 Summary for R3.- 10.8 Exercises.- 10.9 Potential Theory.- 11 E = mc2.- 11.1 Prelude to Maxwell’s Dynamical Theory.-11.2 Flow in Space-Time.- 11.3 Electromagnetic Potential.- 11.4 Exercises.- 11.5 Special Relativity.- 11.6 Exercises.- Appendices.- A An Opportunity Missed 361.- B Bibliography 365.- C Clues and Solutions 367.- Index 382.