eBook Math Explained: The Britannica Guide to Analysis and Calculus, 1st Edition

  • Published By:
  • ISBN-10: 1615302204
  • ISBN-13: 9781615302208
  • DDC: 515
  • Grade Level Range: 9th Grade - 12th Grade
  • 288 Pages | eBook
  • Original Copyright 2011 | Published/Released April 2012
  • This publication's content originally published in print form: 2011
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About

Overview

The dynamism of the natural world means that it is constantly changing, sometimes rapidly, sometimes gradually. By mathematically interpreting the continuous change that characterizes so many natural processes, analysis and calculus have become indispensable to bridging the divide between mathematics and the sciences. This comprehensive volume examines the key concepts of calculus, providing students with a robust understanding of integration and differentiation. Biographies of important figures will leave readers with an increased appreciation for the sometimes competing theories that informed the early history of the field.

Table of Contents

Front Cover.
Half Title Page.
Title Page.
Copyright Page.
Contents.
Introduction.
1: Measuring Continuous Change.
2: Bridging the Gap between Arithmetic and Geometry.
3: Discovery of the Calculus and the Search for Foundations.
4: Numbers and Functions.
5: Number Systems.
6: Functions.
7: The Problem of Continuity.
8: Approximations in Geometry.
9: Infinite Series.
10: The Limit of a Sequence.
11: Continuity of Functions.
12: Properties of the Real Numbers.
13: Calculus.
14: Differentiation.
15: Average Rates of Change.
16: Instantaneous Rates of Change.
17: Formal Definition of the Derivative.
18: Graphical Interpretation.
19: Higher-Order Derivatives.
20: Integration.
21: The Fundamental Theorem of Calculus.
22: Antidifferentiation.
23: The Riemann Integral.
24: Differential Equations.
25: Ordinary Differential Equations.
26: Newton and Differential Equations.
27: Dynamical Systems Theory and Chaos.
28: Partial Differential Equations.
29: Musical Origins.
30: Partial Derivatives.
31: D'Alembert's Wave Equation.
32: Trigonometric Series Solutions.
33: Fourier Analysis.
34: Other Areas of Analysis.
35: Complex Analysis.
36: Formal Definition of Complex Numbers.
37: Extension of Analytic Concepts to Complex Numbers.
38: Some Key Ideas of Complex Analysis.
39: Measure Theory.
40: Functional Analysis.
41: Variational Principles and Global Analysis.
42: Constructive Analysis.
43: Nonstandard Analysis.
44: History of Analysis.
45: The Greeks Encounter Continuous Magnitudes.
46: The Pythagoreans and Irrational Numbers.
47: Zeno's Paradoxes and the Concept of Motion.
48: The Method of Exhaustion.
49: Models of Motion in Medieval Europe.
50: Analytic Geometry.
51: The Fundamental Theorem of Calculus.
52: Differentials and Integrals.
53: Discovery of the Theorem.
54: Calculus Flourishes.
55: Elaboration and Generalization.
56: Euler and Infinite Series.
57: Complex Exponentials.
58: Functions.
59: Fluid Flow.
60: Rebuilding the Foundations.
61: Arithmetization of Analysis.
62: Analysis in Higher Dimensions.
63: Great Figures in the History of Analysis.
64: The Ancient and Medieval Period.
65: Archimedes.
66: Euclid.
67: Eudoxus of Cnidus.
68: Ibn Al-Haytham.
69: Nicholas Oresme.
70: Pythagoras.
71: Zeno of Elea.
72: The 17th and 18th Centuries.
73: Jean Le Rond d'Alembert.
74: Isaac Barrow.
75: Daniel Bernoulli.
76: Jakob Bernoulli.
77: Johann Bernoulli.
78: Bonaventura Cavalieri.
79: Leonhard Euler.
80: Pierre de Fermat.
81: James Gregory.
82: Joseph-Louis Lagrange, Comte de l'Empire.
83: Pierre-Simon, marquis de Laplace.
84: Gottfried Wilhelm Leibniz.
85: Colin Maclaurin.
86: Sir Isaac Newton.
87: Gilles Personne de Roberval.
88: Brook Taylor.
89: Evangelista Torricelli.
90: John Wallis.
91: The 19th and 20th Centuries.
92: Stefan Banach.
93: Bernhard Bolzano.
94: Luitzen Egbertus Jan Brouwer.
95: Augustin-Louis, Baron Cauchy.
96: Richard Dedekind.
97: Joseph, Baron Fourier.
98: Carl Friedrich Gauss.
99: David Hilbert.
100: Andrey Kolmogorov.
101: Henri-Léon Lebesgue.
102: Henri Poincaré.
103: Bernhard Riemann.
104: Stephen Smale.
105: Karl Weierstrass.
106: Concepts in Analysis and Calculus.
107: Algebraic versus Transcendental Objects.
108: Argand Diagram.
109: Bessel Function.
110: Boundary Value.
111: Calculus of Variations.
112: Chaos Theory.
113: Continuity.
114: Convergence.
115: Curvature.
116: Derivative.
117: Difference Equation.
118: Differential.
119: Differential Equation.
120: Differentiation.
121: Direction Field.
122: Dirichlet Problem.
123: Elliptic Equation.
124: Exact Equation.
125: Exponential Function.
126: Extremum.
127: Fluxion.
128: Fourier Transform.
129: Function.
130: Harmonic Analysis.
131: Harmonic Function.
132: Infinite Series.
133: Infinitesimals.
134: Infinity.
135: Integral.
136: Integral Equation.
137: Integral Transform.
138: Integraph.
139: Integration.
140: Integrator.
141: Isoperimetric Problem.
142: Kernel.
143: Lagrangian Function.
144: Laplace's Equation.
145: Laplace Transform.
146: Lebesgue Integral.
147: Limit.
148: Line Integral.
149: Mean-Value Theorem.
150: Measure.
151: Minimum.
152: Newton and Infinite Series.
153: Ordinary Differential Equation.
154: Orthogonal Trajectory.
155: Parabolic Equation.
156: Partial Differential Equation.
157: Planimeter.
158: Power Series.
159: Quadrature.
160: Separation of Variables.
161: Singular Solution.
162: Singularity.
163: Special Function.
164: Spiral.
165: Stability.
166: Sturm-Liouville Problem.
167: Taylor Series.
168: Variation of Parameters.
Glossary.
Bibliography.
Index.