Very Short Introductions: Brilliant, sharp, inspiring
The 17th-century calculus of Newton and Leibniz was built on shaky foundations, and it wasn't until the 18th and 19th centuries that mathematicians--especially Bolzano, Cauchy, and Weierstrass--began to establish a rigorous basis for the subject. The resulting discipline is now known to mathematicians as analysis.
This book, aimed at readers with some grounding in mathematics, describes the nascent evolution of mathematical analysis, its development as a subject in its own right, and its wide-ranging applications in mathematics and science, modelling reality from acoustics to fluid dynamics, from biological systems to quantum theory.
ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
Author(s): Richard Earl
Series: Very Short Introductions
Edition: 1
Publisher: Oxford University Press
Year: 2023
Language: English
Pages: 224
City: New York
Tags: Calculus; Newton; Leibniz; Bolzano; Cauchy; Weierstrass; Analysis; Infinity; Functions; Derivatives; Integrals
Cover
Mathematical Analysis: A Very Short Introduction
Copyright
Dedication
Contents
Acknowledgements
List of illustrations
Chapter 1. Taming infinity
Actual and potential infinities
How is π calculated?
Defining convergence
Countable versus uncountable
Axioms and some early results
Modern analysis
Chapter 2. All change . . . the calculus of Fermat, Newton, and Leibniz
Calculus
The 17th century
Synthetic versus analytic geometry
Analytic geometry and the function concept
Pierre de Fermat
The fundamental theorem of calculus
The calculus of Newton and Leibniz
Newton’s physics
The Newton–Leibniz controversy
Chapter 3. To the limit: analysis in the 18th and 19th centuries
e and the exponential function
Logarithms and powers
Power series and Taylor series
Radians and trigonometry
Euler
Differential equations
Bolzano and Weierstrass
Riemann’s integral
Chapter 4. Should I believe my computer?
Interpolation and extrapolation
Numerical differentiation and integration
Numerical stability and error analysis
Linearization and stability
Chapter 5. Dimensions aplenty
Scalars and vectors
Directional and partial derivatives
Partial differentiation equations
The calculus of variations
Multivariable integration
Surface integrals and flux
Line integrals and work
Stokes’ theorem and the divergence theorem
Chapter 6. I’ll name that tune in . . .
The wave equation
Derivation of the wave equation
Boundary value problems
Fourier analysis
The impact of Fourier’s work
Can you hear the shape of a drum?
The spectral theorem
Distributions
Quantum theory
Chapter 7. Putting the i in analysis
Complex numbers
Cauchy
The complex plane
Two maps of the complex plane
Complex differentiability and the Cauchy–Riemann equations
Holomorphic functions
Complex trigonometric functions and the exponential function
Taylor series and Laurent series
Complex integrals
Cauchy’s residue theorem
Conformal maps and applications
Chapter 8. But there’s more . . .
Lebesgue integration
Measure theory
Analytic number theory
Hyperreal numbers
Epilogue
Appendix
Chapter 1
Ramanujan’s approximation to π
Cantor’s proof that the real numbers are uncountable
The axioms of the real numbers
Chapter 2
The equation of the cissoid of Diocles
Chapter 3
Basic identities of the exponential and logarithmic functions
A trigonometric identity
Rederiving Madhava’s sum
e is irrational
Euler’s first solution of the Basel problem
Chapter 4
Deriving Newton’s method
The stable Lotka–Volterra equilibrium
Chapter 5
Minimizing the least-squares error
Lines are the shortest curves
Stokes’ theorem implies Green’s theorem
Chapter 6
Deriving the Fourier coefficients
Chapter 7
Deriving the Cauchy–Riemann equations
Proving Cauchy’s theorem from Green’s theorem
Complex logarithm and powers
Evaluating an integral with the residue theorem
The real and imaginary parts satisfy Laplace’s equation
Chapter 8
A countable set is null
There are infinitely many primes
Historical timeline
References and further reading
References
Index
Numbers
Statistics
Information
Chaos
