A primer of infinitesimal analysis [E-Book] / John L. Bell.
Bell, J. L., (author)
Second edition.
Cambridge : Cambridge University Press, 2008
1 online resource (xi, 124 pages)
Full Text
Table of Contents:
  • Basic features of smooth worlds
  • Basic differential calculus
  • The derivative of a function
  • Stationary points of functions
  • Areas under curves and the constancy principle
  • The special functions
  • First applications of the differential calculus
  • Areas and volumes
  • Volumes of revolution
  • Arc length; surfaces of revolution; curvature
  • Application to physics
  • Moments of inertia
  • Centres of mass
  • Pappus' theorems
  • Centres of pressure
  • Stretching a spring
  • Flexure of beams
  • The catenary, the loaded chain, and the bollard-rope
  • The Kepler-Newton areal law of motion under a central force
  • Multivariable calculus and applications
  • Partial derivatives
  • Stationary values of functions
  • Theory of surfaces. Spacetime metrics
  • The heat equation
  • The basic equations of hydrodynamics
  • The wave equation
  • The Cauchy-Riemann equations for complex functions
  • The definite integral. Higher-order infinitesimals
  • The definite integral
  • Higher-order infinitesimals and Taylor's theorem
  • The three natural microneighbourhoods of zero
  • Synthetic differential geometry
  • Tangent vectors and tangent spaces
  • Vector fields
  • Differentials and directional derivatives
  • Smooth infinitesimal analysis as an axiomatic system
  • Natural numbers in smooth worlds
  • Nonstandard analysis.