1. Tutorial.- 1.1 A new view of old sets.- 1.2 Using the extended language.- 1.3 Shadows and S-properties.- 1.4 Permanence principles.- 2. Complex analysis.- 2.1 Introduction.- 2.2 Tutorial.- 2.3 Complex iteration.- 2.4 Airy's equation.- 2.5 Answers to exercises.- 3. The Vibrating String.- 3.1 Introduction.- 3.2 Fourier analysis of (DEN).- 3.3 An interesting example.- 3.4 Solutions of limited energy.- 3.5 Conclusion.- 4. Random walks and stochastic differential equations.- 4.1 Introduction.- 4.2 The Wiener walk with infinitesimal steps.- 4.3 Equivalent processes.- 4.4 Diffusions. Stochastic differential equations.- 4.5 Probability law of a diffusion.- 4.6 Ito's calculus - Girsanov's theorem.- 4.7 The "density" of a diffusion.- 4.8 Conclusion.- 5. Infinitesimal algebra and geometry.- 5.1 A natural algebraic calculus.- 5.2 A decomposition theorem for a limited point.- 5.3 Infinitesimal riemannian geometry.- 5.4 The theory of moving frames.- 5.5 Infinitesimal linear algebra.- 6. General topology.- 6.1 Halos in topological spaces.- 6.2 What purpose do halos serve ?.- 6.3 The external definition of a topology.- 6.4 The power set of a topological space.- 6.5 Set-valued mappings and limits of sets.- 6.6 Uniform spaces.- 6.7 Answers to the exercises.- 7. Neutrices, external numbers, and external calculus.- 7.1 Introduction.- 7.2 Conventions; an example.- 7.3 Neutrices and external numbers.- 7.4 Basic algebraic properties.- 7.5 Basic analytic properties.- 7.6 Stirling's formula.- 7.7 Conclusion.- 8. An external probability order theorem with applications.- 8.1 Introduction.- 8.2 External probabilities.- 8.3 External probability order theorems.- 8.4 Weierstrass, Stirling, De Moivre-Laplace.- 9. Integration over finite sets.- 9.1 Introduction.- 9.2 S-integration.-9.3 Convergence in SL1(F).- 9.4 Conclusion.- 10. Ducks and rivers: three existence results.- 10.1 The ducks of the Van der Pol equation.- 10.2 Slow-fast vector fields.- 10.3 Robust ducks.- 10.4 Rivers.- 11. Teaching with infinitesimals.- 11.1 Meaning rediscovered.- 11.2 the evidence of orders of magnitude.- 11.3 Completeness and the shadows concept.- References.- List of contributors.