Infinitesimal CalculusRigorous undergraduate treatment introduces calculus at the basic level, using infinitesimals and concentrating on theory rather than applications. Requires only a solid foundation in high school mathematics. Contents: 1. Introduction. 2. Language and Structure. 3. The Hyperreal Numbers. 4. The Hyperreal Line. 5. Continuous Functions. 6. Integral Calculus. 7. Differential Calculus. 8. The Fundamental Theorem. 9. Infinite Sequences and Series. 10. Infinite Polynomials. 11. The Topology of the Real Line. 12. Standard Calculus and Sequences of Functions. Appendixes. Subject Index. Name Index. Numerous figures. 1979 edition. |
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a₁ an+1 Assume calculus Cauchy Cauchy sequence Chapter closed sets cofinite compact constant continuous function converges curve decimal defined derivative equal equation equicontinuous example EXERCISE Find exists f and g f is continuous f is differentiable finite hyperreal formula func function f function symbol Fundamental Theorem geometry given h₁ hence Hint hyperreal number system implies infinite integer infinite polynomials infinitely close infinitesimal 4x Intermediate Value Theorem interval inverse irrational language least upper bound Leibniz limit point mathematicians mathematics Mean Value Theorem natural numbers nonstandard definition nonstandard number open sets polynomials positive integer positive real primitive of f problem rational number real number rectangles relation symbol Riemann-integrable Rolle's Theorem Sf(x slope Sof(x)dx standard definition Suppose tangent Taylor's Theorem tesimal theorem 7.3 tion true in HR x₁ ΔΧ Σα Дх