Continuum Mathematics

1. Elementary set theory. Basic notions on maps between sets. Relations on sets. Equivalence and order relations. Basics of enumerative combinatorics.
2. The sets N,Z,Q,R of natural numbers, relative integers, rational and real numbers, with their algebraic and order structures. Countability of Q, uncountability of R. Completeness of R. Supremum and infimum of a subset of R. Elementary topological notions in R. The spaces R^n (n=1,2,3, ).
3. A sketch of the abstract notion of vector space. Linear mappings between vector spaces and matrices. Linear equations.
4. Some generalities on real functions from a subset of R to R. Some elementary functions: power functions, exponentials, logarithms, trigonometric functions. Using trigonometric functions in acoustics. An invitation to Fourier series.
5. The notion of limit for functions from a subset of R to R; examples. Limits and algebraic operations on functions: indeterminate forms. The notion of continuity for functions from a subset of R to R. Some facts on continuous functions; the theorems of Darboux and Weierstrass. Continuity of elementary functions. Limits, continuity and composition of functions. Notable special limits. The limit defining the Napier number.
6. Real sequences and their limits. Real series: some examples and convergence criteria.
7. The notion of derivative for a function from a subset of R to R, and its geometrical meaning. An application: the velocity of a particle. Derivatives and algebraic operations on functions. Derivatives of the elementary functions. Derivatives of an inverse function and of the composition of two functions. The theorems of Rolle, Cauchy and Lagrange.
8. Higher order derivatives. An application of the second derivative: the acceleration of a particle. Taylor's formula with reminder in the Peano or in the Lagrange form. Use of Taylor's formula in the calculation of limits, and in the numerical computation of functions. Taylor's series.
9. Using derivatives to determine the maximum and minimum points of a function from a subset of R to R, as well as the intervals where the function is increasing, decreasing, convex or concave. Analyzing other aspects of the graph of such a function; asymptotes.
10. The notion of primitive function, or indefinite integral, for a real function on an interval. Some elementary indefinite integrals. Integration by parts and by substitution in the indefinite case.
11. The notion of Riemann's definite integral for real functions on an interval, and its geometrical meaning. The fundamental theorem of calculus. Integration by parts and by substitution in the definite case. An introduction to improper integrals. Using integrals to estimate finite sums and series.
12. The field C of complex numbers. Modulus, argument and trigonometric representation of a complex number. A sketch of the notions of limit, derivative and integral for complex valued functions. Complex sequences and series. The exponential function in the complex field; Euler's formula. Roots of complex numbers. The fundamental theorem of algebra.