Prof. Dr. Mihai VOICU
1. The field of real numbers. The least upper bound. The greatest lower bound.
Numerical sequences. Cauchy criterion.
2. Numerical series. Criteria of convergence. Series with positive terms. Alternating
series. Approximation of the sum.
3. Sequence of functions. Simple and uniform convergence. Transfer of continuity.
Transfer of integrability.
4. Series of functions. Simple and uniform convergence. Weierstrass criterion.
5. Power series. Convergence set. Radius of convergence. Cauchy-Hadamard theorem.
Term by term derivation and integration of a power series.
6. Taylor’s formula. Taylor series. Expansion of a function in a Taylor
series. Examples.
7. Metric spaces. Elements of topology.
8. Sequences in a metric space. Complete metric spaces. as a metric space.
9. Scalar and vector functions of several variables. Limits and continuity.
10. Compact sets. Remarkable properties of continuous functions defined on compact
sets. Uniformly continuous functions.
11. Connected sets. A geometrical characterization of a connected set.
12. Contractions on metric spaces. Banach’s fixed point theorem. Applications.
13. Normed spaces. Continuous linear mappings.
14. Fréchet differential. Uniqueness. Properties.
15. Derivative with respect to a versor. Partial derivatives. Composite functions.
16. Jacobian matrix. Functional determinant. Diffeomorphism. Changes of variables.
Polar coordinates.
17. Local inversion theorem. Implicit functions. Applications.
18. n-th order partial derivatives. Taylor’s formula for real functions
of several variables.
19. Second order differential. Extreme points for real functions of several
variables. Geometrical applications.
20. Riemann’s integral. Criteria of integrability. Classes of integrable
functions.
21. Mean formulas. Indefinite integral. Change of variables.
22. Lagrange’s interpolation polynomial. Approximative computation of
the definite integral.
23. Improper integrals. Convergence criteria.
24. Integrals depending on a parameter. Continuity and derivability.
25. Euler’s functions (beta and gamma). Remarkable properties.
26. Jordan’s measure. Criteria of measurability.
27. Double integrals. Basic properties. Criteria of integrability.
28. Classes of integrable functions. Mean formulas. Computation. Applications
in geometry and mechanics.
29. Curvilinear integrals of the first and the second kind. Computation. Physical
interpretation. Riemann-Green formula. Applications.
30. Change of variables for the double integral. Triple integral. Computation.
Applications in mechanics.
31. Surface integrals. Basic properties and computation. Integral formulas (Gauss
and Stokes).