Mathematics analysis II

Sequences and series of functions: Sequences of functions: pointwise and uniform convergence. Continuity of the limit. Theorems of passage to the limit under the sign of integral and derivative. Series of functions: definition of pointwise, uniform, absolute and total convergence. Integration and derivation theorems for series. Power series. Criteria of D’Alembert and Cauchy-Hadamard. Taylor series.

Functions of two variables: Outline of the vector space R2 – Cauchy-Schwarz inequality – Elements of topology of R2 – Limits and continuity – Necessary condition for the existence of the limit – Partial derivatives – Higher order  derivatives – Schwarz theorem – Gradient – Differentiability – Differential theorem – Chain rule – Directional derivatives – Directional derivative of a differentiable function – Geometric interpretation of the gradient of a differentiable function – Functions with zero gradient in a connected – Taylor’s formula at the second order with the Lagrange remainder – Relative maxima and minima – Necessary condition of the first order – Sufficient condition of the second order – Absolute maxima and minima.

Differential equations: Cauchy problem: existence and local uniqueness theorem – existence and global uniqueness theorem (no proof) General properties of linear differential equations-General integral of a linear differential equation-Homogeneous linear differential equations-Linear differential equations. Wronskian determinant. Wronskian theorem. Theorem on the general integral of a linear differential equation. Inhomogeneous second order linear differential equations with known term of a particular type-Method of variation of constants. Differential equations with separable variables-Bernoulli’s differential equations. Missing equations of x and y. Curvilinear integrals and differential forms;

Curves in space: Simple, closed and regular curves – Tangent and normal versor – Length of a curve – Oriented curves – Curvilinear abscissa – Curvilinear integral of a function – Independence of the curvilinear integral of a function from the parametric representation of the curve

Differential forms: Exact differential forms – Curvilinear integral of a differential form and its physical interpretation – Integration theorem of exact differential forms – Characterization theorem of exact differential forms – Closed differential forms – Differential forms theorem in a rectangle – Differential forms theorem in a simply connected open

Double and triple integrals: Integrals on normal domains-Integrability of continuous functions-First Guldino theorem- Gauss-Green formulas-Divergence theorem-Stokes formula-Formulas for the calculation of the area of a regular domain. Formula of the change of variables in double integrals – Change of variables in polar coordinates – Other changes of variables – Triple integrals – Formula of change of variables in triple integrals – Change of variables in spherical and cylindrical coordinates

Implicit functions: Introduction to implicit functions – Dini’s theorem for implicit functions of one variable – Constrained maxima and minima Theorem of Lagrange multipliers in two dimensions.

Surfaces: Regular surfaces: definition and examples – Tangent plane and normal versor – Area of a regular surface. Surface integrals. Guldino’s theorem on surfaces of rotation. Divergence theorem and Stokes formula in space (without proof.)