Series; ordinary differential equations; curves; real functions of more variables: differential and integral calculus, optimization.
Course Content - Last names L-Z
Series; ordinary differential equations; curves; integrals on a curve; real functions of more variables: differential and integral calculus, optimization.
To learn to manage series of real numbers and power series. Optimization of functions of two or more variables. To manage curves in the plane and in the space, to calculate their length and integrals over curves. Calculate simple double integrals, areas and volumes.
Learning Objectives - Last names L-Z
To learn lo apply theoretical results to practical situations. In particular:
to manage series of real numbers and power series. Optimization of functions of two or more variables. To manage curves in the plane and in the space, to calculate their length and integrals over curves. Calculate double integrals, areas and volumes.
Prerequisites - Last names A-K
Matematics 1
Prerequisites - Last names L-Z
Matematics 1
Teaching Methods - Last names A-K
Frontal lessons and exercises (individual and group sessions).
Teaching Methods - Last names L-Z
Frontal lessons and exercises
Further information - Last names A-K
Other suggested texts: Fusco-Marcellini-Sbordone, ANALISI MATEMATICA 2 + exercises.
Further information - Last names L-Z
Other suggested texts: Fusco-Marcellini-Sbordone, ANALISI MATEMATICA 2 + exercises.
Type of Assessment - Last names A-K
Two intermediate tests, plus, possibly a written test on the theory. In alternative, a final written exam divided into two parts (exercises and theory).
Type of Assessment - Last names L-Z
2 middle term tests or a final comprehensive written test. It may be necessary a further theory examination.
Course program - Last names A-K
This is the tentative program. For the effective program, please refer to the table of content of the lectures, available on Moodle.
1. Series. A necessary condition for convergence (dem.). Series with non-negative terms. Convergence criteria for series with non-negative terms: comparison test (with dem.), asymptotic comparison test (with dem.), root test, ratio test. Absolute convergence. Leibniz test for series with alternate sign terms. Series of functions (outline, total convergence). Power series: convergence radius, Taylor series, analytic functions.
2. Ordinary differential equations. Introduction: the Malthus model for the growth of an isolated population. Notion of differential equation. First order differential equations, Cauchy problem. Equations with separable variables. Linear first order equations: solution formula (with dem.). Bernoulli equations. Linear differential equations of the second order. Structure of the solution space. Identification of a base of the solution space and solution formula for homogeneous equations with constant coefficients (with dem.). Non-homogeneous case: search for a particular solution for similarity. The Cauchy problem for linear equations of the second order.
3. Curves in the plane (hints of curves in space). Rectifiable curves and regular curves. Curvilinear abscissa. Polar equation of a plane curve. Curvilinear integrals of the first type.
4. Differential calculus for functions of several variables. Open and closed subsets of the n-dimensional Euclidean space. Functions of several variables. Limits. Continuity and basic theorems. Partial derivatives. Differential. Differentiability and continuity (with proof). Equation of the tangent plane to the graph of a function of two variables. Directional derivatives: definition and formula for calculating a directional derivative through the gradient for differentiable functions. Schwarz's theorem for mixed second derivatives.
5.Optimization. Definition of (global and local) maximum and minimum points. Critical points, necessary condition on the gradient (with dem.). Necessary and sufficient conditions on the Hessian matrix at the critical points (dem. of the necessary condition). Search for absolute maxima and minima in compact sets for functions of two variables. Regular constraints in the plane and search for extremes on curves. Lagrange multipliers.
6. Multiple integrals. Definition of integral of a function of two variables on a rectangle and relative reduction formulas. Definition of integral on any domain and area of plan domains. Property of the integrals of functions of two variables. Normal domains of the plan: reduction formula, area of a normal domain. Formula for changing variables. Polar coordinates. Use of polar coordinates in double integrals. Vector fields and curved integrals of the second kind. Conservative fields. Gauss-Green formulas. Divergence theorem and stokes formula (with dim. exploiting Green formulas). Formula for the area of plane domains by curvilinear integral (dim. using Green's formulas).
Course program - Last names L-Z
This is the tentative program. For the effective program, please refer to the lessons register available on Moodle.
1. Series. A necessary condition for convergence (dem.). Series with non-negative terms. Convergence criteria for series with non-negative terms: comparison test (with dem.), asymptotic comparison test (with dem.), root test, ratio test. Absolute convergence. Leibniz test for series with alternate sign terms. Series of functions (outline, total convergence). Power series: convergence radius, Taylor series, analytic functions.
2. Ordinary differential equations. Introduction: the Malthus model for the growth of an isolated population. Notion of differential equation. First order differential equations, Cauchy problem. Equations with separable variables. Linear first order equations: solution formula (with dem.). Bernoulli equations. Linear differential equations of the second order. Structure of the solution space. Identification of a base of the solution space and solution formula for homogeneous equations with constant coefficients (with dem.). Non-homogeneous case: search for a particular solution for similarity. The Cauchy problem for linear equations of the second order.
3. Curves in the plane (hints of curves in space). Rectifiable curves and regular curves. Curvilinear abscissa. Polar equation of a plane curve. Curvilinear integrals of the first type.
4. Differential calculation for functions of several variables. Open and closed subsets of Rn. Functions of several variables. Limits. Continuity and basic theorems. Partial derivatives. Differential. Differentiable implies continuous (with dim.). Equation of the tangent plane to the graph of a function of two variables. Directional derivatives: definition and formula for calculating a directional derivative through the gradient for differentiable functions. Schwarz's theorem for mixed second derivatives.
5.Optimization. Definition of (global and local) maximum and minimum points. Critical points, necessary condition on the gradient (with dem.). Necessary and sufficient conditions on the Hessian matrix at the critical points (dem. of the necessary condition). Search for absolute maxima and minima in compact sets for functions of two variables. Regular constraints in the plane and search for extremes on curves. Lagrange multipliers.
6. Multiple integrals. Definition of integral of a function of two variables on a rectangle and relative reduction formulas. Definition of integral on any domain and area of plan domains. Property of the integrals of functions of two variables. Normal domains of the plan: reduction formula, area of a normal domain. Formula for changing variables. Polar coordinates. Use of polar coordinates in double integrals. Vector fields and curved integrals of the second kind. Conservative fields. Gauss-Green formulas. Divergence theorem and stokes formula (with dim. exploiting Green formulas). Formula for the area of plane domains by curvilinear integral (dim. using Green's formulas).