Teacher

CATTANI Carlo
(syllabus)
DIFFERENTIAL EQUATIONS 1. Equations with separable variables, Cauchy theorem, solution calculation 2. Linear equations of the first order with variable, homogeneous and nonhomogeneous coefficients, solution calculation. 3. Homogeneous equations of the second order with constant coefficients, calculation of the homogeneous solution, characteristic equation, positive, negative and zero discriminant, solution independence, Wronskian, calculation of integration constants with initial conditions (Cauchy problem). 4. Nonhomogeneous equations of the second order with constant coefficients, calculation of the solution as a combination of the homogeneous solution and a particular integral, calculation of the particular integral with the constant variation method, calculation of the particular integral with the method of verisimilitude (comparison), in the case in which the known function is a polynomial of degree n, an exponential function, a trigonometric function. 5. Study of first and second order equations: Malthus equation and logistic equation. 6. First order autonomous equations, analysis of critical points, qualitative analysis 7. Particular equations: of Bernouilli, homogeneous, generalized homogeneous, of D’Alembert, of Clairaut, Implicit differential equations of the type or of the type Pure normal equations of order n of the type, problem of the supported beam 8. Autonomous systems of the first order with constant coefficients, analysis of critical points by eigenvalues and eigenvectors of the system matrix, orbits, classification of the critical point: stable, asymptotically stable, unstable. Systems in 2 dimensions: classification of orbits, node, center, fire,(or saddle point).
DIFFERENTIAL CALCULATION OF ALGEBRAIC CURVES 1. Vectorvalued functions from a 2. Vector calculus: linear dependence and independence, vector spaces, operations between vectors, between vectors and scalars, vector product, scalar product and mixed product. Operations between vectors in intrinsic form and in terms of components. Limits and derivatives of vectors 3. Plane curves in explicit, implicit and parametric form. Curves in polar coordinates. Conics in polar form. Closed, open, simple and regular curves. 4. Correctable and nonrectifiable curve. Length of a rectifiable curve, curvilinear abscissa. Curvilinear integral, calculation of the center of gravity and moment of inertia of a material line. 5. Mobile data: tangent, normal and binormal vector. Curvature and torsion. Formulas for calculating the curvature and torsion according to the curvilinear abscissa and any parameter. Curvature and torsion calculation for the cylindrical propeller. Frenet differential formulas. DIFFERENTIAL CALCULATION FOR MORE VARIABLE FUNCTIONS 1. Functions from to 2. Graphs, level sets and sets of definition of functions of several variables 3. Limits and continuity for functions of several variables. Calculation of limits, restriction method and polar coordinates. 4. Open, closed, unopened or closed sets. Border points. Interior of a whole. Closing of a set. Together simply connected. Convex sets. 5. Weierstrass theorem on the existence of max and min (only statement), theorem of existence of zeros (only statement). Examples. 6. Partial derivatives, tangent plan 7. Differentiability, gradient, directional derivatives. Direction of maximum growth and zero growth of a function 8. Second differential, successive derivatives, Schwarz theorem. Hessian matrix, Taylor's secondorder formula, Lagrange's and Peano's rest. 9. Elementary quadratic forms. Defined, semidefinite and indefinite forms. Fermat's theorem. Max and min points Saddle points. Restriction method for saddle points. 10. Free extremes. Study of the nature of the critical points through the minors of the Hessian or through its eigenvalues. 11. Bound extremes. Lagrange multipliers.
INTEGRAls CALCULATION FOR MORE VARIABLE FUNCTIONS 1. Double integrals: integral of a bounded function defined on a rectangle, reduction formula for rectangular sets, reduction for separable functions defined on rectangular sets. Functions that can be integrated on nonrectangular domains. Normal (simple) sets. Regular sets. Measurable set. Measurement sets zero. Elementary properties of the double integral. 2. Calculation of double integrals: reduction method for normal (simple) domains. Calculation of centroids and moments of inertia. Calculation of double integrals: change of variables. Jacobian. Change from Cartesian to polar coordinates. Double generalized integrals. Improper integrals. Integral of the Gaussian. 3. Calculation of triple integrals. Integration "by wire". "Layered" integration. Change of variable in triple integrals. Geometric and physical interpretations of triple integrals: volume, mass, center of gravity, moment of inertia. VECTOR FIELDS (Chapter 4, 6 of the theory book) 1. Operators: gradient, rotor and divergence. Properties of operators. Iteration of differential operators. Differential identities. 2. Line integral of a vector field. Work of a vector field and circuitry. Line integrals of the first species and second species. Conservative and potential fields. Expression of work for a conservative field as a function of potential. Irrotational fields. Simply connected sets. Sufficient condition on the whole because an irrotational field defined in both conservative. 3. Solenoid fields and vector potential. Maxwell equations. Lorentz condition on the vector potential. Wave equation. Wave equation solution. 4. GaussGreen formulas in the plane. 5. Area and surface integrals: area of a given surface in parametric form, area element on the surface, area of a given surface in Cartesian form, area of a surface of rotation. Surface integral of a scalar function. Calculation of barycentre and moments of inertia. Surface integral of a vector field, flow: orientable and nonorientable surfaces, edge of a surface, Moebius strip, flow of a vector field through a surface. 1. Theorem of divergence (Gauss). Rotor theorem (Stokes). Deduction of the Gauss and Stokes theorem from the GaussGreen formulas.
SERIES OF POWERS AND SERIES OF FOURIER 1. Series of functions. Examples: geometric series and exponential series. Total convergence of a series of functions. Derivability and integrability of a term term series that totally converges. Applications to the calculation of function series using the derived series. 2. Power series. Convergence radius. Root and ratio criteria for calculating the convergence radius. Behavior of a series of powers at the ends of the convergence interval. 3. Trigonometric series. Fourier series. Calculation of the Fourier series coefficients. Fourier series with periods other than. Complex exponential form of the Fourier series.
(reference books)
Analisi Matematica 2, M. Bramanti, C. Pagani, S. Salsa, Zanichelli
