Giving further knowledge and tools of Calculus, required for an adequate understanding of mathematical methods and models relevant for Engineering.
teacher profile teaching materials
Norm and distance in R^n. Functions of several variables. Continuous functions and Weierstrass theorem.
Partial derivatives, directional derivatives and gradient. Functions of class C^1 and C^2. Higher derivatives, Hessian matrix and Schwarz theorem. Differentiation of composed functions. Taylor expansion. Local maxima and minima. Method of Lagrange multipliers to compute local maxima and minima.
Riemann integration and Peano-Jordan measure. Integration of continuous functions, reduction formulae
and iterated integrals; area and volume. Change of variables in integrals and Jacobian matrix: polar, cylindrical and spherical coordinates. Gaussian integral.
Curves in R^n: parametrized and equivalent curves; length of a curve; curve integrals of scalar functions. Work and curve integrals of vector fields. Smooth surfaces in R^3: area of a surface and integrals on surfaces.
Gentile, Introduzione ai Sistemi Dinamici Volume 1, Springer
Marcellini, Sbordone, Esercitazioni di Analisi Matematica Due (vol. I e vol. II), Zanichelli ed.
Programme
Linear first-order differential equations. General first order differential equations. Cauchy problem: local existence and uniqueness. Separation of variables. Systems of first-order equations: linearly independent solutions and Wronskian matrix. Variation of constants. Differential equations with constant coefficients and characteristic polynomial. Matrix exponential and computation for diagonalizable matrices. Some remarkable differential equations: Euler equation and Bernouilli equation.Norm and distance in R^n. Functions of several variables. Continuous functions and Weierstrass theorem.
Partial derivatives, directional derivatives and gradient. Functions of class C^1 and C^2. Higher derivatives, Hessian matrix and Schwarz theorem. Differentiation of composed functions. Taylor expansion. Local maxima and minima. Method of Lagrange multipliers to compute local maxima and minima.
Riemann integration and Peano-Jordan measure. Integration of continuous functions, reduction formulae
and iterated integrals; area and volume. Change of variables in integrals and Jacobian matrix: polar, cylindrical and spherical coordinates. Gaussian integral.
Curves in R^n: parametrized and equivalent curves; length of a curve; curve integrals of scalar functions. Work and curve integrals of vector fields. Smooth surfaces in R^3: area of a surface and integrals on surfaces.
Core Documentation
Bertsch, Dal Passo, Giacomelli, Analisi Matematica , McGraw Hill, II edizioneGentile, Introduzione ai Sistemi Dinamici Volume 1, Springer
Marcellini, Sbordone, Esercitazioni di Analisi Matematica Due (vol. I e vol. II), Zanichelli ed.
Type of delivery of the course
Lectures.Attendance
Attendance is not compulsory but is strongly suggested.Type of evaluation
The exam consists of a written test, to be carried out in 3 hours, and an oral interview, to be carried out subsequently after the correction of the written test. The written test includes 6 exercises, in addition to a preliminary exercise, divided into 4 questions: only if at least 3 out of 4 answers are correct will the rest of the test be evaluated. Passing the written test (with a grade ≥18) allows you to take the oral interview in any session of the same academic year. teacher profile teaching materials
Norm and distance in R^n. Functions of several variables. Continuous functions and Weierstrass theorem.
Partial derivatives, directional derivatives and gradient. Functions of class C^1 and C^2. Higher derivatives, Hessian matrix and Schwarz theorem. Differentiation of composed functions. Taylor expansion. Local maxima and minima. Method of Lagrange multipliers to compute local maxima and minima.
Riemann integration and Peano-Jordan measure. Integration of continuous functions, reduction formulae
and iterated integrals; area and volume. Change of variables in integrals and Jacobian matrix: polar, cylindrical and spherical coordinates. Gaussian integral.
Curves in R^n: parametrized and equivalent curves; length of a curve; curve integrals of scalar functions. Work and curve integrals of vector fields. Smooth surfaces in R^3: area of a surface and integrals on surfaces.
Esercizi: Marcellini, Sbordone, Esercitazioni di Analisi Matematica Due (vol. I e vol. II), Zanichelli ed.
Programme
Linear first-order differential equations. General first order differential equations. Cauchy problem: local existence and uniqueness. Separation of variables. Systems of first-order equations: linearly independent solutions and Wronskian matrix. Variation of constants. Differential equations with constant coefficients and characteristic polynomial. Matrix exponential and computation for diagonalizable matrices. Some remarkable differential equations: Euler equation and Bernouilli equation.Norm and distance in R^n. Functions of several variables. Continuous functions and Weierstrass theorem.
Partial derivatives, directional derivatives and gradient. Functions of class C^1 and C^2. Higher derivatives, Hessian matrix and Schwarz theorem. Differentiation of composed functions. Taylor expansion. Local maxima and minima. Method of Lagrange multipliers to compute local maxima and minima.
Riemann integration and Peano-Jordan measure. Integration of continuous functions, reduction formulae
and iterated integrals; area and volume. Change of variables in integrals and Jacobian matrix: polar, cylindrical and spherical coordinates. Gaussian integral.
Curves in R^n: parametrized and equivalent curves; length of a curve; curve integrals of scalar functions. Work and curve integrals of vector fields. Smooth surfaces in R^3: area of a surface and integrals on surfaces.
Core Documentation
Teoria: Bertsch, Dal Passo, Giacomelli, Analisi Matematica , McGraw Hill, II edizioneEsercizi: Marcellini, Sbordone, Esercitazioni di Analisi Matematica Due (vol. I e vol. II), Zanichelli ed.
Type of delivery of the course
Lectures and integrative teaching.Attendance
Attendance is not compulsory but is strongly suggested.Type of evaluation
The exam consists of a written test and a subsequent oral interview, in which the student will have to discuss the topics treated in class, with reference to the texts used and the notes distributed in class.