ELEMENTS OF MATHEMATICS FOR GEOSCIENCES
Provide the algebraic and analytical tools that allow to deal with the three-dimensional space. In particular, introduce differential and integral calculus in multiple variables, and linear algebra in its relationship with geometric thinking. The course will provide the description of some mathematical models through simple differential equations.
Provide the algebraic and analytical tools that allow to deal with the three-dimensional space. In particular, introduce differential and integral calculus in multiple variables, and linear algebra in its relationship with geometric thinking. The course will provide the description of some mathematical models through simple differential equations.
scheda docente
materiale didattico
Functions of several variables: notions of limit, continuity, and partial derivative. Directional derivatives, gradient. Contour lines and lines of maximum slope. Stationary points, maxima, and minima. Hessian matrix. Calculating eigenvalues and eigenvectors of a symmetric matrix.
Curvilinear integrals of vector fields; conservative and non-conservative fields.
Multiple integrals; changes of variables in multiple integrals
Ordinary differential equations: first-order linear equations; equations with separable variables; equilibrium points and their stability; second-order linear differential equations with constant coefficients, homogeneous, and non-homogeneous.
[BEM] D. Benedetto, M. Degli Esposti, C. Maffei: Matematica per le scienze della vita, Ed. Ambrosiana.
[MS] P. Marcellini, C. Sbordone: Elementi di Calcolo, Liguori Editore. In alternativa, degli stessi autori: Calcolo, Liguori Editori.
Testi di esercizi:
[MSes] P.Marcellini, C.Sbordone: Esercitazioni di matematica, vol. primo, parti prima e seconda, Ed. Liguori.
[SS] S. Salsa, A. Squellati: Esercizi di Matematica - Calcolo infinitesimale e algebra lineare, Zanichelli.
Programma
Integrals of functions of a single variable: definition of definite integrals (limit of Riemann sums, sum of areas with sign); fundamental theorem of integral calculus; primitives and indefinite integrals; integration of elementary functions; integration by parts and by substitution; integration of rational functions; improper integrals.Functions of several variables: notions of limit, continuity, and partial derivative. Directional derivatives, gradient. Contour lines and lines of maximum slope. Stationary points, maxima, and minima. Hessian matrix. Calculating eigenvalues and eigenvectors of a symmetric matrix.
Curvilinear integrals of vector fields; conservative and non-conservative fields.
Multiple integrals; changes of variables in multiple integrals
Ordinary differential equations: first-order linear equations; equations with separable variables; equilibrium points and their stability; second-order linear differential equations with constant coefficients, homogeneous, and non-homogeneous.
Testi Adottati
Testi principale di riferimento:[BEM] D. Benedetto, M. Degli Esposti, C. Maffei: Matematica per le scienze della vita, Ed. Ambrosiana.
[MS] P. Marcellini, C. Sbordone: Elementi di Calcolo, Liguori Editore. In alternativa, degli stessi autori: Calcolo, Liguori Editori.
Testi di esercizi:
[MSes] P.Marcellini, C.Sbordone: Esercitazioni di matematica, vol. primo, parti prima e seconda, Ed. Liguori.
[SS] S. Salsa, A. Squellati: Esercizi di Matematica - Calcolo infinitesimale e algebra lineare, Zanichelli.
Bibliografia Di Riferimento
Testi principale di riferimento: [BEM] D. Benedetto, M. Degli Esposti, C. Maffei: Matematica per le scienze della vita, Ed. Ambrosiana. [MS] P. Marcellini, C. Sbordone: Elementi di Calcolo, Liguori Editore. In alternativa, degli stessi autori: Calcolo, Liguori Editori. Testi di esercizi: [MSes] P.Marcellini, C.Sbordone: Esercitazioni di matematica, vol. primo, parti prima e seconda, Ed. Liguori. [SS] S. Salsa, A. Squellati: Esercizi di Matematica - Calcolo infinitesimale e algebra lineare, Zanichelli.Modalità Frequenza
Non-mandatoryModalità Valutazione
The written exam may be replaced by two midterms that will be offered to the students after approx. 6 weeks and at the end of the course, respectively. The oral exam will be based on a discussion of the written exam and will include some theoretical questions related to the problems in the written test.
scheda docente
materiale didattico
Functions of several variables: notions of limit, continuity, and partial derivative. Directional derivatives, gradient. Contour lines and lines of maximum slope. Stationary points, maxima, and minima. Hessian matrix. Calculating eigenvalues and eigenvectors of a symmetric matrix.
Curvilinear integrals of vector fields; conservative and non-conservative fields.
Multiple integrals; changes of variables in multiple integrals
Ordinary differential equations: first-order linear equations; equations with separable variables; equilibrium points and their stability; second-order linear differential equations with constant coefficients, homogeneous, and non-homogeneous.
[BEM] D. Benedetto, M. Degli Esposti, C. Maffei: Matematica per le scienze della vita, Ed. Ambrosiana.
[MS] P. Marcellini, C. Sbordone: Elementi di Calcolo, Liguori Editore. In alternativa, degli stessi autori: Calcolo, Liguori Editori.
Testi di esercizi:
[MSes] P.Marcellini, C.Sbordone: Esercitazioni di matematica, vol. primo, parti prima e seconda, Ed. Liguori.
[SS] S. Salsa, A. Squellati: Esercizi di Matematica - Calcolo infinitesimale e algebra lineare, Zanichelli.
Programma
Integrals of functions of a single variable: definition of definite integrals (limit of Riemann sums, sum of areas with sign); fundamental theorem of integral calculus; primitives and indefinite integrals; integration of elementary functions; integration by parts and by substitution; integration of rational functions; improper integrals.Functions of several variables: notions of limit, continuity, and partial derivative. Directional derivatives, gradient. Contour lines and lines of maximum slope. Stationary points, maxima, and minima. Hessian matrix. Calculating eigenvalues and eigenvectors of a symmetric matrix.
Curvilinear integrals of vector fields; conservative and non-conservative fields.
Multiple integrals; changes of variables in multiple integrals
Ordinary differential equations: first-order linear equations; equations with separable variables; equilibrium points and their stability; second-order linear differential equations with constant coefficients, homogeneous, and non-homogeneous.
Testi Adottati
Testi principale di riferimento:[BEM] D. Benedetto, M. Degli Esposti, C. Maffei: Matematica per le scienze della vita, Ed. Ambrosiana.
[MS] P. Marcellini, C. Sbordone: Elementi di Calcolo, Liguori Editore. In alternativa, degli stessi autori: Calcolo, Liguori Editori.
Testi di esercizi:
[MSes] P.Marcellini, C.Sbordone: Esercitazioni di matematica, vol. primo, parti prima e seconda, Ed. Liguori.
[SS] S. Salsa, A. Squellati: Esercizi di Matematica - Calcolo infinitesimale e algebra lineare, Zanichelli.
Bibliografia Di Riferimento
Testi principale di riferimento: [BEM] D. Benedetto, M. Degli Esposti, C. Maffei: Matematica per le scienze della vita, Ed. Ambrosiana. [MS] P. Marcellini, C. Sbordone: Elementi di Calcolo, Liguori Editore. In alternativa, degli stessi autori: Calcolo, Liguori Editori. Testi di esercizi: [MSes] P.Marcellini, C.Sbordone: Esercitazioni di matematica, vol. primo, parti prima e seconda, Ed. Liguori. [SS] S. Salsa, A. Squellati: Esercizi di Matematica - Calcolo infinitesimale e algebra lineare, Zanichelli.Modalità Frequenza
Non-mandatoryModalità Valutazione
The written exam may be replaced by two midterms that will be offered to the students after approx. 6 weeks and at the end of the course, respectively. The oral exam will be based on a discussion of the written exam and will include some theoretical questions related to the problems in the written test.