Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Geometry andFirst course of real analysis

The aim of the course is that of providing further mathematical tools commonly employed in the engineering sciences, by means of introducing the basic elements of the differential and integral calculus for real functions of several variables and of the ordinary differential equations.

The many applications of the course topics in the applied sciences, for example in chemistry and in physics, will provide the student with the ability of modeling and solving practical engineering problems; they will also increase the ability of choosing independently the best solution techniques. The course will also provide the student with the ability to use mathematical laws in general scientific problems.

The expertise acquired in this course will be needed in order to study the material of later courses. Individual and collective problem-solving sessions will improve the ability to develop independent thought and learning capabilities. Oral presentations of the topics taught in the course, with the language proper of the basic disciplines of the degree course will help developing communication skills.

Operating with vectors. Scalar product and norm. Cauchy-Schwartz inequality. Neighborhoods, limit points, internal, external and boundary points. Open and closed subsets. Limits for functions of several variables with values in R^n. Limits of components. Limits along a direction. A criterion for the existence of limits. Polar and spherical coordinates. Algebra of limits. Continuous functions and their properties. Connected sets and arc connected sets. Continuous image of a connected set. Zero's and intermediate values theorems. Compact sets. Weierstrass theorem. Differentiable functions. Differentiablity versus continuity. Directional derivatives. The Gradient theorem. Maximal growth direction. The tangent iperplane. C^1 versus differentiability, Continuity e Lipschitzianity of linear functions. Higher order derivatives. Schwartz theorem. The Hessian matrix. Differentiability of vector valued functions and their components. The Jacobian matrix. Rules of differentiations. C^k functions and Schwartz theorem. Taylor formula in several variables with Peano and Lagrange remainder. Local max and min. First order condition. Sign of the Hessian matrix in a max/min/ saddle point. Sufficient condition to recognize stationary points. Sign study of the eigenvalues of a symmetric matrix. Dini's Theorem. Constrained max/min. Smooth points of a variety. Lagrange multiplicators. Differentiable manifolds and tangent space. Gradient of a function and normal line. Smooth curves and their lenght. Integration on curves. Tangent vector and versor. Curvature and normal vector, torsion and binormal versor. Frénét formulae. Measure of a pluri-interval. Measurable sets. Properties of Peano-Jordan measure. Partitions of a mesurable set. Lower and upper sums. The Riemann integral. Integrability of locally constant functions. Measure zero sets and their neglectability. Reduction formula, integrating by slices and leaves. Change of variables. Volumes calculus and Guldino's theorems. Integration on manifolds. Indipendence of parameterization. Guldino's Theorem for surfaces. Integration of vector fields on curves. Vector fields and 1-forms. Conservative and irrotational vector fields. The curl of a vector field in R^3. The Gauss-Green formula. Differential n-forms. The exterior differential. Orientation of a manifold. Manifolds with boundary and boundary orientation of an orientable manifold. Orientation of a hypersurface and the exterior normal. Stokes theorem and consequences: curl and divergence theorems Differential equations. Peano theorem. Locally Lipschitz functions. Cauchy theorem. Maximal interval of existence and its properties. Linear systems. Superposition principle. The fundamental matrix. Variation of constant formula. Similarity method. Exercises will be solved concerning: search of max/min in open subsets and manifolds, integral calculus in R^n, integrals on manifolds, Stokes and the divergence theorems, solving differential equations.

Written proof followed by oral proof. To be admitted to the oral proof the student has to receive a mark of at least 15/30 in the written proof. The student must pass the oral proof in the same session of the written proof. Should the oral proof give a negative result the student must repeat also the written proof.

The student must prove to have understood the topics of the lectures and to know how to use them to solve concrete problems.

At each proof a mark is given between 0/30 and 30/30.

The student will pass the exam if his mark at the written proof is not less than 15/30 and the final mark is at least 18/30. The final mark is obtained adding the 2/5 of the mark obtained in the written proof to the 3/5 of the mark obtained in the oral proof. Full marks and honors is given to the student who deserves full marks and has shown to be particularly brilliant during the oral proof.

1) Bottacin, Zampieri, Analisi 2, Bollati Boringhieri 2) Bramanti, Pagani, Salsa Analisi Matematica 2; Zanichelli - 2) Giusti, Analisi Matematica 2; Bollati Boringhieri - 3) Fusco, Marcellini, Sbordone, Analisi Matematica 2, Liguori - 4) Salsa, Squellati, Esercizi di Matematica, Zanichelli - 5) Giusti, Complementi ed esercizi di Analisi Matematica 2, Bollati Boringhieri

- Ingegneria Edile-Architettura (Corso di Laurea Magistrale con Riconoscimento Europeo (DM 270/04))

**Università Politecnica delle Marche**

P.zza Roma 22, 60121 Ancona

Tel (+39) 071.220.1, Fax (+39) 071.220.2324

P.I. 00382520427