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Analisi Matematica 2 (EA)
Mathematics 2
Flaviano Battelli

Seat Ingegneria
A.A. 2015/2016
Credits 6
Hours 72
Period 1s
Language ENG

Prerequisites
First course of real analysis and Geometry

Learning outcomes
The course is the natural continuation of the course "Analisi Matematica 1" of the first year, it provides more advanced concepts regarding the mathematical tools needed to master, from the analytical point of view, the technical and technological problems underlying the design and building work in architecture.

Program
Vectors in R^n. Operating with vectors. Scalar product and norm. Cauchy-Schwartz inequality. Neighborhoods and 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 curves and along a direction. A criterion for the existence of limits. Polar and spherical coordinates. Algebra of limits. Continuous functions. Algebra of continuous functions. Connected sets and arc connected sets. Continuous image of a connected set. Zero's and intermediate values theorems. Compact sets. Weierstrass theorem. Differential calcolus for functions of several variables. Differentiable functions. Directional derivatives. The gradient and the Gradient theorem. The tangent iperplane to the graph of a function. Uniqueness and existence. C^1 versus differentiability, Continuity e Lipschitzianity of linear functions. Continuity and differentiability. Maximal growth direction. Higher order derivatives. Schwartz theorem. The Hessian matrix. Differentiability of vector valued functions and their components. The Jacobian matrix. The chain rule. C^k functions and Schwartz theorem. Taylor formula in several variables with Peano 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. Tangent space to a manifold at a point. Uniqueness and characterization. Gradient of a function and normal line. Smooth curves. Lenght of a piecewise smooth curve. Integration on curves. Tangent vector and versor. Curvature and normal vector, torsion and binormal versor. Frénét formulae. Norm of the integral and integral of the norm. Measure of a pluri-interval. Exterior and interior measure. Measurable sets. Properties of Peano-Jordan measure. Partitions of a mesurable set. Lower and upper sums. The Riemann integral. Some class of partitions useful for integral calculus. Integrability of locally constant functions. Measure zero sets and their neglectability. Reduction formula and change of variables. Integrating by slices and leaves. Cavalieri's theorem. Volume of the sphere. Guldino's theorems. Integration on manifolds. Indipendence of parameterization. Integration of vector fields on curves. Vector fields and 1-forms. Conservative and irrotational vector fields. Equivalent conditions. The curl of a vector field in R^3. The Gauss-Green formula. The notion of differential n-form. 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. Differential forms and extended Gauss-Green formula. Stokes theorem and consequences: curl and divergence theorems Differential equations in normal form. Peano existence theorem. Locally Lipschitz functions. Cauchy existence and uniqueness theorem. Maximal interval of existence. Properties of the maximal interval. Linear systems. Superposition principle. The vector space of solutions of a homogeneous equation. The fundamental matrix. Variation of constant formula. The case of nth order equations. Wronskian matrix and particular integral. The exponential matrix and its properties. The fundamental matrix for linear equations with constant coefficients. Exponential growth of the solutions of linear equations with constant coefficients. Similarity method.

Development of the examination
LEARNING EVALUATION METHODS
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.

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

LEARNING MEASUREMENT CRITERIA
At each proof a mark is given between 0/30 and 30/30

FINAL MARK ALLOCATION CRITERIA
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.

Recommended reading
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

Courses

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