Electrical Engineering

Sequences and Series of Functions Pointwise and Uniform Convergence with Graphical Interpretation for a Sequence of Functions. Theorems: Continuity of the Limit, Passing the Limit Under the Integral Sign, Passing the Limit Under the Derivative Sign. Series of Functions: Pointwise, Uniform and Total Convergence. Power Series. Theorem on Convergence Radius. Taylor Series. Curves Planar and Spatial Curves with Related Properties. Tangent Vector. Equivalent Curves. Piecewise Regular Curves. Length of a Curve. Differential Calculus for Multivariable Functions Elements of Topology in Rn. Multivariable Functions: Domain, Graph, and Image. Definition of Limit and Continuity. Level Curves. Partial Derivatives. Gradient and Differentiability with Geometric Interpretation. Directional Derivative and Gradient Formula. Overview of Taylor's Formula. Optimization Fermat's Theorem. Study of the Nature of Critical Points with the Hessian Matrix. Absolute Maximums and Minimums on Closed and Bounded Sets, Weierstrass Theorem. Constrained Extrema on a Regular Curve: Direct Method and Lagrange Multipliers Method. Implicit Function Theorem. Double and Triple Integrals Definition of Double Integral according to Riemann and Geometric Interpretation. Reduction Formulas for Rectangles. Normal Domains. Examples of Changes of Variables for Integral Calculation. Mass and Centroid of a Non-Homogeneous Planar Lamina. Solid of Rotation and Guldino First Theorem. Triple Integrals. Mass and Centroid of a Planar Lamina and Solid Body. Integration on Parallelepipeds and First Reduction Formula. Normal Sets with Respect to a Plane. Integration for Wires and Layers (Second and Third Reduction Formula). Change of Variable Theorem in R3. Spherical and Cylindrical Coordinates. Line Integrals, Vector Fields, and Differential Forms Line Integrals of Continuous Functions. Centroid of a Curve. Vector Fields. Line Integrals of Vector Fields (of the Second Kind). Work of a Vector Field. Conservative and Irrotational Vector Fields. Necessary and Sufficient Condition for a Field to be Conservative. The Language of Differential Forms and the Bijective Correspondence between Vector Fields and Differential Forms. Gauss-Green Formulas in Dimension 2. Divergence Theorem in R2. Surface Integrals, Divergence Theorem, and Stokes' Formula Cartesian Surfaces. Rotational Surfaces and their Parametric Representation. Normal Vector and Tangent Plane to a Surface. Regular Parametric Surfaces. Surface Integrals. Guldino Second Theorem for the Area of Rotational Surfaces. Oriented Surfaces. Flux of a Field Through a Regular Surface. Flux Through Closed Surfaces (Canonical Orientation). Divergence Theorem. Stokes' Formula. Fourier Series Trigonometric Series and Fourier Series for Periodic Functions. Fourier Series in Complex Form.

Mathematical Analysis and Geometry

• M. Bertsch, A. Dall'Aglio, L. Giacomelli, , EPSILON 2, Ed. McGraw-Hill • M. Bertsch, R. Dal Passo, L. Giacomelli, Analisi Matematica, Ed. McGraw-Hill • M. Bramanti, C. D. Pagani, S. Salsa, Analisi Matematica 2. Ed. Zanichelli • M. Bramanti, Esercitazioni di Analisi Matematica 2. Ed. Esculapio • G. Catino, F. Punzo, Esercizi svolti di Analisi Matematica e Geometria 2, Amazon Books • P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica Due, Ed. Zanichelli

not mandatory

Written test with exercises and theory (Possible oral revision at the discretion of the teacher)

In Presence