Completed notes of the course Mathematical Methods

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Mathematical Methods Course held by Prof. Giovanni Cipriani Year 2020-2021 Notes by Alessandro M. Ippoliti Augustin-Louis Cauchy 1789-1857. 2 Contents 1 Overview 5 1.1 Euclidian Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Surfaces in R 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 Surface Integrals and the Divergence Theorem . . . . . . . . . 11 1.8.1 Derivation of Archimedes Principle . . . . . . . . . . . 12 2 Fourier Series and Convergence 15 2.1 Conditions for Convergence . . . . . . . . . . . . . . . . . . . 16 2.2 The Norm and Uniform Convegence . . . . . . . . . . . . . . . 17 2.3 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Properties of L 2 19 3.1 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Bilinear Functionals and the Lax-Milgram Theorem . . . . . . 23 3.4.1 Applications of the Lax-Milgram Theorem . . . . . . . 24 4 Transport Equations 27 4.1 Derivation of the transport Equation from a Conservation Law 27 4.1.1 Construction of the Solution to the T.Eq in the Drift Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.2 Construction of the Solution to the T.Eq in the Drift Case in the Presence of Sources . . . . . . . . . . . . . 30 4.1.3…