Part I. Linear spaces and operators: We shall start with vectors, curves, and linear transformations on the plane. Then we move on to three-dimensional space and Euclidean space. Finally we shall arrive at infinite dimensional spaces such as Hilbert space and functional space with different examples of linear operators and scalar products. The idea of orthogonal decomposition over our journey through the space will be emphasized. The concept of decomposition will lead us to Fourier and spectral analyses, two of the major techniques in sciences and engineering. Besides, the following concepts and methods will be studied: elements of matrix algebra and linear operators in Euclidean space, eigenvalue and eigenvector, best fit in linear space, elements of optimization of multivariate function, constrained optimization and Lagrange multipliers, vector fields, divergence and curl vectors.
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Part II. Partial differential equations: Partial Differential Equations (PDEs) are the major tool for modeling space-dynamic processes in physics and engineering. During the course we shall learn where PDEs come from and how to solve them. Two major types of PDEs will be considered: diffusion (heat) and hyperbolic (wave). We shall start with diffusion-type problem (heat-flow equation) under different boundary conditions. Several methods to solve PDEs will be discussed: separation of variables, Fourier and Laplace transform, D'Alembert solution, among them. The idea of linear operators and particularly Fourier series from Part I will be employed. We shall start with the simplest version of the wave equation and then consider possible generalizations. Again, the connection to linear spaces and operators and particularly function eigenvalue will be discovered. Also, we shall discuss other types of PDEs (PDEs classification) and major methods to solve them. At last, I will introduce calculus of variation as the functional optimization problem and its connection to PDE.
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