Mathematical Methods in Physics - NUFY027
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Last update: T_UTF (02.05.2001)
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Last update: T_KVOF (16.05.2003)
[1] M.Brdička, A.Hladík: Teoretická mechanika , Academia, Praha, 1987. [2] L.D.Landau, E.M.Lifšic: Mechanika , Fizmatgiz, Moskva, 1958. [3] J.Horský, J.Novotný, M.Štefaník: Mechanika ve fyzice, Academia, Praha, 2001 [4] J.Kvasnica a kol.: Mechanika , Academia, Praha, 1988. [5] J.W.Leech: Klasická mechanika , SNTL, Praha, 1970. [6] K.R.Symon: Mechanics , Addison-Wesley, Reading, 1971. [7] M.Brdička: Teorie kontinua , NČSAV, Praha, 1959. [8] J.Slavík: Teoretická mechanika , skripta ZČU, Plzeň, 1994. |
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Last update: T_KVOF (06.05.2003)
Physical motivation for introducing scalars, vectors, and tensors. Geometrical vs. algebraic vectors, abstract definition of a vector space, particular examples. Linear combination of vectors, base and components. Scalar and vector products. Coordinate systems. The most common coordinates in plane and space: Cartesian, polar, cylindrical and spherical. Definition and motivation: planetary motion ... Function and its derivative. Recalling functions and limits. Differentiation and elementary methods of calculus. Physical applications, differential equations and examples (radioactive decay, discharging, harmonic oscillations). Three important generalizations: higher-order derivatives (Taylor expansion of functions), differentiation of functions of several variables (partial derivative), derivatives of vectors (velocity and acceleration in non-cartesian coordinates). Integration. A primitive function (motivation: shape of water surface in a rotating glass), indefinite integral. Elementary rules a methods of calculus (per partes method, substitution, partial fractions). Definite integrals and their properties. Newton-Leibniz formula. Various physical and geometrical applications. Unbounded integrals: Euler-Poisson-Laplace integral and velocities of molecules. Integration of functions of several variables. Volume (double, triple) integral (definition, evaluation using the Fubini theorem in various coordinates, applications). Integration of the first kind along curves and surfaces. Integration of the second kind along curves and surfaces (conservative fields, circulation of the vector field along the curve, flow of the vector across the surface, conservation laws). Operators. Physical meaning and definition of grad, div, rot and Delta. The Gauss and the Stokes law including main applications. Explicit form of the operators in curvilinear coordinates (the Lame coefficients). Illustrations taken from the electromagnetic theory, a short comment on Maxwell's equations and electromagnetic waves. Tensors. The transformation matrix for rotation in a plane, relations of orthogonality, and transforming a vector. Definitions of scalar, vector and tensor using transformation properties of their components. Physical applications: definition and meaning of the tensor of inertia, permitivity. Basic operations with tensors.
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