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1. Calculus:
Real and complex numbers, complete induction, sets, functions. Series of real numbers, convergence, Cauchy series. Steadiness,
theorems on continuous functions, polynoms, nulls, rational functions.
Inverse functions: exponential functions and logarithms, trigonometric functions, hyperbola functions.
Differential calculus:
characteristics of differentiable functions and differentiation rules, derivatives of basic functions, median theorem, extrema, Taylor’s and L 'Hospital’s rules.
Integral calculus: antiderivative,
indefinite integrals, substitutional rules, partial integration, factoring of polynomials, Riemann integrals, examples on continuous and monoton functions, main clause of differential and integral calculus.
Indefinite integrals, gamma function, Stirling’s formula.
Infinite series: criterions on convergence, power series, monotoneous convergence, differentiation and integration of individual terms, examples on
Taylor series, criteria of convergence with respect to Fourier series.
Arc length, curvature, convergence criteria; power series, uniform convergence, differentiation and integration by segments, examples for
Taylor series;
Fourier series: questions of convergence, Bessel's inequality
Taylor's formula;
extrema of functions in several variables, least squares method, Lagrange multipliers;
integration in Rn:
integral over domains, iterated integrals (Fubini), volume, substitution rule: polar and sphere coordinates, calculation of concrete domain integrals
2. Algebra:
Euclidean vector spaces in R2, R3: vectors, scalar product, matrices, linear maps in R2, vector product in R3, analytic geometry in R2,
R3 vector spaces: linear independence, basis, dimension, linear maps and matrices, rank; linear systems of equations: solvability, Gauss' algorithm, L-R-factorization, inverse matrix, Cramer's rule, determinants;
eigenvalues and eigenvectors, characteristic polynomial, scalar product and norm, Schwarz inequality, orthonormalization, Legendre polynomials, orthogonal and unitary maps;
linear transformations: eigenvalues and
eigenvectors of symmetric and orthogonal matrices, quadratic forms
some topology in Rn: open, closed, tangent plane, directional derivative, special partial derivatives, gradient, direction of biggest slope
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