Linear Algebra
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Matrices, Determinants, and Systems of Linear Equations.
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Elements of Vector Algebra and Linear Space Theory.
Matrices. Operations with matrices.
Symmetric, nonsymmetric, orthogonal, and inverse matrices.
Elementary transformations. Matrix rank. Solving homogeneous systems of equations.
Solving systems of linear equations using matrix method.
Gaussian elimination method for solving systems of linear equations.
Operations with geometric vectors.
Linear combinations, linear dependence of vectors. Collinear and coplanar vectors.
Basis of a linear space. Decomposition of a vector by basis.
Scalar product of vectors, properties. Length of vectors. Angle between vectors.
Projections of a vector. Direction cosines. Cauchy-Schwarz inequality.
Coordinate transformation. Transition matrix.
Vector and Mixed Product of Vectors.
Polar, cylindrical, and spherical coordinate systems. Transition formulas.
Eigenvalues and Eigenvectors of Matrices. Methods for Finding Them.
Analytical Geometry
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Algebraic Lines of the First Order in the Plane and in Space.
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Algebraic Curves of the Second Order in the Plane and in Space.
Line on a Plane, Various Equations.
Division of a segment in a given ratio (vector and coordinate methods).
The plane in space, various equations, the distance from a point to a plane.
Mutual arrangement of planes, angle between planes.
Line in space, all possible equations.
The distance between two intersecting lines.
Ellipse, hyperbola, parabola. Director property of ellipse and hyperbola.
The equation of an ellipse, hyperbola, and parabola in the polar coordinate system.
Reducing the quadratic form to canonical form.
General equation of a curve of the second order. Canonical equation of a curve of the second order.
Mathematical Analysis
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Limit of a Function
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Differentiability of a Function, Its Differential, and Derivative.
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Higher Order Derivatives and Differentials. Taylor's Formula.
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The indefinite integral
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Definite integral and its applications
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Numerical series
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Metric space ℝ^n. Multivariable calculus.
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Extrema of multivariable functions
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Double integrals
Sequence. Boundedness and monotonicity of a sequence. Limit of a sequence.
The limit of a function, computing limits.
The derivative of a function. Calculating first-order derivatives.
The differential of a function. First-order differentials.
The geometric interpretation of the derivative.
Approximate calculations using differentials
The derivative of an inverse function.
Calculation of higher-order derivatives.
Calculation of higher-order differentials.
Function analysis using derivatives.
Antiderivative and indefinite integral
Integration using variable substitution
Integration of rational functions.
Integration of irrational functions
Integration of trigonometric functions.
Evaluation of definite integrals.
Multiple and iterated limits of functions of several variables.
Partial derivatives, differential, directional derivative, gradient.
Tangent plane and normal to an explicitly defined surface.
Higher-order derivatives and differentials.
Differentiation of complex and implicitly defined functions.
Substitution of variables in differential expressions.
Investigation of functions for local extrema.
Investigation of functions for constrained extrema.
Complex Analysis
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Complex Numbers
The geometric interpretation of complex numbers.
Trigonometric and exponential forms of a complex number.
The Euler and de Moivre's formulas. The nth root of a complex number.
Solution of quadratic equations with real coefficients and a complex variable.
Logic and Set Theory
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Some Introductory Concepts of Mathematical Logic and Set Theory.
Logical symbolism. Necessary and sufficient conditions.
Countability and Uncountability of Sets. Equinumerosity of Sets.
Boundedness of numerical sets, their exact boundaries, and limit points of numerical sets.
