| Learning outcomes of course unit |
Component provides mathematical training for acquiring integral competence of the educational program: "The ability to solve complex specialized tasks and practical problems in the field of shipping and ship engineering, that involves the application of theories and methods of sciences about ship structure, navigation, mechanical and electrical engineering, operation of means of transport , resource management". In this regard, the successful completion of the program of the academic discipline "Higher Mathematics" requires the cadet (student) to obtain the following learning outcomes.
Knowledge:
- Concept of matrix and determinant. Properties and methods of calculating determinants. Methods of solving systems of linear algebraic equations. Coordinate systems in space and on the plane. Vectors and linear operations on them in coordinate form. Operations on vectors, their definitions, properties and applications. Formulas for the transformation of coordinates in space and on the plane.
- Definition and general equations of a surface and a line in space and on a plane. The equation of a plane and a line in space. Definitions, properties and standard equations of conics (ellipse, circle, hyperbola, parabola). The equation of the sphere. The equation of surfaces of revolution of the second order. The ratio between the angles and sides of a spherical triangle. Basic theorems of spherical trigonometry.
- Definition of one- and multivariable function. Definitions of the limit of one- and multivariable function, and their properties. Infinitesimals and their comparisons. Continuity of functions, properties of continuous functions. Types of discontinuities.
- Definition of partial derivatives of a multivariable function. Rules of differentiation. Derivatives of basic elementary functions. The concept of differentiability and differential of a multivariable function. Derivative of parametric and implicit functions. Directional derivative and gradient. The concept of the extremum of one- and multivariable functions. Conditions of concavity for the function graph. Basic geometric applications of derivatives. Taylor and Maclaurin polynomials. Complex numbers and operations on them. Polar form of a complex number.
- Definition of antiderivative and indefinite integral. Integrals of basic elementary functions. Methods of integration: substitution, integration by parts. Definition of definite integral. Newton-Leibnitz formula. Improper integrals. Definition of double integrals and their calculation. Line integrals with respect to s and with respect to x,y,and z and their calculation. Green-Ostrogradsky formula and the conditions for independence of line integral from the path of integration. Basic applications of integrals.
- Definition of the n-th-order differential equation, general and partial solution, Initial-value problem. Linear differential equations, theorems on general solutions of homogeneous and inhomogeneous equations. Definition of a system of differential equations, general and partial solution, initial-value problem. Systems of linear differential equations and their matrix notation. Characteristic polynomial of a linear system with constant coefficients.
- Definition of numerical and functional infinite series and their sums. The Divergence Test. The sufficient tests for infinite nonnegative series. Concepts of absolute and conditional convergence, convergence tests for alternating series. Definition of a power series and its radius of convergence. Taylor and Maclaurin series, Maclaurin series of basic elementary functions. Definition of Fourier series, formulas for the coefficients of Fourier series. The convergence tests of Fourier series.
- Definition of probability. Algebra of events. Basic theorems of probability. Definition of a random variable, its distribution laws. Determination of the distribution function and density. Basic numerical characteristics of random one- and two-dimensional normal distribution laws. Concept of sampling, variation series, histogram, statistical law of distribution. Sampling estimates of distribution parameters. Interval estimates of distribution parameters. Basic methods of statistical processing of measurement results.
Skills:
- Calculate determinants. Operations over matrices. Investigate and solve systems of linear algebraic equations. Define vectors by coordinates, operations over vectors in coordinate form.
- Find the equation of a plane, a line in space and on a plane. Investigate the mutual arrangement of planes, lines, line and plane. Calculate distances from a point to a line and a plane. Solve problems about the conics given by standard equations.
- Find the limits of functions. Determine the discontinuity of functions.
- Finding the derivatives and partial derivatives. Finding the differentials of one- and multivariable functions. Obtaining the directional derivatives and gradient. Expanding a function by Taylor and Maclaurin polynomials. Investigating a function and sketch it’s graph. Finding the operations over the complex numbers, expressing them in polar form.
- Find indefinite integrals using basic integration methods. Calculate definite integrals. Investigate convergence and calculate improper integrals. Calculate double and line integrals.
- Solve the first-order differential equations: separable, with a homogeneous function, linear, Bernoulli. Solve linear homogeneous and nonhomogeneous differential equations with constant coefficients. Solve systems of linear differential equations.
- Investigate the convergence of infinite series with non-negative terms. Determine the radius of convergence for the power series. Apply Taylor and Maclaurin series for approximate calculations. Express the functions by Fourier series.
Calculate probabilities of random variables. Find numerical characteristics of random variables. Carry out selective and interval estimations of distribution parameters. Propose and test hypotheses about the distribution of a random variable. Evaluate random errors in the measurement of navigation parameters. |
