Mathematics

obligatory courses

(in Polish) Student powinien posiadać wiedzę związaną z:

- działaniami na liczbach rzeczywistych,

- działaniami na wyrażeniach algebraicznych, wzorami uproszczonego mnożenia,

- znajomością podstawowych funkcji i ich wykresów, w tym funkcji liniowej, kwadratowej, wykładniczej, logarytmicznej, trygonometrycznych,

- rozwiązywaniu równań, nierówności opartych na wyżej wymienionych funkcjach,

- pojęciem funkcji i ich własności (parzystość, nieparzystość, monotoniczność),

- elementów geometrii analitycznej na płaszczyźnie, równania prostej, okręgu, paraboli, hiperboli, elipsy,

- pojęciami i własnościami ciągów algebraicznych, geometrycznych sumy ciągów,

- powinien posiadać odpowiednią technikę rachunkową.

Number sets: complex numbers. Matrixes, determinants, systems of linear algebraic equations, matrix calculus, row of a matrix. Vector calculus in Rn, vector product in R3, straight line and plane in R3. Functions as a relation, inverse functions. Limits of a sequences in Rn, sequence of numbers, the e number. Limits and continuity of functions. Derivative of a function: limits of indeterminate expressions, monotonicity, extrema, variability of function, derivative functions of several variables and its applications. Sequences of numbers and their convergence, power sequences. Indefinite integrals. Indefinite integrals and their applications. Double integrals and their applications. Ordinary differential equations: with separated variables, homogeneous, linear. Higher order equations. Elements of field theory: gradient, divergence, rotation, directional derivative.

The lecture is designed to acquaint a student with:

• number sets and feasible operations, complex numbers and their geometric interpretation, the de Moivre'a formula, roots of complex numbers

• the concept of a matrix, determinants of the nth degree, the use of determinants to solve systems of linear algebraic equations, Cramer's theorem (multiplication, inverse), the matrix method of solving systems of equations, row of a matrix and Kronecker- Cappelli’s theorem

• distance in Rn, vectors in Rn, operations of vectors (sum, scalar product) in Rn, vector product in R3, perpendicularity and parallelism of vectors, straight line and plane in R3,

• the concept of relations, functions and transformations, composite function, inverse function, inverse function to trigonometric functions, field and set of values of function

• the concept of a limit of sequence, properties, the theorem of limits of sequences, the e number

• sequences of points in Rk, the theorem of convergence of coordinates,

• the concept of limit of functions (sequence definition), the theorem of limits, continuity of function, the existence of solutions of the equation f (x) = 0,

• derivative of function of one variable, definition, interpretation: geometric and physical models for derivatives, the theorem of derivative, the aplications of derivatives: a study of monotonicity, extrema of functions, calculation of limits of a function, Taylor's theorem,

• partial derivatives, definition, method of calculation, application to the calculation of approximate values of functions of several variables,

• sequences of numbers, sum of numerical sequences, convergence of numerical sequences, comparative criteria, sequences of d'Alambert Cauchy, alternating sequences, power sequences, determining the radius of convergence of power series,

• the concept of indefinite integral, formulas for elementary integrals, the theorem of integration by parts, by substitution, typical substitution, integration of rational functions, recurrence formulas for the integrals, integrals of functions dependent to trigonometric functions

• definite integrals, Newton's formula, the properties of integral calculus, applications of integral calculus to calculate the fields of plane figures, volumes of solids of revolution, the arc length, improper integrals,

• double integrals and their calculation, the applications of double integrals in geometry, mechanics,

• elements of first-order ordinary differential equations, equations with separated variables, homogeneous, linear equations, elements of the nth order equations,

• the concept of scalar and vector fields, gradient, divergence, rotation, and their physical interpretation, derivative in the direction of the vector, calculation of directional derivatives.

After completing the course (lectures and exercises) a student:

- can single-handedly solve tasks related to the learning program

- can understand and interpret the results obtained by computer calculations,

- is prepared to understand the mathematical models introduced in other subjects (soil mechanics, geomechanics, hydrogeology, statistics, etc.).

A presence of the lecture is not obligatory.

Requirements for examination:

- Knowledge of the material presented in the lecture

- Practical application of given theorems during the lectures to solve the problems

- Practical knowledge gained during the exercise.

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