ECTS
ECTS Course Catalogue
Course details
Course code: IWS10168o14Semester: 2014/2015 summer
Name: Mathematical Analysis
Major: Water Engineering and Managment
Study Type: first cycle
Course type: compulsory
Study Semester: 2
ECTS points: 8
Hours (Lectures / Tutorials / Other): 30 / 45 / 0
Lecturer: dr hab. Ryszard Deszcz
Language of instruction: Polish
Learning outcomes: Knowledge He knows the basic theorems of the known branches of mathematics, knows the basics of differential and integral calculus of functions of one and the basis of differential calculus and integral calculus of functions of several variables, with particular emphasis on the function of two variables; familiar elements of classical differential geometry of curves and surfaces and elements of vector analysis. Knowledge Uses calculus for investigating the functions of one variable, use calculus of functions of one variable to calculate some geometrical quantities, solve the differential equations of selected types, either the extremes of function of two variables, used integral calculus of functions of two and three variables to calculate some geometrical quantities, calculate the curvature and torsion of the curve, determines a plane tangent and normal at the point of a regular surface, the metric tensor determines the coordinates of the surface.
Competences: He knows his own limitations of knowledge and understands the need for further education, understands and appreciates the importance of intellectual honesty in the activities of their own and other people, act ethically.
Prerequisites: algebra.
Course content: Within the boundary, continuity and derivatives of functions of one variable, the Lagrange theorem, the rule de L'Hospital, Taylor and Maclaurin formulas, the study of a function of one variable, series of numbers, the criteria of convergence, power series, indefinite integrals, definite integrals, Leibniz formula -Newton, improper integrals, ordinary differential equations of first order, ordinary differential equations of the second order, the Cauchy problem and applications. Functions of two or more variables, elements of classical differential geometry of curves and surfaces, double integrals, triple integrals, line integrals, Green's theorem, surface integrals, Stokes' theorem, theorem Gauss-Ostrogradski, elements of vector analysis: gradient, divergence, rotation.
Recommended literature:
Assessment methods: Assessment of exercise on the basis of the results of ongoing tests and assessments. Written exam. The final score consist of score from classes (50%) and lecture (50%).
Comment: Lecture 1 Real numbers, rational numbers, irrational numbers. Sequences, limit of a sequence, the basic method of calculating the limits of sequences, the number of e functions of one variable, monotonicity, periodicity. The inverse function. Lecture 2 Elementary functions. Limits and continuity of functions of one variable, the basic method for calculating limits of functions. Derivatives of functions, calculation of derivatives of functions. 3rd Lecture Geometric interpretation of the first order derivative. Lagrange's theorem. Extremes of function, a function inflection points, convexity and concavity features. 4th Lecture Unmarked expression, the rule de L'Hospital. Differential function. Taylor and Maclaurin formulas and applications. Examination of a function. Lecture fifth Series of numbers, the criteria for convergence, power series, differentiation of a power series. Lesson 6 Indefinite integrals, the fundamental patterns of calculus, integration by substitution and by parts. Integrals of rational functions. Lecture 7th Definite integrals, Leibniz-Newton formula. Integration of power series. Improper integrals. Lecture 8th Geometric Applications of definite integrals, calculation of area of plane figures, length of curves and the volume and surface areas of solids of revolution. Lecture 9th Ordinary differential equations of first order, the general solution, the Cauchy problem, the differential equation with separated variables, linear differential equations, first order, Bernoulli's equation, applications. Ordinary differential equations of the second order, second order linear equations with constant coefficients, applications. Lecture 10th Functions of two or more variables, limit and continuity, partial derivatives. Differential complete. Geometric interpretation of partial derivatives of the first order. Directional derivative, gradient. Taylor's formula. Lecture 11th Determination of extreme functions of two variables; application. Implicit functions, derivatives of implicit function. Cartesian coordinate system. Curvilinear coordinate systems, cylindrical and spherical coordinates. A surface tangent plane and simple medium, the first form of a square surface. 12th Lecture Double integrals, applications. Triple integrals, applications. Lesson 13 Curves in space, the length of the curve, the natural parameter of the curve, the curvature and torsion of the curve. Line integrals, applications. Lecture 14th Green's Theorem. Elements of field theory, divergence and rotation vector. 15th Lecture Integral powierzchniowa.Twierdzenie Stokes. Theorem Gauss-Ostrogradski. The nature and scope of the exercise: exercise Auditorium