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Functions of several variables

Class at Faculty of Education |


* Introduction

- repetition - linear vector spaces, scalar, vector and outer product (geometric meaning, determinants), lines - general form, slope-intercept form, parametric form, parametrization corresponding with longitude, planes, functions

- convergency, neighbourhood, distance of points (metrics, norm - euclid, sum, maximum), points - inner, outer, border, limit, isolated, sets - open, closed, bounded, convex, connex, compact, area.

* Differential calculus

- real functions of several variables (R2->R), domain, level sets, cross-sections, limit (over a set, over domain), continuity

- derivative in direction(Gâteaux differential and derivative), partial derivative, total differential (Frechet derivative), interrelations, theorems on derivatives and differential (counterexamples), gradient (V) - geometric meaning

- higher order derivatives (exchange of mixed second derivatives), second differential, Taylor theorem

- extremes local, absolut, constraint extremes (substitut method and Lagrange multipliers)

- Banach fixed point theorem, implicit function theorem, calculating of derivatives, differentials, tangents, tangent planes transformation of coordinates (R2->R2, R3->R3) - polar, (cylindric), spheric

* Integral calculus

- multiple (double, triple) integral, calculating of an area (disc), volume (ball, cone), centre of gravity (triangle, tetrahedron), moments, Fubini theorem, substitute theorem - connection of determinants with volume and area

- curves in R2 (explicit, implicit, parametric form), tangent, normal, longitude of a curve (circle), divergence, (3. coordinate of curl), curve integral, Green theorem

- křivky v R3 (vyjádření parametrické), tečna, hlavní normála, binormála surfaces in R3 (explicit, implicit, parametric form), tangent plane, normal, area (of a sphere, lateral area of a cone), points on surface (eliptic, hyperbolic,..., asymptotic directions), divergence, curl, surface integral, Stokes, Gauss-Ostrogradsky theorem.


Vector spaces, neighbourhood of a point, convergence, functions of several variables, limits, continuity, directional derivative, partial derivatives, differential, tangent planes, normals, implicit function, curves, surfaces, transformation of coordinates, multidimensional integral, substitution, Fubini theorem, curvilinear and surface integrals, application.