Syllabus
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.
Annotation
Vector spaces, convergence, functions of several variables, limits, continuity, derivative, partial derivative, differential, tangent planes, normals, implicitly defined functions, curves, surfaces, transformation of coordinates, multiple integrals, substitution, Fubini theorem, line and surface integrals, their use.