BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry concave function – derived set concave function   A function is concave if the chord connecting any two points of its graph lies entirely below the graph.Cf. convex function. cone    ARTICLE   The infinite surface of revolution generated as shownThe term also refers to the solid bounded by one of the nappes and a flat elliptical base. If in this case the base is circular (at right angles to the axis), the cone is called a right circular cone. The surface area S (excluding the base) and volume V of a right circular cone are given byCf. conic section. conic section    ARTICLE   A plane curve, either the ellipse, parabola, or hyperbola, which results from the intersection of a plane with a cone. See the article for a full expositionCf. Dandelin's Spheres. connected   A topological space X is connected if there are no two open sets of X whose union is X and whose intersection is empty. constant function   A constant function f is one whose value is the same at all points of its domain. continuous   Analysis: A function f is continuous at a point x of its domain if, whenever we are given a number e greater than 0, we may find a d greater than 0 so that whenever y is within a d-neighborhood of x, then f(y) is within an e-neighborhood of f(x).Topology: A transformation of one topological space into another is continuous if the inverse image of every open set is open, or equivalently if the inverse image of every closed set is closed.Cf. uniformly continuous, equicontinuous, absolutely continuous. convergent sequence   See sequence. convergent series   See series. Related article: Series convex   Naively, a region of space is convex if the line segement joining any two points of the region lies wholly within it. Thus, a polygon is convex if every line segment joining any two points on its sides lies entirely within the polygon. (This is equivalent to the condition that all its interior angles be less than 180°.)More generally, a region in a real vector space is convex if whenever two points x and y are in the region then so is any point tx + (1 - t)y, where t lies in the interval [0, 1]. See the immediately following entries for additional uses of the descriptor “convex.”Cf. concave. convex function   A function is convex if the chord joining any two points of its graph lies entirely above the graph.Cf. concave function. countable   A set is countable if it is finite, or if it is infinite and bijective to the set of natural numbers (finite ordinals), i.e., if there exists a complete one-to-one mapping of the set in question onto the set N. Sets that are both countable and infinite are sometimes called denumerable. Georg Cantor proved that sets may be uncountably infinite, for example the set of real numbers. Related MiniText: Infinity -- You Can't Get There From Here... countably infinite   See countable cover   Topology: a collection of sets which contains a given set. If the sets in the covering collection are open sets, the cover is called an open cover.Partially ordered sets: If x and y are elements of a partially ordered set such that x y, and such that there is no z such that x z y, then we say that x covers y.Graph Theory: An edge or vertex of a graph is said to cover (verb) those vertices or edges, respectively, that it is incident on. A set of edges or vertices is said to cover any vertex or edge covered by any element of that set. A set of edges or vertices that covers all the vertices or edges, respectively, of the graph is called a cover (noun), usually with a specification of whether it consists of edges or vertices. See edge cover, vertex cover. cross product   See vector product. Dandelin’s Spheres    ARTICLE   A proof by the 17th century French mathematician Germinal Dandelin of the equivalence of the plane-geometry and conic-section definitions of the ellipse, parabola, and hyperbola. See the article for a full exposition. definite integral   See: integral. degenerate conic   A conic section in which the intersecting plane passes through the vertex. Related article: Conics dense   Given a space X and a subset A of X, we say that A is dense in X if the intersection of every open set of X with A is non-empty. E.g., the rational numbers are dense in the real numbers. denumerable   A set is denumerable if it is infinite and countable. derivative   For a function f of a single real variable x, the derivative is defined as the limit of the difference quotientprovided this limit exists. In practice, the derivative is interpreted as the instantaneous rate of change of the function at x. Graphically, the derivative returns the slope of the tangent at x.Cf. differentiation rules. derived set   Given a set X, the derived set of X is the set of accumulation points of X. The second derived set is the derived set of the derived set, and so on.Cf. Cantor-Bendixson Theorem. concave function – derived set
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