A geometric series is the sum of the numbers in a geometric progression. For example: + + + = + + +. Letting a be the first term (here 2), n be the number of terms (here 4), and r be the constant that each term is multiplied by to get the next term (here 5), the sum is given by: ()In the example above, this gives: + + + = = = The formula works for any real numbers a and r (except r = 1, …

Area is the quantity that expresses the extent of a region on the plane or on a curved surface.The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.Area can be understood as the amount of material with a given thickness that would be necessary to …

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional space is called a half-line or ray.. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space.

For any , ⁡ is a derivation on , which follows from the Jacobi identity.Any such derivation is called an inner derivation.This derivation extends to the universal enveloping algebra of the Lie algebra.. Examples. If is a unital algebra, then () = since () = = + (). For example, in a differential field of characteristic zero , the rationals are always a subfield of the field of constants of .

An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. It is similar to the two-dimensional quadrant and the one-dimensional ray.. The generalization of an octant is …

Yantra (यन्त्र) (literally "machine, contraption") is a geometrical diagram, mainly from the Tantric traditions of the Indian religions.Yantras are used for the worship of deities in temples or at home; as an aid in meditation; used for the benefits given by their supposed occult powers based on Hindu astrology and tantric texts. They are also used for adornment of temple floors ...

Linear isometry. Given two normed vector spaces and , a linear isometry is a linear map: that preserves the norms: ‖ ‖ = ‖ ‖ for all . Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are surjective.. In an inner product space, the above definition reduces to , = , for all , which is equivalent to saying that † = ⁡ .

Disphenoidal or seesaw (also known as sawhorse) is a type of molecular geometry where there are four bonds to a central atom with overall C 2v molecular symmetry.The name "seesaw" comes from the observation that it looks like a playground seesaw.Most commonly, four bonds to a central atom result in tetrahedral or, less commonly, square planar geometry. ...

Counter-electromotive force (counter EMF, CEMF, back EMF), is the electromotive force (EMF) manifesting as a voltage that opposes the change in current which induced it. CEMF is the EMF caused by electromagnetic induction.. Details. For example, the voltage appearing across an inductor or coil is due to a change in current which causes a change in the magnetic field within …

The Cool S (also known as the Stussy S, Super S, Superman S, Universal S, Pointy S, Middle School S, Graffiti S, and by many other names) is a graffiti sign in popular culture that is typically doodled on children's notebooks or graffitied on walls. The exact origin of the Cool S is unknown, but an instance was found in a late 1800s geometry textbook and it became prevalent around …

In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a curvature vector); its ...

Here, for the ellipse and the hyperbola, a is the length of the semi-major axis and b is the length of the semi-minor axis. When the conic section is given in the general quadratic form + + + + + =, the following formula gives the eccentricity e if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse, and not an …

In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high ...

SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics.. The first version of SageMath was released on 24 February 2005 as free and open-source …