Definite quadratic formIn mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively.
Sesquilinear formIn mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half".
Canadian Armed ForcesThe Canadian Armed Forces (CAF; Forces armées canadiennes, FAC) are the unified military forces of Canada, including sea, land, and air elements referred to as the Royal Canadian Navy, Canadian Army, and Royal Canadian Air Force. Personnel may belong to either the Regular Force or the Reserve Force, which has four sub-components: the Primary Reserve, Supplementary Reserve, Cadet Organizations Administration and Training Service, and the Canadian Rangers.
LengthLength is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in.
Multilinear formIn abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map that is separately -linear in each of its arguments. More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces. A multilinear -form on over is called a (covariant) -tensor, and the vector space of such forms is usually denoted or .
GeometryGeometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Arc lengthArc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. A rectifiable curve has a finite number of segments in its rectification (so the curve has a finite length). If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function) , then the curve is rectifiable (i.
Differential geometryDifferential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky.
Differential formIn mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) dx is an example of a 1-form, and can be integrated over an interval [a, b] contained in the domain of f: Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a surface S: The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms.
Non-Euclidean geometryIn mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries.
Elliptic geometryElliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
Euclidean geometryEuclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.
