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Multivector
In multilinear algebra, a multivector, sometimes called Clifford number or multor, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (also known as decomposable -vectors or -blades) of the form : v_1\wedge\cdots\wedge v_k, where v_1, \ldots, v_k are in . A -vector is such a linear combination that is ''homogeneous'' of degree (all terms are -blades for the same ). Depending on the authors, a "multivector" may be either a -vector or any element of the exterior algebra (any linear combination of -blades with potentially differing values of ). In differential geometry, a -vector is usually a vector in the exterior algebra of the tangent vector space of a smooth manifold; that is, it is an antisymmetric tensor obtained by taking linear combinations of the exterior product of tangent vectors, for some integer . A differential -form is a -vector in the exterior al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blade (geometry)
In the study of geometric algebras, a -blade or a simple -vector is a generalization of the concept of scalars and vectors to include ''simple'' bivectors, trivectors, etc. Specifically, a -blade is a -vector that can be expressed as the exterior product (informally ''wedge product'') of 1-vectors, and is of '' grade'' . In detail: * A 0-blade is a scalar. * A 1-blade is a vector. Every vector is simple. * A 2-blade is a ''simple'' bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors and : *: a \wedge b . * A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors , , and : *: a \wedge b \wedge c. * In a vector space of dimension , a blade of grade is called a ''pseudovector'' or an '' antivector''. *The highest grade element in a space is called a ''pseudoscalar'', and in a space of dimension is an -blade. * In a vector space of dimension , there are di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Algebra
Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concepts such as Matrix (mathematics), matrices, tensors, multivectors, System of linear equations, systems of linear equations, Higher-dimensional space, higher-dimensional spaces, Determinant, determinants, inner product, inner and outer product, outer products, and Dual space, dual spaces. It is a mathematical tool used in engineering, machine learning, physics, and mathematics. Origin While many theoretical concepts and applications involve Vector space, single vectors, mathematicians such as Hermann Grassmann considered structures involving pairs, triplets, and multivectors that generalize Vector (mathematics and physics), vectors. With multiple combinational possibilities, the space of multivectors expands to 2''n'' dimensions, where ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. Geometrically, a simple bivector can be interpreted as characterizing a directed plane segment (or oriented plane segment), much as vectors can be thought of as characterizing '' directed line segments''. The bivector has an ''attitude'' (or direction) of the plane spanned by and , has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (mathematics And Physics)
In mathematics and physics, vector is a term that refers to physical quantity, quantities that cannot be expressed by a single number (a scalar (physics), scalar), or to elements of some vector spaces. Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacement (geometry), displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers. The term ''vector'' is also used, in some contexts, for tuples, which are finite sequences (of numbers or other objects) of a fixed length. Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set (mathematics), set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the abov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade (geometry), blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude (mathematics), magnitude of a bivector, -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the Parallelepiped#Parallelotope, parallelotope defined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was a British mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics, geometry, and computing. Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression ''mind-stuff''. Biography Born in Exeter, William Clifford was educated at Doctor Templeton's Academy on Bedford Circus and showed great promise at school. He went on to King's College London (at age 15) and Trinity College, Cambridge, where he was elected fello ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wendell Fleming
Wendell Helms Fleming (March 7, 1928 – February 18, 2023) was an American mathematician, specializing in geometrical analysis and stochastic differential equations. Fleming received his PhD in 1951 under Laurence Chisholm Young at the University of Wisconsin–Madison with a thesis entitled ''Boundary and related notions for generalized parametric surfaces''. Fleming was a professor at Brown University, where he retired in 2009 as professor emeritus. Fleming was with Herbert Federer a pioneer of geometric measure theory. Later in his career, he worked on stochastic processes, stochastic differential equations and their applications in control theory. In 1976–1977 he was a Guggenheim Fellow. In 1982 he gave a plenary address (''Optimal control of Markov Processes'') at the ICM in Warsaw. Awards and honors In 1987 he received with Federer the Leroy P. Steele Prize of the American Mathematical Society. In 1994 he won the Reid Prize from the Society for Industrial and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In 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 -form, and can be integrated over an interval ,b/math> contained in the domain of f: \int_a^b f(x)\,dx. Similarly, the expression f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz is a -form that can be integrated over a surface S: \int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right). The symbol \wedge denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form f(x,y,z) \, dx \wedge dy \wedge dz represents a volume element that can be integrated over a region of space. In general, a -form is an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric Lengyel
Eric Lengyel is a computer scientist specializing in game engine development, computer graphics, and geometric algebra. He holds a Ph.D. in computer science from the University of California, Davis and a master's degree in mathematics from Virginia Tech, where he also competed in cross-country and track and field. Lengyel is an expert in font rendering technology for 3D applications and is the inventor of the Slug font rendering algorithm, which allows glyphs to be rendered directly from outline data on the GPU with full resolution independence. Lengyel is also the inventor of the Transvoxel algorithm, which is used to seamlessly join multiresolution voxel data at boundaries between different levels of detail that have been triangulated with the Marching cubes algorithm. Among his many written contributions to the field of game development, Lengyel is the author of the four-volume book series ''Foundations of Game Engine Development''. The first volume, covering the mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plücker Coordinates
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, . Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in and points on a quadric in (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe -dimensional linear subspaces, or ''flats'', in an -dimensional Euclidean space), Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control. Geometric intuition A line in 3-dimensional Euclidean space is determined by two distinct points that it contains, or by two distinct planes that contain it (a plane-plane intersection). Consider the first case, with points x=(x_1,x_2,x_3) and y=( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (projective Geometry)
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by Point (geometry), points and Line (geometry), lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special Map (mathematics), mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to ''space duality'' and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
