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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. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector can be thought of as being 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 quaternions in three dimensions. They can be used to generate rotations in any number of dimensions, and are a useful tool for classifying such rotations. They are also used in physics, tying together a number of otherwise unrelated quantities. Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as a single exterior product. More precisely, a bivector that can be expressed as an exterior product is called ''simpl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivector (complex)
In mathematics, a bivector is the vector part of a biquaternion. For biquaternion , ''w'' is called the biscalar and is its bivector part. The coordinates ''w'', ''x'', ''y'', ''z'' are complex numbers with imaginary unit h: :x = x_1 + \mathrm x_2,\ y = y_1 + \mathrm y_2,\ z = z_1 + \mathrm z_2, \quad \mathrm^2 = -1 = \mathrm^2 = \mathrm^2 = \mathrm^2 . A bivector may be written as the sum of real and imaginary parts: :(x_1 \mathrm + y_1 \mathrm + z_1 \mathrm) + \mathrm (x_2 \mathrm + y_2 \mathrm + z_2 \mathrm) where r_1 = x_1 \mathrm + y_1 \mathrm + z_1 \mathrm and r_2 = x_2 \mathrm + y_2 \mathrm + z_2 \mathrm are vectors. Thus the bivector q = x \mathrm + y \mathrm + z \mathrm = r_1 + \mathrm r_2 . Link from David R. Wilkins collection at Trinity College, Dublin The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if ''r''1 and ''r''2 are right versors so that r_1^2 = -1 = r_2^2, then the biquaternion curve traces over and over the unit circle in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions. The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. Biography Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium, where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics. Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wedge Product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a force applied to its blunt end into forces perpendicular ( normal) to its inclined surfaces. The mechanical advantage of a wedge is given by the ratio of the length of its slope to its width..''McGraw-Hill Concise Encyclopedia of Science & Technology'', Third Ed., Sybil P. Parker, ed., McGraw-Hill, Inc., 1992, p. 2041. Although a short wedge with a wide angle may do a job faster, it requires more force than a long wedge with a narrow angle. The force is applied on a flat, broad surface. This energy is transported to the pointy, sharp end of the wedge, hence the force is transported. The wedge simply transports energy in the form of friction and collects it to the pointy end, consequently breaking the item. History Wedges have e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Attitude (geometry)
In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement. It may be necessary to add an imaginary translation, called the object's location (or position, or linear position). The location and orientation together fully describe how the object is placed in space. The above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its location does not change when it rotates. Euler's rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis. This gives one common way of representing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are ''not'' linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, '' Vector Analysis''. In the conventional form using cross products, vector calculus does not generalize to higher di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, and rewrote Maxwell's equations in the form commonly used today. He significantly shaped the way Maxwell's equations are understood and applied in the decades following Maxwell's death. His formulation of the telegrapher's equations became commercially important during his own lifetime, after their significance went unremarked for a long while, as few others were versed at the time in his novel methodology. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science. Biography Early life Heaviside was born in Camden Town, London, at 55 Kings Street (now Plender Street), the youngest of three children of Thomas, a draughtsman and wood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous inductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of Statistical ensemble (mathematical physics), ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he invented modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period). In 1863, Yale University, Yale awarded Gibbs the first American Doctor of Philosophy, doctorate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, movies/videos, moving images, and millions of books. In addition to its archiving function, the Archive is an activist organization, advocating a free and open Internet. , the Internet Archive holds over 35 million books and texts, 8.5 million movies, videos and TV shows, 894 thousand software programs, 14 million audio files, 4.4 million images, 2.4 million TV clips, 241 thousand concerts, and over 734 billion web pages in the Wayback Machine. The Internet Archive allows the public to upload and download digital material to its data cluster, but the bulk of its data is collected automatically by its web crawlers, which work to preserve as much of the public web as possible. Its web archive, the Wayback Machine, contains hundreds of b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Forder
Henry George Forder (27 September 1889 – 21 September 1981) was a New Zealand mathematician. Academic career Born in Shotesham All Saints, near Norwich, he won a scholarships first to a Grammar school and then to University of Cambridge. After teaching mathematics at a number of schools, he was appointed to the chair of mathematics at Auckland University College in New Zealand in 1933. He was very critical of the state of the New Zealand curriculum and set about writing a series of well received textbooks. His ''Foundations of Euclidean Geometry'' (1927) was reviewed by F.W. Owens, who noted that 40 pages are devoted to "concepts of classes, relations, linear order, non archimedean systems, ..." and that order axioms together with a continuity axiom and a Euclidean parallel axiom are the required foundation. The object achieved is a "continuous and rigorous development of the uclideandoctrine in the light of modern investigations." In 1929 Forder obtained drawings and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.see for ex. Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
