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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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ireland, living at Dunsink Observatory. Hamilton's scientific career included the study of geometrical optics, ideas from Fourier analysis, and his work on quaternions which made him one of the founders of modern linear algebra. He made major contributions in optics, classical mechanics and abstract algebra. His work was fundamental to modern theoretical physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. It is now central both to electromagnetism and to quantum mechanics. Early life Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819),Bruno (2003) who ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annalen Der Physik
''Annalen der Physik'' (English: ''Annals of Physics'') is one of the oldest scientific journals on physics; it has been published since 1799. The journal publishes original, peer-reviewed papers on experimental, theoretical, applied, and mathematical physics and related areas. The editor-in-chief is Stefan Hildebrandt. Prior to 2008, its ISO 4 abbreviation was ''Ann. Phys. (Leipzig)'', after 2008 it became ''Ann. Phys. (Berl.)''. The journal is the successor to , published from 1790 until 1794, and ', published from 1795 until 1797. The journal has been published under a variety of names (', ', ', ''Wiedemann's Annalen der Physik und Chemie'') during its history. History Originally, was published in German, then a leading scientific language. From the 1950s to the 1980s, the journal published in both German and English. Initially, only foreign authors contributed articles in English but from the 1970s German-speaking authors increasingly wrote in English in order to reach an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Silberstein Vector
In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector or Weber vector named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, (or sometimes ambiguously called the "electromagnetic field") is a complex vector that combines the electric field E and the magnetic field B. History Heinrich Martin Weber published the fourth edition of "The partial differential equations of mathematical physics according to Riemann's lectures" in two volumes (1900 and 1901). However, Weber pointed out in the preface of the first volume (1900) that this fourth edition was completely rewritten based on his own lectures, not Riemann's, and that the reference to "Riemann's lectures" only remained in the title because the overall concept remained the same and that he continued the work in Riemann's spirit. In the second volume (1901, §138, p. 348), Weber demonstrated how to consolidate Maxwell’s equations using \mathfrak + i\ \mathfrak. The real and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexified
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers. Formal definition Let V be a real vector space. The of is defined by taking the tensor product of V with the complex numbers (thought of as a 2-dimensional vector space over the reals): :V^ = V\otimes_ \Complex\,. The subscript, \R, on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, V^ is only a real vector space. However, we can make V^ into a complex vector space by defining complex multiplication as follows: :\alpha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludwik Silberstein
Ludwik Silberstein (1872 – 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmillan in 1914 with a second edition, expanded to include general relativity, in 1924. Life Silberstein was born May 17, 1872 in Warsaw to Samuel Silberstein and Emily Steinkalk. He was educated in Krakow, Heidelberg, and Berlin. To teach he went to Bologna, Italy from 1899 to 1904. Then he took a position at Sapienza University of Rome. In 1907 Silberstein described a bivector approach to the fundamental electromagnetic equations. When \mathbf and \mathbf represent electric and magnetic vector fields with values in \mathbb^3, then Silberstein suggested \mathbf + i \mathbf would have values in \mathbb^3, consolidating the field description with complexification. This contribution has been described as a crucial step in modernizing Maxwell's equations, while \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew-Hermitian Matrix
__NOTOC__ In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relation where A^\textsf denotes the conjugate transpose of the matrix A. In component form, this means that for all indices i and j, where a_ is the element in the j-th row and i-th column of A, and the overline denotes complex conjugation. Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers., §4.1.2 The set of all skew-Hermitian n \times n matrices forms the u(n) Lie algebra, which corresponds to the Lie group U(n). The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm. Note that the adjoint of an operator depends on the scalar product considered on the n dimensional c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). It is often denoted as \boldsymbol^\mathrm or \boldsymbol^* or \boldsymbol'. H. W. Turnbull, A. C. Aitken, "An Introduction to the Theory of Canonical Matrices," 1932. For real matrices, the conjugate transpose is just the transpose, \boldsymbol^\mathrm = \boldsymbol^\mathsf. Definition The conjugate transpose of an m \times n matrix \boldsymbol is formally defined by where the subscript ij denotes the (i,j)-th entry, for 1 \le i \le n and 1 \le j \le m, and the overbar denotes a scalar complex conjugate. This definition can also be written as :\boldsymbol^\mathrm = \left(\overline\right)^\mathsf = \overline where \boldsymbol^\mathsf denotes the transpose and \overline denotes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (linear Algebra)
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the ''dimension'' of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Definition A basis of a vector space over a field (such as the real numbers or the complex numbers ) is a linearly independent subset of that spans . This me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or ''modulus'' of the product is the product of the two absolute values, or moduli, and the angle or ''argument'' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes known as the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol ''z'', which can be separated into its real (''x'') and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Diameters
In geometry, two diameters of a conic section are said to be conjugate if each chord parallel to one diameter is bisected by the other diameter. For example, two diameters of a circle are conjugate if and only if they are perpendicular. Of ellipse For an ellipse, two diameters are conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called a bounding parallelogram (skewed compared to a bounding rectangle). In his manuscript De motu corporum in gyrum, and in the ' Principia', Isaac Newton cites as a lemma proved by previous authors that all (bounding) parallelograms for a given ellipse have the same area. It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any bounding parallelogram. For example, in proposition 14 of Book VIII of his ''Collection'', Pappus of Alexandria g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
