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Vector Analysis (Gibbs Wilson)
''Vector Analysis'' is a textbook by Edwin Bidwell Wilson, first published in 1901 and based on the lectures that Josiah Willard Gibbs had delivered on the subject at Yale University. The book did much to standardize the vector notation, notation and vocabulary of three-dimensional linear algebra and vector calculus, as used by physicists and mathematicians. It was reprinted by Yale in 1913, 1916, 1922, 1925, 1929, 1931, and 1943. The work is now in the public domain. It was reprinted by Dover Publications in 1960. Contents The book carries the subtitle "A text-book for the use of students of mathematics and physics. Founded upon the lectures of J. Willard Gibbs, Ph.D., LL.D." The first chapter covers Vector (geometric), vectors in three spatial dimensions, the concept of a (real) scalar (mathematics), scalar, and the product of a scalar with a vector. The second chapter introduces the dot product, dot and cross products for pairs of vectors. These are extended to a Triple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edwin Bidwell Wilson
Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist Paul Samuelson. Wilson had a distinguished academic career at Yale and MIT, followed by a long and distinguished period of service as a civilian employee of the US Navy in the Office of Naval Research. In his latter role, he was awarded the Distinguished Civilian Service Award, the highest honorary award available to a civilian employee of the US Navy. Wilson made broad contributions to mathematics, statistics and aeronautics, and is well-known for producing a number of widely used textbooks. He is perhaps best known for his derivation of the eponymously named Wilson score interval, which is a confidence interval used widely in statistics. Life Edwin Bidwell Wilson was born in Hartford, Connecticut to Edwin Horace Wilson (a teacher ... [...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] |
James Mills Peirce
James Mills Peirce (May 1, 1834 – March 21, 1906) was an American mathematician and educator. He taught at Harvard University for almost 50 years. Early life and family He was the eldest son of Sarah Hunt (Mills) Peirce and Benjamin Peirce (1809–1880), a professor of astronomy and mathematics at Harvard University. The family was considered part of the Boston Brahmin elite class. The surname is pronounced to rhyme with "". Benjamin Peirce's father, also named Benjamin, was librarian at Harvard. James had four younger siblings; one brother was philosopher, logician and professor Charles Sanders Peirce (1839–1914). Another brother was Herbert Henry Davis Peirce (1849–1916) who was the First Secretary of the American Embassy in Saint Petersburg, Russia, at the end of the 19th century. J. M. Peirce graduated from Harvard College in 1853. While an undergraduate at Harvard, he was a member of the Hasty Pudding Club. He attended Harvard's law school for one year. In 1857 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Principles In Statistical Mechanics
''Elementary Principles in Statistical Mechanics'', published in March 1902, is a work of scientific literature by Josiah Willard Gibbs which is considered to be the foundation of modern statistical mechanics. Its full title was ''Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics''. Overview In this book, Gibbs carefully showed how the laws of thermodynamics would arise exactly from a generic classical mechanical system, if one allowed for a certain natural uncertainty about the state of that system. The themes of thermodynamic connections to statistical mechanics had been explored in the preceding decades with Clausius, Maxwell, and Boltzmann, together writing thousands of pages on this topic. One of Gibbs' aims in writing the book was to distill these results into a cohesive and simple picture. Gibbs wrote in 1892 to his colleague Lord Rayleigh He had been working on this topic for some time, at lea ... [...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] |
Scalar Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elements Of Dynamic
''Elements of Dynamic'' is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positively, has remained a standard reference since its appearance, and is now available online as a ''Historical Math Monograph'' from Cornell University. On page 95 Clifford deconstructed the quaternion product of William Rowan Hamilton into two separate "products" of two vectors: vector product and scalar product, anticipating the complete severance seen in ''Vector Analysis'' (1901). ''Elements of Dynamic'' was the debut of the term cross-ratio for a four-argument function frequently used in geometry. Clifford uses the term ''twist'' to discuss (pages 126 to 131) the screw theory that had recently been introduced by Robert Stawell Ball. Reviews A review in the '' Philosophical Magazine'' explained for prospective readers that kinematics i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linea ... [...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] |
Stationary Wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect to time, and the oscillations at different points throughout the wave are in phase. The locations at which the absolute value of the amplitude is minimum are called nodes, and the locations where the absolute value of the amplitude is maximum are called antinodes. Standing waves were first noticed by Michael Faraday in 1831. Faraday observed standing waves on the surface of a liquid in a vibrating container. Franz Melde coined the term "standing wave" (German: ''stehende Welle'' or ''Stehwelle'') around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings. This phenomenon can occur because the medium is moving in the direction opposite to the movement of the wave, or it can arise in a stationary medium ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit n-sphere, -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
