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Chandrasekhar–Wentzel Lemma
In vector calculus, Chandrasekhar–Wentzel lemma was derived by Subrahmanyan Chandrasekhar and Gregor Wentzel in 1965, while studying the stability of rotating liquid drop. The lemma states that ''if \mathbf S is a surface bounded by a simple closed contour C, then'' :\mathbf L = \oint_C \mathbf x\times(d\mathbf x\times\mathbf n) = -\int_(\mathbf x\times\mathbf n)\nabla\cdot\mathbf n\ dS. Here \mathbf x is the position vector and \mathbf n is the unit normal on the surface. An immediate consequence is that if \mathbf S is a closed surface, then the line integral tends to zero, leading to the result, : \int_(\mathbf x\times\mathbf n)\nabla\cdot\mathbf n\ dS =0, or, in index notation, we have :\int_x_j\nabla\cdot\mathbf n\ dS_k = \int_ x_k \nabla \cdot \mathbf n\ dS_j. That is to say the tensor :T_ = \int_x_j\nabla\cdot\mathbf n\ dS_i defined on a closed surface is always symmetric, i.e., T_=T_. Proof Let us write the vector in index notation, but summation convention In m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-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 the theory of quaternions 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'', though earlier mathematicians such as Isaac Newton pioneered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar (; 19 October 1910 – 21 August 1995) was an Indian Americans, Indian-American theoretical physicist who made significant contributions to the scientific knowledge about the structure of stars, stellar evolution and black holes. He was awarded the 1983 Nobel Prize in Physics along with William Alfred Fowler, William A. Fowler for theoretical studies of the physical processes of importance to the structure and evolution of the stars. His mathematical treatment of stellar evolution yielded many of the current theoretical models of the later evolutionary stages of massive stars and black holes. Many concepts, institutions and inventions, including the Chandrasekhar limit and the Chandra X-ray Observatory, Chandra X-Ray Observatory, are named after him. Chandrasekhar worked on a wide variety of problems in physics during his lifetime, contributing to the contemporary understanding of stellar structure, white dwarfs, stellar dynamics, stochastic process, r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregor Wentzel
Gregor Wentzel (17 February 1898 – 12 August 1978) was a German physicist known for development of quantum mechanics. Wentzel, Hendrik Kramers, and Léon Brillouin developed the Wentzel–Kramers–Brillouin approximation in 1926. In his early years, he contributed to X-ray spectroscopy, but then broadened out to make contributions to quantum mechanics, quantum electrodynamics, superconductivity and meson theory. Biography Early life and family Gregor Wentzel was born in Düsseldorf, Germany, as the first of four children of Joseph and Anna Wentzel. He married Anna Lauretta Wielich and his only child, Donat Wentzel, was born in 1934. The family moved to the United States in 1948 until he and Anny returned to Ascona, Ascona, Switzerland in 1970. Education and academia Wentzel began his university education in mathematics and physics in 1916, at the University of Freiburg. During 1917 and 1918, he served in the armed forces during World War I. He then resumed his education ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Notation
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916. Introduction Statement of convention According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set , y = \sum_^3 x^i e_i = x^1 e_1 + x^2 e_2 + x^3 e_3 is simplified by the convention to: y = x^i e_i The upper indices are not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stokes's Theorem
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on \R^3. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: : The line integral of a vector field over a loop is equal to the surface integral of its ''curl'' over the enclosed surface. Stokes' theorem is a special case of the generalized Stokes theorem. In particular, a vector field on \R^3 can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form. Theorem Let \Sigma be a smooth oriented surface in \R^3 with boundary \partial \Sigma \equiv \Gamma . If a vector field \mathbf(x,y,z) = (F_x(x, y, z), F_y(x, y, z), F_z(x, y, z)) is defined and has continuous first ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
