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Tensor Calculus
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., tensor calculus, tensor analysis, or Ricci calculus In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... is an extension of vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in ... to tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and num ...
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Mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ... and number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...), formulas and related structures (algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...), shapes and spaces in which they are contained (geometry Geometry (from the grc, ...
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Computer Science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines, such as algorithms, theory of computation, and Information theory, information theory, to Applied science, practical disciplines including the design and implementation of Computer architecture, hardware and Computer programming, software. Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures have been called the heart of computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. Cryptography and computer security study the means for secure communication and prevent Vulner ...
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Einstein Summation
In mathematics, especially in applications of linear algebra to physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation [esn]) 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 space, tangent and cotangent space, 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 Addend, 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 (mathematics), set , : y = \sum_^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3 is simplified by the convention to: : y = c_i x^ ...
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Covariance And Contravariance Of Vectors
In multilinear algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... and tensor analysis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ..., covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and .... In physics, a basis is sometimes thought ...
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Shiing-Shen Chern
Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geometry" and is widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century, winning numerous awards and recognition including the Wolf Prize in Mathematics, Wolf Prize and the inaugural Shaw Prize. In memory of Shiing-Shen Chern, the International Mathematical Union established the Chern Medal in 2010 to recognize "an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics". Chern worked at the Institute for Advanced Study (1943–45), spent about a decade at the University of Chicago (1949-1960), and then moved to University of California, Berkeley, where he co-founded the world-renowned Mathematical Sciences Research Instit ...
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Elie Cartan
Elie and Earlsferry is a coastal town and former royal burgh in Fife, and parish, Scotland, situated within the East Neuk beside Chapel Ness on the north coast of the Firth of Forth, eight miles east of Leven, Fife, Leven. The burgh comprised the linked villages of Elie ( ) to the east and to the west Earlsferry, which were formally merged in 1930 by the Local Government (Scotland) Act 1929. To the north is the village of Kilconquhar and Kilconquhar Loch. The civil parish has a population of 861 (in 2011).Census of Scotland 2011, Table KS101SC – Usually Resident Population, publ. by National Records of Scotland. Web site http://www.scotlandscensus.gov.uk/ retrieved March 2016. See "Standard Outputs", Table KS101SC, Area type: Civil Parish 1930 Ancient times Earlsferry, the older of the two villages, was first settled in time immemorial. It is said that Clan MacDuff, MacDuff, the Earl of Fife, crossed the Forth here in 1054 while fleeing from King Macbeth. In particular the l ...
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Exterior Calculus
In the mathematics, mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. 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 from one-variable calculus is an example of a ''1-form, -form'', and can be integral, integrated over an oriented interval in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that has a surface integral over an oriented Surface (topology), surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume ...
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Machine Learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data, known as training data, in order to make predictions or decisions without being explicitly programmed to do so. Machine learning algorithms are used in a wide variety of applications, such as in medicine, email filtering, speech recognition, and computer vision, where it is difficult or unfeasible to develop conventional algorithms to perform the needed tasks.Hu, J.; Niu, H.; Carrasco, J.; Lennox, B.; Arvin, F.,Voronoi-Based Multi-Robot Autonomous Exploration in Unknown Environments via Deep Reinforcement Learning IEEE Transactions on Vehicular Technology, 2020. A subset of machine learning is closely related to computational statistics, which focuses on making predictions using computers; but not all machine learning is statistical lear ...
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Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called Quantum, quanta) of their underlying quantum field (physics), fields, which are more fundamental than the particles. Interactions between particles are described by interaction terms in the Lagrangian (field theory), Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory (quantum mechanics), perturbation theory in quantum mechanics. History Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of int ...
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Mathematics Of General Relativity
The mathematics of general relativity refers to various mathematics, mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity. The main tools used in this geometry, geometrical theory of gravitation are tensor fields defined on a Pseudo-Riemannian manifold#Lorentzian manifold, Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity. ''Note: General relativity articles using tensors will use the abstract index notation''. Tensors The General covariance, principle of general covariance was one of the central principles in the development of general relativity. It states that the laws of physics should take the same mathematical form in all Frame of reference, reference frames. The term 'general covariance' was used in the early formulation of general relativity, but the principle is now often referred to as 'General covariance, diffeomorphism cov ...
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General Relativity
General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ... theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, G ... of gravitation Gravity (), or gravitation, is a natural phenomenon Types of natural phenomena include: Weather, fog, thunder, tornadoes; biological processes, decomposition, germination seedlings, three days after germination. Germination is th ... published by Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely ac ...
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Mathematical Descriptions Of The Electromagnetic Field
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ..., one of the four fundamental interaction In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...s of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking. Vector field approach The most common description of the electromagnetic field uses two three-dimensional vector field In vector ca ...
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