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Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892.
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Tensor and calculus
Tensor calculus has many real-life applications, especially in the field of general relativity.
Tensor and was
It was during this period that he developed his doctoral dissertation, " The Application of Tensor Methods to Riemannian Manifolds.
For this body of work, Basser was awarded the 2008 International Society for Magnetic Resonance in Medicine Gold Medal for " his pioneering and innovative scientific contributions in the development of Diffusion Tensor Imaging ( DTI ).
Tensor and by
* Tensor – vector – scalar gravity ( TeVeS ) ( 2004 ), a relativistic modification of MOND by Jacob Bekenstein
* Tensor tympani ( innervated by the mandibular nerve of CN V )
* Tensor veli palatini ( innervated by the mandibular nerve of CN V )
Their anterior fibers, by drawing the greater trochanter forward, rotate the thigh inward, in which action they are also assisted by the Tensor fasciæ latæ.
* " Omnilingual ", by H. Beam Piper by Tenser, said the Tensor
* Duffy Wyg & ( read " Wygand "), the composer of the "' Tenser ,' said the Tensor " jingle, is portrayed as the essential " modern career girl-the virgin seductress " who seems to crave either humiliation or conquest by men.
Tensor and differential
Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences and engineering.
* Tensor density in differential geometry
* The Tensor algebra is a DGA with differential similar to that of the Koszul complex.
Tensor and Ricci
ca: Tensor de Ricci
es: Tensor de Ricci
pt: Tensor de curvatura de Ricci
In three dimensions this condition is met when the Ricci Tensor () is equal to the metric times the Ricci Scalar (, not to be confused with the R of the previous section ).
Tensor and .
Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.
Tensor order is the number of indices required to write a tensor, and thus matrices all have tensor order 2.
More precisely, matrices are tensors of type ( 1, 1 ), having one row index and one column index, also called covariant order 1 and contravariant order 1 ; see Tensor ( intrinsic definition ) for details.
* Tensor fields, which associate a tensor to every point in space.
Manifolds, Tensor Analysis, and Applications, 2nd edn.
Tensor contraction can be seen as a generalization of the trace.
Tensor – vector – scalar gravity ( TeVeS ) is a proposed relativistic theory that is equivalent to Modified Newtonian dynamics ( MOND ) in the non-relativistic limit, which purports to explain the galaxy rotation problem without invoking dark matter.
Tensor product of fields.
The co / contra-variant nature of vector coordinates has been treated as an elementary characterisation in Tensor Analysis.
Chou, N. J. Pagano, Elasticity: Tensor, Dyadic, and Engineering Approaches, Dover, ISBN 0-486-66958-0
It is known as TeVeS for Tensor / Vector / Scalar and it introduces three different fields in space time to replace the one gravitational field.
In the mid 1970s Dziewonski started the Harvard Global Centroid Moment Tensor Catalog.
Tractography is performed using Diffusion Tensor Imaging, an MR technique which is sensitive to the diffusion of water in the body, and can be used to reveal its 3D shape.
This shape is often modeled as an ellipsoid, and the technique is then called Diffusion Tensor Imaging.
The gluteus medius and gluteus minimus abduct the thigh, when the limb is extended, and are principally called into action in supporting the body on one limb, in conjunction with the Tensor fasciæ latæ.
Newer studies such as Diffusion Tensor Imaging are able to demonstrate the degree of white matter fiber tract injury even when the standard MRI is negative.
The Tensor Contraction Engine, or TCE, provides most of the functionality for the correlated methods, and can be used to develop additional many-body methods using a Python interface.
This led Moffat to propose Metric Skew Tensor Gravity ( MSTG ), in which a skew symmetric tensor field postulated as part of the gravitational action.
calculus and was
Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.
This was done by Alonzo Church in 1936 with the concept of " effective calculability " based on his λ calculus and by Alan Turing in the same year with his concept of Turing machines.
The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that subject.
More than one century after Euler's paper on the bridges of Königsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from differential calculus to study a particular class of graphs, the trees.
His first major work, a textbook on calculus, was published in 1884 and was credited to Genocchi.
In fact, he became so determined to teach his new mathematical symbols that the calculus in his course was neglected.
In early calculus the use of infinitesimal quantities was unrigorous and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley.
A recent study argues that the force of Berkeley's criticisms has been overestimated ; that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof ; and that Leibniz's system for differential calculus was free of logical contradictions.
Several mathematicians, including Maclaurin and d ' Alembert, attempted to prove the soundness of using limits, but it would be 150 years later, through the work of Augustin Louis Cauchy and Karl Weierstrass, where a means was finally found to avoid mere " notions " of infinitely small quantities, that the foundations of differential and integral calculus were made firm.
After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by Abraham Robinson in the 1960s.
The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis — family friends of Euler — were responsible for much of the early progress in the field.
Lisp was originally created as a practical mathematical notation for computer programs, influenced by the notation of Alonzo Church's lambda calculus.
Although propositional logic ( which is interchangeable with propositional calculus ) had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus and expanded by the Stoics.
Lazy evaluation was introduced for the lambda calculus by and for programming languages independently by and.
Historically, the most important system was the untyped lambda calculus.
The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.
She is credited with writing the first book discussing both differential and integral calculus and was an honorary member of the faculty at the University of Bologna.
Newman subsequently arranged for Turing to visit Princeton where Alonzo Church was working on the same problem but using his Lambda calculus.
Historically, ML stands for metalanguage: it was conceived to develop proof tactics in the LCF theorem prover ( whose language, pplambda, a combination of the first-order predicate calculus and the simply typed polymorphic lambda calculus, had ML as its metalanguage ).
Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity.
File: Evangelista Torricelli2. jpg | Evangelista Torricelli ( 1608-1647 ): invented the barometer ( a glass tube of mercury inverted into a dish ), found that the change of height of the mercury each day was from atmospheric pressure, worked in geometry and developed integral calculus, published findings on fluid and projectile motion in his 1644 Opera Geometrica ( Geometric Works )
Roger Porter, another architect of the program, acknowledges that the program was weakened by the many hands that changed the President's calculus, such as Congress.
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