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Journal articles on the topic 'Tensor Calculus'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 June 2025

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1

Javaloyes, Miguel Angel. "Anisotropic tensor calculus." International Journal of Geometric Methods in Modern Physics 16, supp02 (2019): 1941001. http://dx.doi.org/10.1142/s0219887819410019.

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We introduce the anisotropic tensor calculus, which is a way of handling tensors that depends on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application we show how to define derivations using anisotropic linear connections in a manifold. In particular, we show that the Chern connection of a Finsler metric can be interpreted as the Levi-Civita connection and we introduce the anisotropic curvature tensor. We also relate the concept of anisotropic connection with the classical concept of linear connections in the vertical bundle. Furthermore, we also introduce the concept of anisotropic Lie derivative.

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2

Hackbusch, Wolfgang. "Numerical tensor calculus." Acta Numerica 23 (May 2014): 651–742. http://dx.doi.org/10.1017/s0962492914000087.

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The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid withngrid points per direction, a spatial dimensiondyieldsndgrid points. A grid function defined on such a grid is an example of a tensor of orderd. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly ind. In this way, data of sizend= 10001000can also be treated.This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.

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3

Lee, Wha-Suck, Johann Engelbrecht, and Rita Moller. "Tensor calculus: unlearning vector calculus." International Journal of Mathematical Education in Science and Technology 49, no. 2 (2017): 293–304. http://dx.doi.org/10.1080/0020739x.2017.1357849.

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4

N. J. Essa. "A Study in Tensor Analysis on Fractional Differential Implications." Advances in Nonlinear Variational Inequalities 28, no. 5s (2025): 31–44. https://doi.org/10.52783/anvi.v28.3541.

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This research investigates the synthesis of fractional calculus with tensor analysis in order to establish a comprehensive mathematical structure that successfully overcomes the limitations of classical tensor methods on systems having non-local and memory-dependent behavior. Classical tensor calculus, which relies on integer ordered calculus, often fails when applied to physical and engineering systems where interactions are not localized within immediate space and time neighborhood. Though it generalizes classical calculus by allowing derivatives and integrals of any real or complex order, fractional calculus provides a suitable toolkit for capturing such interactions as it captures processes that depend on their history. The inclusion of these fractional derivatives into tensor calculus would lead to a paradigm shift in our understanding and modeling capability of complicated systems spanning across multiple fields. By including fractional derivatives, existing theories based on classical derivatives can be extended with tensor fields for more accurate descriptions of some phenomena. These include various forms of physics and engineering such as materials science, biomechanics or geophysical modeling which have systems characterized by anomalous diffusion, viscoelastic material properties, irregular geometrical domains etc. The study’s primary objectives are to provide a strict definition for the operations involving fractional tensors together with investigating their consequences in different physical situations. It is hypothesized that applying fractional tensors will reveal new aspects of physical phenomena leading to better account for dynamics interaction between them. Among other things this means looking at changes in geometric characteristics (like curvature and torsion) due to fractional derivatives in differential geometry; how do solutions of field equations change in theoretical physics? By constructing computational techniques and theoretical frameworks, this work aims at equipping scientists and engineers with strong models that enable them model complex interacting systems more precisely thereby creating new frontiers in scientific disciplines. Such an improved mathematical framework could also make significant advancements towards understanding fundamentals underlying the universe or developing technologies based upon newly gained insights.

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5

Kopský, V. "Tensor Calculus of Twinning." Acta Crystallographica Section A Foundations of Crystallography 56, s1 (2000): s153. http://dx.doi.org/10.1107/s010876730002359x.

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6

Dods, Victor. "Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus." Mathematics 10, no. 18 (2022): 3231. http://dx.doi.org/10.3390/math10183231.

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In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.

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7

Saukova, Y. N., and M. A. Hundzina. "Tensor Calculus in Digital Colorimetry." Devices and Methods of Measurements 13, no. 3 (2022): 216–27. http://dx.doi.org/10.21122/2220-9506-2022-13-3-216-227.

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Any object can have many implementations in the form of digital images and any digital image can be processed many times increasing or decreasing accuracy and reliability. Digital colorimetry faces the need to work out issues of ensuring accuracy, metrological traceability and reliability. The purpose of this work was to generalize approaches to the description of multidimensional quantized spaces and show the possibilities of their adaptation to digital colorimetry. This approach will minimize the private and global risks in measurements.For color identification digital colorimetry uses standard color models and spaces. Most of them are empirical and are improved during the transition from standard to real observation conditions taking into account the phenomena of vision and the age of observers. From the point of view of measurement, a digital image can be represented by a combinatorial model of an information and measurement channel with the appearance of the phenomenon of a color covariance hypercube requiring a significant amount of memory for data storage and processing. The transition from the covariance hypercube to high-dimensional matrices and tensors of the first, second and higher ranks provides the prospect of optimizing the color parameters of a digital image by the criterion of information entropy.Tensor calculus provides opportunities for expanding the dynamic range in color measurements describing multidimensional vector fields and quantized spaces with indexing tensors and decomposing them into matrices of low orders.The proposed complex approach based on tensor calculus. According to this approach the color space is a set of directed vector fields undergoing sampling, quantization and coding operations. Also it is a dynamic open system exchanging information with the environment at a given level and to identify color with specified levels of accuracy, reliability, uncertainty and entropy.

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8

Chorlton, Frank, and A. N. Srivastava. "Tensor Calculus: Theory and Problems." Mathematical Gazette 79, no. 484 (1995): 245. http://dx.doi.org/10.2307/3620129.

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9

Аbiev, N. А., and Zh K. Askerbekova. "USING MAPLE TO TENSOR CALCULUS." Theoretical & Applied Science 83, no. 03 (2020): 175–80. http://dx.doi.org/10.15863/tas.2020.03.83.36.

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10

Vassilevich, D. V. "Tensor calculus on noncommutative spaces." Classical and Quantum Gravity 27, no. 9 (2010): 095020. http://dx.doi.org/10.1088/0264-9381/27/9/095020.

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11

Not Available, Not Available. "Tensor calculus and analytical dynamics." Archive of Applied Mechanics (Ingenieur Archiv) 69, no. 7 (1999): 499–500. http://dx.doi.org/10.1007/s004190050238.

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12

Damm, Carsten, Markus Holzer, and Pierre McKenzie. "The complexity of tensor calculus." Computational Complexity 11, no. 1-2 (2002): 54–89. http://dx.doi.org/10.1007/s00037-000-0170-4.

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13

Boulanger, Nicolas. "A Weyl-covariant tensor calculus." Journal of Mathematical Physics 46, no. 5 (2005): 053508. http://dx.doi.org/10.1063/1.1896381.

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14

WESTBURY, BRUCE W. "Invariant tensors for the spin representation of (7)." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 1 (2008): 217–40. http://dx.doi.org/10.1017/s0305004107000722.

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AbstractWe give a graphical calculus for the invariant tensors of the eight dimensional spin representation of the quantum groupUq(B3). This leads to a finite confluent presentation of the centraliser algebras of the tensor powers of this representation and a construction of a cellular basis.

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15

Altenbach, H. "Some remarks on the paper ‘A unified tensor approach to the analysis of mechanical systems’ by A. A. Fogarasy and M. R. Smith." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 223, no. 1 (2008): 143–55. http://dx.doi.org/10.1243/09544062jmes1141.

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The tensor calculus is a powerful mathematical tool in the analysis of mechanical problems (or generally, of problems in physics). The aim of Fogarasy and Smith's ‘A unified tensor approach to the analysis of mechanical systems’ is to demonstrate this fact considering a problem of dynamical analysis of mechanisms. Fogarasy and Smith are applying the traditional approach in tensor calculus, which comes originally from the mathematics. This implies the a priori introduction of coordinates, which is usual in the analysis of mechanisms. Later the tensor calculus is used for the discussion of the basic equations and their mathematical handling. In this sense, the results obtained by Fogarasy and Smith are mathematically correct, and the interested reader can now have many fruitful and stimulating ideas.

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16

Laue, Sören, Matthias Mitterreiter, and Joachim Giesen. "A Simple and Efficient Tensor Calculus." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (2020): 4527–34. http://dx.doi.org/10.1609/aaai.v34i04.5881.

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Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of these expressions. Recently, an algorithm for computing higher order derivatives of tensor expressions like Jacobians or Hessians has been introduced that is a few orders of magnitude faster than previous state-of-the-art approaches. Unfortunately, the approach is based on Ricci notation and hence cannot be incorporated into automatic differentiation frameworks like TensorFlow, PyTorch, autograd, or JAX that use the simpler Einstein notation. This leaves two options, to either change the underlying tensor representation in these frameworks or to develop a new, provably correct algorithm based on Einstein notation. Obviously, the first option is impractical. Hence, we pursue the second option. Here, we show that using Ricci notation is not necessary for an efficient tensor calculus and develop an equally efficient method for the simpler Einstein notation. It turns out that turning to Einstein notation enables further improvements that lead to even better efficiency.

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17

Makokha, A. N., and T. E. Tyshlyar. "AUTOMATION OF TENSOR CALCULATIONS BASED ON NEURAL NETWORKS." Sovremennaya nauka i innovatsii, no. 2 (34) (2021): 19–40. http://dx.doi.org/10.37493/2307-910x.2021.2.2.

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18

Ou, Miao-jung Yvonne. "Teaching multivariable calculus and tensor calculus with computer algebra software." ACM Communications in Computer Algebra 54, no. 4 (2020): 134–35. http://dx.doi.org/10.1145/3465002.3465005.

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To go from calculus of scalar functions of one variable to multivariate calculus of vector-valued functions is a steep learning curve for many students. It takes a lot of practice to get used to the new concepts such as the directional derivatives, the di.erentiability, the many types of first order di.erential operators, parameterization of surfaces and the fundamental theorems of integrals, e.g. the Divergence Theorem and the Stokes Theorem. Along the learning process of mastering the skills, the students often need to check whether the intermediate steps in the tedious calculations are correct. Unfortunately, this is beyond the capability of an ordinary calculator and the answers provided at the end of the books. This is where computer algebra software, such as Mathematica, can come to students' help. With the developed symbolic computation tools, the students can tweak a given problem, solve a new one by hand and then check the answer against the result obtained by using the computer algebra software.

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19

MACFARLANE, A. J., and HENDRYK PFEIFFER. "DEVELOPMENT OF A UNIFIED TENSOR CALCULUS FOR THE EXCEPTIONAL LIE ALGEBRAS." International Journal of Modern Physics A 19, no. 02 (2004): 287–316. http://dx.doi.org/10.1142/s0217751x04017562.

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The uniformity of the decomposition law, for a family ℱ of Lie algebras which includes the exceptional Lie algebras, of the tensor powers ad ⊗n of their adjoint representations ad is now well known. This paper uses it to embark on the development of a unified tensor calculus for the exceptional Lie algebras. It deals explicitly with all the tensors that arise at the n=2 stage, obtaining a large body of systematic information about their properties and identities satisfied by them. Some results at the n=3 level are obtained, including a simple derivation of the dimension and Casimir eigenvalue data for all the constituents of ad ⊗3. This is vital input data for treating the set of all tensors that enter the picture at the n=3 level, following a path already known to be viable for [Formula: see text]. The special way in which the Lie algebra [Formula: see text] conforms to its place in the family ℱ alongside the exceptional Lie algebras is described.

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20

BARRETT, JOHN W. "Skein spaces and spin structures." Mathematical Proceedings of the Cambridge Philosophical Society 126, no. 2 (1999): 267–75. http://dx.doi.org/10.1017/s0305004198003168.

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This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter −A.There is an application to Penrose's binor calculus, which is related to the tensor calculus of representations of SU(2). The perspective developed here is that this tensor calculus is actually a calculus of spinors on the plane and the matrices are determined by a type of spinor transport which generalizes to links in any 3-manifold.A second application shows that there is a skein space which is the algebra of functions on the set of spin structures for the 3-manifold.

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21

Xu, Hao. "Bergman kernel and Kähler tensor calculus." Pure and Applied Mathematics Quarterly 9, no. 3 (2013): 507–46. http://dx.doi.org/10.4310/pamq.2013.v9.n3.a7.

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22

Castellví, P., X. Jaén, and E. Llanta. "TTC: Symbolic tensor and exterior calculus." Computers in Physics 8, no. 3 (1994): 360. http://dx.doi.org/10.1063/1.168499.

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23

Castellví, P., X. Jaén, and E. Llanta. "Symbolic tensor calculus using index notation." Computers in Physics 9, no. 3 (1995): 335. http://dx.doi.org/10.1063/1.168532.

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24

Balfagón, A., and X. Jaén. "TTC: Symbolic tensor calculus with indices." Computers in Physics 12, no. 3 (1998): 286. http://dx.doi.org/10.1063/1.168656.

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25

Brandt, F., J. Simón, and U. Theis. "Exotic gauge theories from tensor calculus." Classical and Quantum Gravity 17, no. 7 (2000): 1627–36. http://dx.doi.org/10.1088/0264-9381/17/7/304.

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26

Joyal, André, and Ross Street. "The geometry of tensor calculus, I." Advances in Mathematics 88, no. 1 (1991): 55–112. http://dx.doi.org/10.1016/0001-8708(91)90003-p.

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27

Pollock, D. S. G. "Tensor products and matrix differential calculus." Linear Algebra and its Applications 67 (June 1985): 169–93. http://dx.doi.org/10.1016/0024-3795(85)90194-6.

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28

Fujita, Tomoyuki, and Keisuke Ohashi. "Superconformal Tensor Calculus in Five Dimensions." Progress of Theoretical Physics 106, no. 1 (2001): 221–47. http://dx.doi.org/10.1143/ptp.106.221.

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29

Lenzen, Manfred, and Bonnie McBain. "Using tensor calculus for scenario modelling." Environmental Modelling & Software 37 (November 2012): 41–54. http://dx.doi.org/10.1016/j.envsoft.2012.02.020.

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30

BEJANCU, AUREL. "A NEW APPROACH FOR SPACE-TIME-MATTER THEORY." International Journal of Geometric Methods in Modern Physics 10, no. 04 (2013): 1350004. http://dx.doi.org/10.1142/s0219887813500047.

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This is the first paper in a series of three papers on a new approach for space-time-matter (STM) theory. The main purpose of this approach is to replace the Levi-Civita connection on the space-time from the classical Kaluza–Klein theory by what we call the Riemannian horizontal connection on the general Kaluza–Klein space. This is done by a development of a 4D tensor calculus whose geometrical objects live in a 5D space. The 4D tensor calculus and the Riemannian horizontal connection enable us to define in a 5D space some 4D differential operators: horizontal differential, horizontal gradient, horizontal divergence and horizontal Laplacian, which have a great role in the presentation of the STM theory in a covariant form. Finally, we introduce and study the horizontal electromagnetic tensor field, the horizontal Ricci tensor and the horizontal Einstein gravitational tensor field, which replace the well-known tensor fields from the classical Kaluza–Klein theory.

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31

Laue, Sören, Matthias Mitterreiter, and Joachim Giesen. "A Simple and Efficient Tensor Calculus for Machine Learning." Fundamenta Informaticae 177, no. 2 (2020): 157–79. http://dx.doi.org/10.3233/fi-2020-1984.

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Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of these expressions. Recently, an algorithm for computing higher order derivatives of tensor expressions like Jacobians or Hessians has been introduced that is a few orders of magnitude faster than previous state-of-the-art approaches. Unfortunately, the approach is based on Ricci notation and hence cannot be incorporated into automatic differentiation frameworks from deep learning like TensorFlow, PyTorch, autograd, or JAX that use the simpler Einstein notation. This leaves two options, to either change the underlying tensor representation in these frameworks or to develop a new, provably correct algorithm based on Einstein notation. Obviously, the first option is impractical. Hence, we pursue the second option. Here, we show that using Ricci notation is not necessary for an efficient tensor calculus and develop an equally efficient method for the simpler Einstein notation. It turns out that turning to Einstein notation enables further improvements that lead to even better efficiency. The methods that are described in this paper for computing derivatives of matrix and tensor expressions have been implemented in the online tool www.MatrixCalculus.org.

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32

Robidoux, Nicolas, and Stanly Steinberg. "A Discrete Vector Calculus in Tensor Grids." Computational Methods in Applied Mathematics 11, no. 1 (2011): 23–66. http://dx.doi.org/10.2478/cmam-2011-0002.

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Abstract Mimetic discretization methods for the numerical solution of continuum mechanics problems directly use vector calculus and differential forms identities for their derivation and analysis. Fully mimetic discretizations satisfy discrete analogs of the continuum theory results used to derive energy inequalities. Consequently, continuum arguments carry over and can be used to show that discrete problems are well-posed and discrete solutions converge. A fully mimetic discrete vector calculus on three dimensional tensor product grids is derived and its key properties proven. Opinions regarding the future of the field are stated.

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33

Shun, Carson, Lam Kai. "A Mathematical Proof to the HKLam Theory by Linear/Tensor Algebra and Analysis." International Journal of Mathematics and Statistics Studies 10, no. 5 (2022): 1–14. http://dx.doi.org/10.37745/ijmss.13/vol10n5114.

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In the previous papers, I have mentioned several times of HKLam Theory and their everyday usage but without the abstract mathematical proof. In order to remediate the flaws, I am now trying to proof the theory through both Tensor Algebra and Analysis as well as the statistical inference in this present paper. Indeed, people always say that mathematicians are linear animals or participate much in the subject of linear algebra while the British Scientist Newton observed a falling apple and discovered the gravity together with the development of calculus. In a similar case, my proof in the part of tensor algebra will be an analogy to the linear mapping, transformation etc while there are the corresponding corollary real physical life cases – 2 to 3 dimensional vectors calculus or even higher dimension of tensor analysis. Indeed, my proof will be based on the order two tensor but the HKLam theory may be extended up to nth order tensor but NOT applicable to the topic of the planned politics or even economics etc.The main aim is to show the proof of HKLam Theory by linear/Tensor algebra together with some applications in fluid dynamic and stress tensor field etc.

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34

Popescu, Gelu. "!COMMUTANT LIFTING, TENSOR ALGEBRAS, AND FUNCTIONAL CALCULUS." Proceedings of the Edinburgh Mathematical Society 44, no. 2 (2001): 389–406. http://dx.doi.org/10.1017/s0013091598001059.

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AbstractA non-commutative multivariable analogue of Parrott’s generalization of the Sz.-Nagy–Foia\c{s} commutant lifting theorem is obtained. This yields Tomita-type commutant results and interpolation theorems (e.g. Sarason, Nevanlinna–Pick, Carathéodory) for $F_n^\infty\,\bar{\otimes}\,\M$, the weakly-closed algebra generated by the spatial tensor product of the non-commutative analytic Toeplitz algebra $F_n^\infty$ and an arbitrary von Neumann algebra $\M$. In particular, we obtain interpolation theorems for bounded analytic functions from the open unit ball of $\mathbb{C}^n$ into a von Neumann algebra.A variant of the non-commutative Poisson transform is used to extend the von Neumann inequality to tensor algebras, and to provide a generalization of the functional calculus for contractive sequences of operators on Hilbert spaces. Commutative versions of these results are also considered.AMS 2000 Mathematics subject classification: Primary 47L25; 47A57; 47A60. Secondary 30E05

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35

Nikabadze, M. U. "On some problems of tensor calculus. I." Journal of Mathematical Sciences 161, no. 5 (2009): 668–97. http://dx.doi.org/10.1007/s10958-009-9595-8.

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36

Nikabadze, M. U. "On some problems of tensor calculus. II." Journal of Mathematical Sciences 161, no. 5 (2009): 698–733. http://dx.doi.org/10.1007/s10958-009-9596-7.

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37

Blundell, Stephen J. "Tensor calculus for physics, by Dwight E.Neuenschwander." Contemporary Physics 57, no. 2 (2016): 263–64. http://dx.doi.org/10.1080/00107514.2015.1133710.

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38

Ivanov, E. A., та A. A. Kapustnikov. "Towards a tensor calculus for κ-supersymmetry". Physics Letters B 267, № 2 (1991): 175–82. http://dx.doi.org/10.1016/0370-2693(91)91244-p.

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39

Kugo, T., and K. Ohashi. "Supergravity Tensor Calculus in 5D from 6D." Progress of Theoretical Physics 104, no. 4 (2000): 835–65. http://dx.doi.org/10.1143/ptp.104.835.

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40

Mironov, A., and A. Morozov. "Correlators in tensor models from character calculus." Physics Letters B 774 (November 2017): 210–16. http://dx.doi.org/10.1016/j.physletb.2017.09.063.

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41

Batchelor, P. G., M. Moakher, D. Atkinson, F. Calamante, and A. Connelly. "A rigorous framework for diffusion tensor calculus." Magnetic Resonance in Medicine 53, no. 1 (2004): 221–25. http://dx.doi.org/10.1002/mrm.20334.

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42

Brandt, Friedemann. "Properties of an Alternative Off-Shell Formulation of 4D Supergravity." Symmetry 13, no. 4 (2021): 620. http://dx.doi.org/10.3390/sym13040620.

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This article elaborates on an off-shell formulation of D = 4, N = 1 supergravity whose auxiliary fields comprise an antisymmetric tensor field without gauge degrees of freedom. In particular, the relation to new minimal supergravity, a supercovariant tensor calculus and the construction of invariant actions including matter fields are discussed.

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43

Wood, Christopher J., Jacob D. Biamonte, and David G. Cory. "Tensor networks and graphical calculus for open quantum systems." Quantum Information and Computation 15, no. 9&10 (2015): 759–811. http://dx.doi.org/10.26421/qic15.9-10-3.

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We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate fidelity and entanglement fidelity of a channel in terms of each of the different representations of the channel.

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44

Lu, Yiyu, Peng Yue, and Sibei Wei. "Extension of Calculus Operations in Cartesian Tensor Analysis." Mathematics 8, no. 4 (2020): 561. http://dx.doi.org/10.3390/math8040561.

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In this paper, we derive and propose basic differential operations and generalized Green’s integral theorems applicable to multidimensional spaces based on Cartesian tensor analysis to solve some nonlinear problems in smooth spaces in the necessary dimensions. In practical applications, the theorem can be applied to numerical analysis in the conservation law, effectively reducing the dimensions of high-dimensional problems and reducing the computational difficulty, which can be effectively used in the solution of complex dimensional mechanical problems.

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45

Aguila, F. del, and R. Pittau. "Recursive numerical calculus of one-loop tensor integrals." Journal of High Energy Physics 2004, no. 07 (2004): 017. http://dx.doi.org/10.1088/1126-6708/2004/07/017.

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46

Mignani, R., E. Pessa, and G. Resconi. "Commutative diagrams and tensor calculus in Riemann spaces." Il Nuovo Cimento B 108, no. 12 (1993): 1319–31. http://dx.doi.org/10.1007/bf02755186.

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47

Nikabadze, M. U. "Topics on Tensor Calculus with Applications to Mechanics." Journal of Mathematical Sciences 225, no. 1 (2017): 1–194. http://dx.doi.org/10.1007/s10958-017-3467-4.

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48

Vasil, Geoffrey M., Keaton J. Burns, Daniel Lecoanet, Sheehan Olver, Benjamin P. Brown, and Jeffrey S. Oishi. "Tensor calculus in polar coordinates using Jacobi polynomials." Journal of Computational Physics 325 (November 2016): 53–73. http://dx.doi.org/10.1016/j.jcp.2016.08.013.

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49

Gourgoulhon, Éric, and Marco Mancini. "Symbolic tensor calculus on manifolds: a SageMath implementation." Les cours du CIRM 6, no. 1 (2018): 1–54. http://dx.doi.org/10.5802/ccirm.26.

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Kugo, T., and K. Ohashi. "Superconformal Tensor Calculus on an Orbifold in 5D." Progress of Theoretical Physics 108, no. 1 (2002): 203–28. http://dx.doi.org/10.1143/ptp.108.203.

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