Books: 'Euler's theorem'

1

Chen, Jingkai. Nonlocal Euler–Bernoulli Beam Theories. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-69788-4.

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2

1707-1783, Euler Leonhard, ed. Leonhard Euler et la découverte progressive des sommations des séries infinies. Paris: Compagnie littéraire, 2010.

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3

Turner, J. C. Number trees for Pythagoras, Plato, Euler, and the modular group. Hamilton, N.Z: University of Waikato, 1990.

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4

Hakfoort, Casper. Optics in the age of Euler: Conceptions of the nature of light, 1700-1795. Cambridge: Cambridge University Press, 1995.

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5

Euler through time: A new look at old themes. Providence, R.I: American Mathematical Society, 2006.

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6

Diskin, Boris. New factorizable discretizations for the Euler equations. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 2002.

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7

Shimura, Gorō. Euler products and Eisenstein series. Providence, R.I: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1997.

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8

Deshpande, Suresh M. A second-order accurate kinetic-theory-based method for inviscid compressible flows. Hampton, Va: Langley Research Center, 1986.

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9

Optica in de eeuw van Euler: Opvattingen over de natuur van het licht, 1700-1795 = Optics in the age of Euler : conceptions of the nature of light, 1700-1795. Amsterdam: Rodopi, 1986.

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10

Tadmor, Eitan. A minimum entropy principle in the gas dynamics equation. Hampton, Va: ICASE, 1986.

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11

De fis van Euler: Een nieuwe visie op de muziek van Schubert, Beethoven, Mozart en Bach. Bloemendaal: Aramith, 1989.

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12

Diskin, Boris. Analysis of boundary conditions for factorizable discretizations of the Euler equations. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 2002.

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13

De Piero, Alvise, translator, writer of added commentary and Euler Leonhard 1707-1783, eds. Il Tentamen novae theoriae musicae di Leonhard Euler (Pietroburgo 1739): Traduzione e introduzione. Torino, Italia: Accademia delle scienze di Torino, 2010.

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14

Lui, Shiu-Hong. Entropy analysis of kinetic flux vector splitting schemes for the compressible Euler equations. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1999.

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15

Ono, Takashi. Variations on a theme of Euler: Quadratic forms, elliptic curves, and Hopf maps. New York: Plenum Press, 1994.

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16

Ramūnas, Garunkštis, ed. The Lerch zeta-function. Dordrecht: Kluwer Academic Publishers, 2002.

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17

Goodrich, John W. An approach to the development of numerical algorithms for first order linear hyperbolic systems in multiple space dimensions: The constant coefficient case. [Washington, D.C.]: National Aeronautics and Space Administration, 1995.

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18

Regularised integrals, sums, and traces: An analytic point of view. Providence, R.I: American Mathematical Society, 2012.

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19

Invitation to classical analysis. Providence, R.I: American Mathematical Society, 2012.

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20

Hyperbolic partial differential equations and geometric optics. Providence, R.I: American Mathematical Society, 2012.

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21

Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. Oxford University Press, 2019.

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22

Wilson, Robin J. Euler's pioneering equation: The most beautiful theorem in mathematics. 2018.

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23

Graph Theory: Euler's Rich Legacy (Contemporary Applied Mathematics). Everyday Learning Corporation, 1987.

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24

Isett, Philip. Main Lemma Implies the Main Theorem. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691174822.003.0011.

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This chapter shows that the Main Lemma implies the main theorem. It proves Theorem (10.1) by inductively applying the Main Lemma in order to construct a sequence of solutions of the Euler-Reynolds system. At each stage of the induction, an energy function is chosen along with a parameter whose choice determines the growth of the frequency parameter and the decay of the energy level. A base case lemma is then established, after which the proof of the Main Theorem (10.1) is presented so that the Main Lemma implies the Main Theorem. The Main Lemma is employed to approximately prescribe the energy increment of the correction. The solution obtained at the end of the process is nontrivial.

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25

Rajeev, S. G. Hamiltonian Systems Based on a Lie Algebra. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0010.

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There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is not invariant, on it. The concepts of a vector space, Lie algebra, and inner product are reviewed. A hamiltonian dynamical system is derived from each metric Lie algebra. The Virasoro algebra (famous in string theory) is shown to lead to the KdV equation; and in a limiting case, to the Burgers equation for shocks. A hamiltonian formalism for two-dimensional Euler equations is then developed in detail. A discretization of these equations (using a spectral method) is then developed using mathematical ideas from quantum mechanics. Then a hamiltonian formalism for the full three-dimensional Euler equations is developed. The Clebsch variables which provide canonical pairs for fluid dynamics are then explained, in analogy to angular momentum.

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26

Rubin, Karl. Euler Systems. Princeton University Press, 2000.

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27

Euler Systems. Princeton University Press, 2000.

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28

Rubin, Karl. Euler Systems. (Am-147). Princeton University Press, 2014.

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29

Rubin, Karl. Euler Systems. (AM-147), Volume 147. Princeton University Press, 2014.

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30

Isett, Philip. The Main Iteration Lemma. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691174822.003.0010.

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This chapter properly formalizes the Main Lemma, first by discussing the frequency energy levels for the Euler-Reynolds equations. Here the bounds are all consistent with the symmetries of the Euler equations, and the scaling symmetry is reflected by dimensional analysis. The chapter proceeds by making assumptions that are consistent with the Galilean invariance of the Euler equations and the Euler-Reynolds equations. If (v, p, R) solve the Euler-Reynolds equations, then a new solution to Euler-Reynolds with the same frequency energy levels can be obtained. The chapter also states the Main Lemma, taking into account dimensional analysis, energy regularity, and Onsager's conjecture. Finally, it introduces the main theorem (Theorem 10.1), which states that there exists a nonzero solution to the Euler equations with compact support in time.

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31

Reciprocity Laws: From Euler to Eisenstein (Springer Monographs in Mathematics). Springer, 2000.

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32

Livermore, Roy. The Paving Stone Theory of World Tectonics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198717867.003.0003.

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Tuzo Wilson introduces the concept of transform faults, which has the effect of transforming Earth Science forever. Resistance to the new ideas is finally overcome in the late 1960s, as the theory of moving plates is established. Two scientists play a major role in quantifying the embryonic theory that is eventually dubbed ‘plate tectonics’. Dan McKenzie applies Euler’s theorem, used previously by Teddy Bullard to reconstruct the continents around the Atlantic, to the problem of plate rotations on a sphere and uses it to unravel the entire history of the Indian Ocean. Jason Morgan also wraps plate tectonics around a sphere. Tuzo Wilson introduces the idea of a fixed hotspot beneath Hawaii, an idea taken up by Jason Morgan to create an absolute reference frame for plate motions.

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33

A general multiblock Euler code for propulsion integration. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1991.

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34

A, Appleby R., Chen H. C, and Langley Research Center, eds. A general multiblock Euler code for propulsion integration. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1991.

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35

A general multiblock Euler code for propulsion integration. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1991.

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36

Hakfoort, Casper. Optics in the Age of Euler: Conceptions of the Nature of Light, 17001795. Cambridge University Press, 2006.

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37

Euler, Leonhard. Commentationes physicae ad theoriam caloris, electricitatis et magnetismi pertinentes (Leonhard Euler, Opera Omnia). Birkhäuser Basel, 1992.

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38

(Adapter), George A. Anastassiou, ed. The History of Approximation Theory: From Euler to Bernstein. Birkhäuser Boston, 2005.

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39

Farb, Benson, and Dan Margalit. Presentations and Low-dimensional Homology. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691147949.003.0006.

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This chapter presents explicit computations of the first and second homology groups of the mapping class group. It begins with a simple proof, due to Harer, of the theorem of Mumford, Birman, and Powell; the proof includes the lantern relation, a relation in Mod(S) between seven Dehn twists. It then applies a method from geometric group theory to prove the theorem that Mod(Sɡ) is finitely presentable. It also provides explicit presentations of Mod(Sɡ), including the Wajnryb presentation and the Gervais presentation, and gives a detailed construction of the Euler class, the most basic invariant for surface bundles, as a 2-cocycle for the mapping class group of a punctured surface. The chapter concludes by explaining the Meyer signature cocycle and the important connection of this circle of ideas with the theory of Sɡ-bundles.

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40

Isett, Philip. A Main Lemma for Continuous Solutions. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691174822.003.0005.

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This chapter introduces the Main Lemma that implies the existence of continuous solutions. According to this lemma, there exist constants K and C such that the following holds: Let ϵ‎ > 0, and suppose that (v, p, R) are uniformly continuous solutions to the Euler-Reynolds equations on ℝ x ³, with v uniformly bounded⁷ and suppR ⊆ I x ³ for some time interval. The Main Lemma implies the following theorem: There exist continuous solutions (v, p) to the Euler equations that are nontrivial and have compact support in time. To establish this theorem, one repeatedly applies the Main Lemma to produce a sequence of solutions to the Euler-Reynolds equations. To make sure the solutions constructed in this way are nontrivial and compactly supported, the lemma is applied with e(t) chosen to be any sequence of non-negative functions whose supports are all contained in some finite time interval.

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41

Rajeev, S. G. Fluid Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.001.0001.

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Starting with a review of vector fields and their integral curves, the book presents the basic equations of the subject: Euler and Navier–Stokes. Some solutions are studied next: ideal flows using conformal transformations, viscous flows such as Couette and Stokes flow around a sphere, shocks in the Burgers equation. Prandtl’s boundary layer theory and the Blasius solution are presented. Rayleigh–Taylor instability is studied in analogy with the inverted pendulum, with a digression on Kapitza’s stabilization. The possibility of transients in a linearly stable system with a non-normal operator is studied using an example by Trefethen et al. The integrable models (KdV, Hasimoto’s vortex soliton) and their hamiltonian formalism are studied. Delving into deeper mathematics, geodesics on Lie groups are studied: first using the Lie algebra and then using Milnor’s approach to the curvature of the Lie group. Arnold’s deep idea that Euler’s equations are the geodesic equations on the diffeomorphism group is then explained and its curvature calculated. The next three chapters are an introduction to numerical methods: spectral methods based on Chebychev functions for ODEs, their application by Orszag to solve the Orr–Sommerfeld equation, finite difference methods for elementary PDEs, the Magnus formula and its application to geometric integrators for ODEs. Two appendices give an introduction to dynamical systems: Arnold’s cat map, homoclinic points, Smale’s horse shoe, Hausdorff dimension of the invariant set, Aref ’s example of chaotic advection. The last appendix introduces renormalization: Ising model on a Cayley tree and Feigenbaum’s theory of period doubling.

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42

Mann, Peter. Differential Equations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0035.

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This chapter presents the general formulation of the calculus of variations as applied to mechanics, relativity and field theories. The calculus of variations is a common mathematical technique used throughout classical mechanics. First developed by Euler to determine the shortest paths between fixed points along a surface, it was applied by Lagrange to mechanical problems in analytical mechanics. The variational problems in the chapter have been simplified for ease of understanding upon first introduction, in order to give a general mathematical framework. This chapter takes a relaxed approach to explain how the Euler–Lagrange equation is derived using this method. It also discusses first integrals. The chapter closes by defining the functional derivative, which is used in classical field theory.

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43

Isett, Philip. Gluing Solutions. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691174822.003.0012.

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This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

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44

König, Denes. Theorie der endlichen und unendlichen Graphen (Teubner-Archiv zur Mathematik) (German Edition). Springer, 1986.

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45

Tretkoff, Paula. Topological Invariants and Differential Geometry. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691144771.003.0002.

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This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.

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46

Tretkoff, Paula. Riemann Surfaces, Coverings, and Hypergeometric Functions. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691144771.003.0003.

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This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.

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47

Leonhardi Euleri Opera Omnia: Series Secunda: Commentationes Astronomicae Ad Theoriam Perturbationum Pertinentes - 2nd Part: Opera Mechanica Et Astronomica. Birkhauser Verlag AG, 2001.

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48

Chi-Wang, Shu, and Langley Research Center, eds. Uniform high order spectral methods for one and two dimensional Euler equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1991.

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49

Research Institute for Advanced Computer Science (U.S.), ed. Control theory based airfoil design using the Euler equations. [Moffett Field, Calif.]: Research Institute for Advanced Computer Science, NASA Ames Research Center, 1994.

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50

Isett, Philip. The Divergence Equation. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691174822.003.0006.

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This chapter introduces the divergence equation. A key ingredient in the proof of the Main Lemma for continuous solutions is to find special solutions to this divergence equation, which includes a smooth function and a smooth vector field on ³, plus an unknown, symmetric (2, 0) tensor. The chapter presents a proposition that takes into account a condition relating to the conservation of momentum as well as a condition that reflects Newton's law, which states that every action must have an equal and opposite reaction. This axiom, in turn, implies the conservation of momentum in classical mechanics. In view of Noether's theorem, the constant vector fields which act as Galilean symmetries of the Euler equation are responsible for the conservation of momentum. The chapter shows proof that all solutions to the Euler-Reynolds equations conserve momentum.

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Books on the topic 'Euler's theorem'

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