16,028 research outputs found
Approaching Cauchy’s Theorem
We hope to initiate a discussion about various methods for introducing Cauchy’s Theorem. Although Cauchy’s Theorem is the fundamental theorem upon which complex analysis is based, there is no “standard approach.” The appropriate choice depends upon the prerequisites for the course and the level of rigor intended. Common methods include Green’s Theorem, Goursat’s Lemma, Leibniz’ Rule, and homotopy theory, each of which has its positives and negatives
Cauchy's infinitesimals, his sum theorem, and foundational paradigms
Cauchy's sum theorem is a prototype of what is today a basic result on the
convergence of a series of functions in undergraduate analysis. We seek to
interpret Cauchy's proof, and discuss the related epistemological questions
involved in comparing distinct interpretive paradigms. Cauchy's proof is often
interpreted in the modern framework of a Weierstrassian paradigm. We analyze
Cauchy's proof closely and show that it finds closer proxies in a different
modern framework.
Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation;
uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc
An elementary proof of Euler formula using Cauchy's method
The use of Cauchy's method to prove Euler's well-known formula is an object
of many controversies. The purpose of this paper is to prove that Cauchy's
method applies for convex polyhedra and not only for them, but also for
surfaces such as the torus, the projective plane, the Klein bottle and the
pinched torus
Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow
Two prized papers, one by Augustin Cauchy in 1815, presented to the French
Academy and the other by Hermann Hankel in 1861, presented to G\"ottingen
University, contain major discoveries on vorticity dynamics whose impact is now
quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal
incompressible flow in terms of three invariants that generalize to three
dimensions the now well-known law of conservation of vorticity along fluid
particle trajectories for two-dimensional flow. This has very recently been
used to prove analyticity in time of fluid particle trajectories for 3D
incompressible Euler flow and can be extended to compressible flow, in
particular to cosmological dark matter. Hankel showed that Cauchy's formulation
gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux
invariants and, in the middle of the proof, derived an intermediate result
which is the conservation of the circulation of the velocity around a closed
contour moving with the fluid. This circulation theorem was to be rediscovered
independently by William Thomson (Kelvin) in 1869. Cauchy's invariants were
only occasionally cited in the 19th century --- besides Hankel, foremost by
George Stokes and Maurice L\'evy --- and even less so in the 20th until they
were rediscovered via Emmy Noether's theorem in the late 1960, but reattributed
to Cauchy only at the end of the 20th century by Russian scientists.Comment: 23 pages, 6 figures, EPJ H (history), in pres
Ten Misconceptions from the History of Analysis and Their Debunking
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note:
text overlap with arXiv:1108.2885 and arXiv:1110.545
Discrete complex analysis on planar quad-graphs
We develop a linear theory of discrete complex analysis on general
quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon,
Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our
approach based on the medial graph yields more instructive proofs of discrete
analogs of several classical theorems and even new results. We provide discrete
counterparts of fundamental concepts in complex analysis such as holomorphic
functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss
discrete versions of important basic theorems such as Green's identities and
Cauchy's integral formulae. For the first time, we discretize Green's first
identity and Cauchy's integral formula for the derivative of a holomorphic
function. In this paper, we focus on planar quad-graphs, but we would like to
mention that many notions and theorems can be adapted to discrete Riemann
surfaces in a straightforward way.
In the case of planar parallelogram-graphs with bounded interior angles and
bounded ratio of side lengths, we construct a discrete Green's function and
discrete Cauchy's kernels with asymptotics comparable to the smooth case.
Further restricting to the integer lattice of a two-dimensional skew coordinate
system yields appropriate discrete Cauchy's integral formulae for higher order
derivatives.Comment: 49 pages, 8 figure
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