differential geometry book that uses infinitesimals

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We will see that this result applies to a vast class of PDE, e.g. Generalized smooth functions (GSF) are a minimal branch of Colombeau’s theory that allows for all these possibilities, see preprints at ∼giordap7, and the result is hence proved in this framework.

Finally, to define Fréchet-like spaces of GF, we needed a good notion of compact set for GF and for their norms. Therefore, we also needed GF defined only on infinitesimal sets. But if we have an infinitesimal h, then the ODE y'(1 + y)h = −t has solution defined only in the infinitesimal interval. But in that case, the Lipschitz constants could be infinite so we needed a language with infinitesimal and infinite numbers. Since that’s not simple if we define GF as functionals, we followed Cauchy and Dirac and thought GF as smooth set-theoretical functions that can take infinite values and derivatives.

DIFFERENTIAL GEOMETRY BOOK THAT USES INFINITESIMALS FREE

Free composition of generalized functions (GF) is one of those. To realize this goal, some necessary requirements needed to be satisfied. In case of PDE, the proof is based on a generalization of the Banach fixed point theorem, but with loss of derivatives, without the uniqueness part, and starting the iterates exactly from the initial condition. Both the right hand side of the DE and the initial conditions can be, e.g., Schwartz distributions. We already proved a Picard-Lindelöf theorem for normal singular nonlinear ODE and PDE. More precisely, if A is a standard point of a standard curve γ and B, C are points of γ defined by) (α γ + = t B and) (β γ + = t C where α and β are infinitesimals, we intend to calculate the quantity Our goal is to give a nonstandard proof of this fact.

It is known that the radius of curvature of a plane curve () t γ is the limit of the radius of a circle circumscribed to a triangle ABC, where B and C are points of γ infinitely close to A. Tied with Hooke’s law using the language of nilpotent infinitesimals.

DIFFERENTIAL GEOMETRY BOOK THAT USES INFINITESIMALS HOW TO

In particular, an interesting rigorous deduction of the wave equation is given, which clarifies how to formalize the approximations Used in physics, and all these calculations now become rigorous, and at the same time, formally equal to the informal ones. Finally, we present several applications to informal classical calculations

Using nilpotent infinitesimals, every smooth function becomes a polynomialīecause the remainder in Taylor’s formulas is now zero. The construction is highly constructive, and every Fermat real admits a clearĪnd order-preserving geometrical representation. Product of powers of nilpotent infinitesimals vanishes, study the identity principle for polynomials, and discuss the definitionĪnd properties of the total order relation. We face the problem of deciding whether or not a Only, the theory of Fermat reals is consistent with the classical logic. In particular, in contrast to SIA, which admits models in intuitionistic logic Without any background in mathematical logic. The construction is inspired by Smooth Infinitesimal Analysis (SIA) and provides a powerful theory of actual infinitesimals We introduce a ring of the so-called Fermat reals, which is an extension of the real field containing nilpotent infinitesimals.

Differential geometry book that uses infinitesimals