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The forward difference operator () = (+) ()is a delta operator. Differentiation with respect to x, written as D, is also a delta operator.; Any operator of the form = (where D n (ƒ) = ƒ (n) is the n th derivative) with is a delta operator.
In calculus, the differential represents a change in the linearization of a function.. The total differential is its generalization for functions of multiple variables.; In traditional approaches to calculus, differentials (e.g. dx, dy, dt, etc.) are interpreted as infinitesimals.
Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of have the form , where p is a prime number).
Demonstration of this result is fairly straightforward under the assumption that () is differentiable near the neighborhood of and ′ is continuous at with ′ ().To begin, we use the mean value theorem (i.e.: the first order approximation of a Taylor series using Taylor's theorem):
(Figure 2) Illustration of numerical integration for the equation ′ =, = Blue is the Euler method; green, the midpoint method; red, the exact solution, =. The step size is =
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Formally, a Δ-set is a sequence of sets {} = together with maps : + for each and =,, …, +, that satisfy + = + whenever <.Often, the superscript of is omitted for brevity.. This definition generalizes the notion of a simplicial complex, where the are the sets of n-simplices, and the are the associated face maps, each mapping the -th face of a simplex in + to a simplex in .
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