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Taylor dispersion or Taylor diffusion is an apparent or effective diffusion of some scalar field arising on the large scale due to the presence of a strong, confined, zero-mean shear flow on the small scale. Essentially, the shear acts to smear out the concentration distribution in the direction of the flow, enhancing the rate at which it ...
In mathematics, Itô's lemma or Itô's formula is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process.It serves as the stochastic calculus counterpart of the chain rule.
That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...
The notes that Marx took have been collected into four independent treatises: On the Concept of the Derived Function, On the Differential, On the History of Differential Calculus, and Taylor's Theorem, MacLaurin's Theorem, and Lagrange's Theory of Derived Functions, along with several notes, additional drafts, and supplements to these four ...
The above arithmetic can be generalized to calculate second order and higher derivatives of multivariate functions. However, the arithmetic rules quickly grow complicated: complexity is quadratic in the highest derivative degree. Instead, truncated Taylor polynomial algebra can be used. The resulting arithmetic, defined on generalized dual ...
An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
In 3 dimensions, a differential 0-form is a real-valued function (,,); a differential 1-form is the following expression, where the coefficients are functions: + +; a differential 2-form is the formal sum, again with function coefficients: + +; and a differential 3-form is defined by a single term with one function as coefficient: .
This so-called "differential method" [9] will be described next. (For a derivation of Eq(13) and (14), see this section , below.) As is usual in applied mathematics, one approach for avoiding complexity is to approximate a function with another, simpler, function, and often this is done using a low-order Taylor series expansion.