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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".
The following procedure can be used to easily test if any source code is derivative code or not. Delete the code in question; Build (or compile) the project; If the build process simply replaces the source code which has been deleted, it is (obviously) code which has been derived from something else and is therefore, by definition, derivative code.
Note that linear functionals (multilinear 1-forms over ) are trivially alternating, so that () = =, while, by convention, 0-forms are defined to be scalars: () = =. The determinant on n × n {\displaystyle n\times n} matrices, viewed as an n {\displaystyle n} argument function of the column vectors, is an important example of an alternating ...
The set of all such test functions forms a vector space that is denoted by () or (). Most commonly encountered functions, including all continuous maps f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } if using U := R , {\displaystyle U:=\mathbb {R} ,} can be canonically reinterpreted as acting via " integration against a test function."
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In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague , in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
Hence Φ ∗ is a (linear) operator from multilinear forms on W to multilinear forms on V. As a special case, note that if F is a linear form (or (0,1)-tensor) on W , so that F is an element of W ∗ , the dual space of W , then Φ ∗ F is an element of V ∗ , and so pullback by Φ defines a linear map between dual spaces which acts in the ...
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f: