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In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of ...
Rigorously, a subderivative of a convex function : at a point in the open interval is a real number such that () for all .By the converse of the mean value theorem, the set of subderivatives at for a convex function is a nonempty closed interval [,], where and are the one-sided limits = (), = + ().
In the mathematical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem states that a representation of an algebraic group over the adeles is automorphic whenever the L-functions of various twists of it are well-behaved.
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) [9] named after Alexis Clairaut and Hermann Schwarz, states that for a function : defined on a set , if is a point such that some neighborhood of is contained in and has continuous second partial derivatives on that neighborhood of , then for all i ...
A version of the fundamental theorem of calculus holds for the Gateaux derivative of , provided is assumed to be sufficiently continuously differentiable. Specifically: Specifically: Suppose that F : X → Y {\displaystyle F:X\to Y} is C 1 {\displaystyle C^{1}} in the sense that the Gateaux derivative is a continuous function d F : U × X → Y ...
A function is invertible if and only if its converse relation is a function, in which case the converse relation is the inverse function. The converse relation of a function f : X → Y {\displaystyle f:X\to Y} is the relation f − 1 ⊆ Y × X {\displaystyle f^{-1}\subseteq Y\times X} defined by the graph f − 1 = { ( y , x ) ∈ Y × X : y ...
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.