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The directional derivative of a scalar function f with respect to a vector v at a point ... Proof of the last equation. In standard single-variable calculus, ...
The first step of the proof is to show that, for any fixed unit vector v, the v-directional derivative of u exists almost everywhere. This is a consequence of a special case of the Fubini theorem : a measurable set in R n has Lebesgue measure zero if its restriction to every line parallel to v has (one-dimensional) Lebesgue measure zero.
The original theorem given by J. M. Danskin in his 1967 monograph [1] provides a formula for the directional derivative of the maximum of a (not necessarily convex) directionally differentiable function. An extension to more general conditions was proven 1971 by Dimitri Bertsekas.
The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, ... Proof. The vectors x and c can be written as = ...
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
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 ...
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then
From this follows that the directional derivative is the inner product of its direction by the vector derivative. All needs to be observed is that the direction a {\displaystyle a} can be written a = ( a ⋅ e i ) e i {\displaystyle a=(a\cdot e^{i})e_{i}} , so that: