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In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. [citation needed]The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction ...
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.
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 ...
Toggle Del formula subsection. 4.1 Calculation rules. 5 Cartesian derivation. 6 Cylindrical derivation. 7 Spherical derivation. ... Directional derivative (A ⋅ ∇ ...
2.4 Directional derivative. 2.5 Laplacian. 2.6 Hessian matrix. 2.7 Tensor derivative. 3 Product rules. ... The formula for the vector product is slightly less ...
These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. [1] The directional derivative provides a systematic way of finding these derivatives. [2]
If all the partial derivatives of exist and are continuous at , then they determine the directional derivative of in the direction by the formula: [43] = =. Total derivative, total differential and Jacobian matrix
Hence the formula () is regarded as the directional derivative at point in the direction of . This is analogous to vector calculus, where the inner product of a vector v {\displaystyle v} with the gradient gives the directional derivative in the direction of v {\displaystyle v} .