Directional derivative

Common card · No. 220067

In multivariable calculus, the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given point. If the vector is multiplied by a scalar, the corresponding directional derivative is multiplied by the same scalar. Some elementary texts instead use the phrase "directional derivative in the direction of v" for the rate of change of a function per unit distance in that direction. In that convention the nonzero vector v is first normalized to the unit vector 𝐯̂=𝐯/‖𝐯‖, where the normalized vector is denoted with a circumflex (hat) symbol: ^. The directional derivative of a scalar function f with respect to a vector v may be denoted by any of the following: ∇𝐯f(𝐱)=f'𝐯(𝐱)=D𝐯f(𝐱)=Df(𝐱)(𝐯)=∂𝐯f(𝐱)=∂f(𝐱)∂𝐯=𝐯⋅∇f(𝐱)=𝐯⋅∂f(𝐱)∂𝐱. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. In functional analysis, the analogous...

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