Normal derivative

In mathematics, the directional derivative of a multivariate differentiable 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 velocity specified by v. 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.

The directional derivative is a special case of the Gâteaux derivative.

Directional derivatives can be also denoted by:

where v is a parameterization of a curve to which v is tangent and which determines its magnitude.

The directional derivative of a scalar function

along a vector

is the function ${\displaystyle \nabla _{\mathbf {v}}{f}}$ defined by the limit

This definition is valid in a broad range of contexts, for example where the norm of a vector (and hence a unit vector) is undefined.

If the function f is differentiable at x, then the directional derivative exists along any vector v, and one has

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