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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 coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gâteaux derivative. == Definition == The directional derivative of a scalar function : along a vector : is the function defined by the limit : In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector. Without the restriction, 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.〔The applicability extends to functions over spaces without a metric and to differentiable manifolds, such as in general relativity.〕 If the function ''f'' is differentiable at x, then the directional derivative exists along any vector v, and one has : where the on the right denotes the gradient and is the dot product.〔Technically, the gradient ∇''f'' is a covector, and the "dot product" is the action of this covector on the vector v (or equivalently, the duality pairing of the covector and the vector).〕 Intuitively, the directional derivative of ''f'' at a point x represents the rate of change of ''f'' with respect to time when moving past x at velocity v. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「directional derivative」の詳細全文を読む スポンサード リンク
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