Directional derivative
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In mathematics, the directional derivative of a multivariate differentiable function along a given unit vector intuitively represents the rate of change of the function in the direction of that vector. It therefore generalizes the notion of a partial derivative in which the direction is always taken along one of the coordinate axes.
Definition
The directional derivative of a differentiable function <math>f(\vec{x}) = f(x_1, x_2, \ldots, x_n)<math> along a unit vector <math>\vec{v} = (v_1, \ldots, v_n)<math> is the function defined by the limit
- <math>D_{\vec{v}}{f} = \lim_{h \rightarrow 0}{\frac{f(\vec{x} + h\vec{v}) - f(\vec{x})}{h}}<math>
It can be written in terms of the gradient <math>\nabla(f)<math> of <math>f<math> by
- <math>D_{\vec{v}}{f} = \nabla(f) \cdot \vec{v}<math>
where <math>\cdot<math> denotes the dot product (Euclidean inner product). At any point <math>p<math>, the directional derivative of <math>f<math> intuitively represents the rate of change in <math>f<math> in the direction of <math>\vec{v}<math> at the point <math>p<math>.
The directional derivative in differential geometry
A vector field at a point <math>p<math> naturally gives rise to linear functionals defined on <math>p<math> by evaluating the directional derivative of a differentiable function <math>f<math> along the unit vector <math>\vec{v}/||\vec{v}||<math> where <math>\vec{v}<math> is the vector of the tangent space at <math>p<math> assigned by the vector field. The value of the functional is then defined as the value of the corresponding directional derivative at <math>p<math> in the direction of <math>\vec{v}<math>.