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Categories: Multivariate calculus | Differential geometry

Directional derivative

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 $f(\vec{x}) = f(x_1, x_2, \ldots, x_n)$ along a unit vector $\vec{v} = (v_1, \ldots, v_n)$ is the function defined by the limit

$D_{\vec{v}}{f} = \lim_{h \rightarrow 0}{\frac{f(\vec{x} + h\vec{v}) - f(\vec{x})}{h}}$

It can be written in terms of the gradient $\nabla(f)$ of $f$ by

$D_{\vec{v}}{f} = \nabla(f) \cdot \vec{v}$

where $\cdot$ denotes the dot product (Euclidean inner product). At any point $p$ , the directional derivative of $f$ intuitively represents the rate of change in $f$ in the direction of $\vec{v}$ at the point $p$ .

The Directional Derivative in Differential Geometry

A vector field at a point $p$ naturally gives rise to linear functionals defined on $p$ by evaluating the directional derivative of a differentiable function $f$ along the unit vector $\vec{v}/||\vec{v}||$ where $\vec{v}$ is the vector of the tangent space at $p$ assigned by the vector field. The value of the functional is then defined as the value of the corresponding directional derivative at $p$ in the direction of $\vec{v}$ .

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