Directional derivative
From Free net encyclopedia
In mathematics, the directional derivative of a multivariate differentiable function along a given 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 parallel to one of the coordinate axes.
Contents |
Definition
The directional derivative of a scalar function <math>f(\vec{x}) = f(x_1, x_2, \ldots, x_n)</math> along a 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>
If the function is differentiable, 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 vector <math>\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>.
Normal derivative
A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space, or more generally along a normal vector field orthogonal to some hypersurface. See for example Neumann boundary condition.