Directional derivative

From Wikipedia, the free encyclopedia

(Redirected from Normal derivative)
Jump to: navigation, search

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the direction is always taken parallel to one of the coordinate axes.

Contents

The directional derivative of a scalar function f(\vec{x}) = f(x_1, x_2, \ldots, x_n) along a vector \vec{v} = (v_1, \ldots, v_n) is the function defined by the limit

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

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

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

where \nabla denotes the gradient and \cdot is the Euclidean inner product. At any point p, the directional derivative of f intuitively represents the rate of change in f along \vec{v} at the point p. Usually directions are taken to be normalized, so \vec{v} is a unit vector, although the definition above works for arbitrary (even zero) vectors.

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 vector \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}.

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. If the normal direction is denoted by \vec{n}, then the directional derivative of a function ƒ is sometimes denoted as \frac{ \partial f}{\partial n}.

Retrieved from "http://en.wikipedia.org/wiki/Directional_derivative"
Advanced Search
Included Web Search Engines


Safe Search

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.