Directional derivative
From Wikipedia, the free encyclopedia
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.
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[edit] Definition
The directional derivative of a scalar function 
along a vector 
is the function defined by the limit
If the function f is differentiable at 
, then the directional derivative exists along any vector 
and one has
where 
denotes the gradient and 
is the Euclidean inner product. At any point p, the directional derivative of f intuitively represents the rate of change in f along 
at the point p. Usually directions are taken to be normalized, so is a unit vector, although the definition above works for arbitrary (even zero) vectors.
[edit] 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 vector where 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 .
[edit] 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.
