Directional derivative
From Exampleproblems
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 parallel to of the coordinate axes.
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Definition
The directional derivative of a differentiable function 
along a unit vector 
is the function defined by the limit
It can be written in terms of the gradient 
of f by
where 
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 
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 
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 .
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 unit normal vector field orthogonal to some hypersurface. See for example Neumann boundary condition.
