What is directional derivative?

Directional Derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.

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Some articles on directional derivative:

Derivatives in Higher Dimensions - Total Derivative, Total Differential and Jacobian Matrix
... When f is a function from an open subset of Rn to Rm, then the directional derivative of f in a chosen direction is the best linear approximation to f at that point and in that direction ... But when n > 1, no single directional derivative can give a complete picture of the behavior of f ... The total derivative, also called the (total) differential, gives a complete picture by considering all directions at once ...
Directional Derivative - In The Continuum Mechanics of Solids - Derivatives of Tensor Valued Functions of Second-order Tensors
... Then the derivative of with respect to (or at ) in the direction is the fourth order tensor defined as for all second order tensors ...
Del - Notational Uses - Directional Derivative
... The directional derivative of a scalar field f(x,y,z) in the direction is defined as This gives the change of a field f in the direction of a ... dynamics uses this convention extensively, terming it the convective derivative—the "moving" derivative of the fluid ...
Differentiable Manifold - Differentiable Functions - Differentiation of Functions
... There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative ... The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine structure with which to define vectors ... The directional derivative therefore looks at curves in the manifold instead of vectors ...
Differentiable Manifold - Differentiable Functions - Differentiation of Functions - Directional Differentiation
... f on an m dimensional differentiable manifold M, the directional derivative of f at a point p in M is defined as follows ... Then the directional derivative of f at p along γ is If γ1 and γ2 are two curves such that γ1(0) = γ2(0) = p, and in any coordinate chart φ, then, by the chain rule, f has the same ... This means that the directional derivative depends only on the tangent vector of the curve at p ...

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