## 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:

... 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 ...

**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 ...

... 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 ...

... 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 ...

### Famous quotes containing the word derivative:

“When we say “science” we can either mean any manipulation of the inventive and organizing power of the human intellect: or we can mean such an extremely different thing as the religion of science the vulgarized *derivative* from this pure activity manipulated by a sort of priestcraft into a great religious and political weapon.”

—Wyndham Lewis (1882–1957)