Directional derivative on arvau

Wikipedia / Directional derivative

In multivariable calculus, the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given point. If the vector is multiplied by a scalar, the corresponding directional derivative is multiplied by the same scalar.

Some elementary texts instead use the phrase "directional derivative in the direction of " for the rate of change of a function per unit distance in that direction. In that convention the nonzero vector , where the normalized vector is denoted with a circumflex.

The directional derivative of a scalar function f with respect to a vector v

It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. In functional analysis, the analogous notion for functions between topological vector spaces is the Gateaux derivative.

Definition

The directional derivative of a scalar function is the function defined by the limit

This definition is valid in a broad range of contexts, for example, where the norm of a vector is defined. In finite dimensions, it does not depend on the choice of norm, since all norms are equivalent. Its applicability extends to functions on finite-dimensional vector spaces without a metric and to differentiable manifolds, such as in general relativity.

For differentiable functions

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

on the right denotes the gradient is the dot product.

It can be derived by using the property that all directional derivatives at a point make up a single tangent plane which can be defined using partial derivatives. This can be used to find a formula for the gradient vector and an alternative formula for the directional derivative, the latter of which can be rewritten as shown above for convenience.

Using only direction of vector

In a Euclidean space, some authors define the directional derivative to be with respect to an arbitrary nonzero vector v after normalization, thus being independent of its magnitude and depending only on its direction.

This definition gives the rate change of a function per unit of distance moved in the direction given by or in case f is differentiable at x

Properties

Many of the familiar properties of the ordinary derivative hold for the directional derivative. These include, for any functions f and g defined in a neighborhood of, and differentiable at, p:

sum rule
constant factor rule: For any constant c
product rule (or Leibniz's rule
chain rule: If g is differentiable at p and h is differentiable at g(p

In differential geometry

Let be a differentiable manifold and a point of . Suppose that is a function defined in a neighborhood of , and differentiable at . If is a tangent vector to at , then the directional derivative of along , denoted variously as (see Exterior derivative (see Covariant derivative (see Lie derivative (see ), can be defined as follows. Let be a differentiable curve with and This definition can be proven independent of the choice of , provided is selected in the prescribed manner so that and .

The Lie derivative

The Riemann tensor

Directional derivatives are often used in introductory derivations of the Riemann curvature tensor. Instead of building the directional derivative using partial derivatives, we use the covariant derivative It can be argued is the Riemann curvature tensor and the sign depends on the sign convention of the author.

In group theory

Translations

In the Poincaré algebra, we can define an infinitesimal translation operator P (the i ensures that P is a self-adjoint operator) For a finite displacement λ, the unitary Hilbert space representation for translations is This is a translation operator in the sense that it acts on multivariable functions f(x

Rotations

The rotation operator also contains a directional derivative. The rotation operator for an angle θ, i.e. by an amount θ = |θ Here L is the vector operator that generates SO(3) It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector x Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:

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, then the normal derivative of a function f

In the continuum mechanics of solids

Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors. The directional derivative provides a systematic way of finding these derivatives.