Directional derivative

In mathematics, the directional derivative of a multivariate differentiable (scalar) 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.

The directional derivative of a scalar function f with respect to a vector v at a point (e.g., position) x may be denoted by any of the following:

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. The directional derivative is a special case of the Gateaux derivative.

This definition is valid in a broad range of contexts, for example where the norm of a vector (and hence a unit vector) is undefined.[2]

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

Intuitively, the directional derivative of f at a point x represents the rate of change of f, in the direction of v with respect to time, when moving past x.

In a Euclidean space, some authors[4] 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.[5]

This definition gives the rate of increase of f per unit of distance moved in the direction given by v. In this case, one has

In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector. With this restriction, both the above definitions are equivalent.[6]

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:

This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ′(0) = v.

Directional derivatives are often used in introductory derivations of the Riemann curvature tensor. Consider a curved rectangle with an infinitesimal vector δ along one edge and δ′ along the other. We translate a covector S along δ then δ′ and then subtract the translation along δ′ and then δ. Instead of building the directional derivative using partial derivatives, we use the covariant derivative. The translation operator for δ is thus

It can be argued[7] that the noncommutativity of the covariant derivatives measures the curvature of the manifold:

where R is the Riemann curvature tensor and the sign depends on the sign convention of the author.

In the Poincaré algebra, we can define an infinitesimal translation operator P as

(the i ensures that P is a self-adjoint operator) For a finite displacement λ, the unitary Hilbert space representation for translations is[8]

By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative:

This is a translation operator in the sense that it acts on multivariable functions f(x) as

In standard single-variable calculus, the derivative of a smooth function f(x) is defined by (for small ε)

It is evident that the group multiplication law[10] U(g)U(f)=U(gf) takes the form

So suppose that we take the finite displacement λ and divide it into N parts (N→∞ is implied everywhere), so that λ/N=ε. In other words,

The actual operators on the Hilbert space are represented by unitary operators U(T(ξ)). In the above notation we suppressed the T; we now write U(λ) as U(P(λ)). For a small neighborhood around the identity, the power series representation

is quite good. Suppose that U(T(ξ)) form a non-projective representation, i.e.,

After expanding the representation multiplication equation and equating coefficients, we have the nontrivial condition

with C the structure constant. The generators for translations are partial derivative operators, which commute:

This implies that the structure constants vanish and thus the quadratic coefficients in the f expansion vanish as well. This means that f is simply additive:

It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector x by

Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:[12]

Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors.[13] The directional directive provides a systematic way of finding these derivatives.

The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.

Let f(v) be a real valued function of the vector v. Then the derivative of f(v) with respect to v (or at v) is the vector defined through its dot product with any vector u being

for all vectors u. The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction.

Let f(v) be a vector valued function of the vector v. Then the derivative of f(v) with respect to v (or at v) is the second order tensor defined through its dot product with any vector u being

for all vectors u. The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u.