In mathematics, the directional derivative of a multivariate differentiable function along a given vectorv 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 curvilinearcoordinate curves, all other coordinates being constant.
A contour plot of , showing the gradient vector in blue, and the unit vector scaled by the directional derivative in the direction of in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.
In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector. Without the restriction, 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.
If the function f is differentiable at x, then the directional derivative exists along any vector v, and one has
where the on the right denotes the gradient and is the dot product. Intuitively, the directional derivative of f at a point x represents the rate of change of f with respect to time when moving past x at velocity v.
Variation using only direction of vector for Euclidean space
The angle α between the tangent A and the horizontal will be maximum if the cutting plane contains the direction of the gradient A.
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 of increase of f per unit of distance moved in the given direction. In this case, one has
or in case f is differentiable at x,
This definition is not equivalent with the one above except when v is a unit vector.
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
and for δ′
The difference between the two paths is then
It can be argued that the noncommutativity of the covariant derivatives measures the curvature of the manifold:
with R the Riemann tensor of course and the sign depending on the sign convention of the author.
As a technical note, this procedure is only possible because the translation group forms an Abeliansubgroup (Cartan subalgebra) in the Poincaré algebra. In particular, the group multiplication law U(a)U(b)=U(a+b) should not be taken for granted. We also note that Poincaré is a connected Lie group. It is a group of transformations T(ξ) that are described by a continuous set of real parameters . The group multiplication law takes the form
Taking =0 as the coordinates of the identity, we must have
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. that
The expansion of f to second power is
After expanding the representation multiplication equation and equating coefficients, we have the nontrivial condition
Since is by definition symmetric in its indices, we have the standard Lie algebra commutator:
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:
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. If the normal direction is denoted by , then the directional derivative of a function f is sometimes denoted as . In other notations
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 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.
Derivatives of scalar valued functions of vectors
Let be a real valued function of the vector . Then the derivative of with respect to (or at ) in the direction is defined as
for all vectors .
Derivatives of vector valued functions of vectors
Let be a vector valued function of the vector . Then the derivative of with respect to (or at ) in the direction is the second-order tensor defined as
for all vectors .
Derivatives of scalar valued functions of second-order tensors
Let be a real valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the second order tensor defined as
for all second order tensors .
Derivatives of tensor valued functions of second-order tensors
Let be a second order tensor valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the fourth order tensor defined as
^If the dot product is undefined, the gradient is also undefined; however, for differentiable f, the directional derivative is still defined, and a similar relation exists with the exterior derivative.
^Thomas, George B. Jr.; and Finney, Ross L. (1979) Calculus and Analytic Geometry, Addison-Wesley Publ. Co., fifth edition, p. 593.
^This typically assumes a Euclidean space – for example, a function of several variables typically has no definition of the magnitude of a vector, and hence of a unit vector.
^Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. p. 341. ISBN9780691145587.
^Weinberg, Steven (1999). The quantum theory of fields (Reprinted (with corr.). ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN9780521550017.
^Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. ISBN9780691145587.
^Mexico, Kevin Cahill, University of New (2013). Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press. ISBN978-1107005211.
^Edwards, Ron Larson, Robert, Bruce H. (2010). Calculus of a single variable (9th ed.). Belmont: Brooks/Cole. ISBN9780547209982.
^Shankar, R. (1994). Principles of quantum mechanics (2nd ed.). New York: Kluwer Academic / Plenum. p. 318. ISBN9780306447907.
^J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.