In mathematics, the directional derivative of a multivariable differentiable (scalar) 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.
The directional derivative of a scalar functionf with respect to a vector v at a point (e.g., position) x may be denoted by any of the following:
A contour plot of , showing the gradient vector in black, 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.
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.
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
where the on the right denotes the gradient, is the dot product and v is a unit vector. This follows from defining a path and using the definition of the derivative as a limit which can be calculated along this path to get:
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.
Using only direction of vector
The angle α between the tangent A and the horizontal will be maximum if the cutting plane contains the direction of the gradient A.
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 of increase of f per unit of distance moved in the direction given by v. In this case, one has
or in case f is differentiable at x,
Restriction to a unit vector
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.
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.
The Lie derivative
The Lie derivative of a vector field along a vector field is given by the difference of two directional derivatives (with vanishing torsion):
In particular, for a scalar field , the Lie derivative reduces to the standard directional derivative:
The Riemann tensor
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:
where R is the Riemann curvature tensor and the sign depends on the sign convention of the author.
In group theory
In the Poincaré algebra, we can define an infinitesimal translation operator P as
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 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.,
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:
and thus for abelian groups,
The rotation operator also contains a directional derivative. The rotation operator for an angle θ, i.e. by an amount θ = |θ| about an axis parallel to is
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 by
So we would expect under infinitesimal rotation:
It follows that
Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional 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. If the normal direction is denoted by , then the normal derivative of a function f is sometimes denoted as . In other notations,
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 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 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.
Derivatives of vector valued functions of vectors
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.
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.