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A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point.

The **directional derivative** of a multivariable 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.

The *directional derivative* of a scalar function

along a vector

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 (and hence a unit vector) is undefined.^{[2]}

If the function *f* is differentiable at **x**, then the directional derivative exists along any unit 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.^{[3]} 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**.

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

or in case

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**:

**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**), then

See also: Tangent space § Tangent vectors as directional derivatives |

Let *M* be a differentiable manifold and **p** a point of *M*. Suppose that *f* is a function defined in a neighborhood of **p**, and differentiable at **p**. If **v** is a tangent vector to *M* at **p**, then the **directional derivative** of *f* along **v**, denoted variously as *df*(**v**) (see Exterior derivative), (see Covariant derivative), (see Lie derivative), or (see Tangent space § Definition via derivations), can be defined as follows. Let *γ* : [−1, 1] → *M* be a differentiable curve with *γ*(0) = **p** and *γ*′(0) = **v**. Then the directional derivative is defined by

This definition can be proven independent of the choice of

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:

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

and for ,

The difference between the two paths is then

It can be argued

where 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

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

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

This can be rearranged to find

It follows that is a translation operator. This is instantly generalized

Here is the directional derivative along the infinitesimal displacement

It is evident that the group multiplication law

So suppose that we take the finite displacement

Then by applying

We can now plug in our above expression for U(

Using the identity

we have

And since

Q.E.D.

As a technical note, this procedure is only possible because the translation group forms an Abelian subgroup (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

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

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

and thus for abelian groups,

Q.E.D.

The rotation operator also contains a directional derivative. The rotation operator for an angle * θ*, i.e. by an amount

Here

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

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,

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.

Properties:

- If then
- If then
- If then

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.

Properties:

- If then
- If then
- If then

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 .

Properties:

- If then
- If then
- If then

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

for all second order tensors .

Properties:

- If then
- If then
- If then
- If then