# Directional derivative

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In mathematics, the **directional derivative** of a multivariate differentiable 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**. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.

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

## Contents

## Definition

### Generally applicable definition

The directional derivative of a scalar function

along a vector

is the function defined by the limit^{[1]}

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.^{[2]} Intuitively, the directional derivative of *f* at a point **x** represents the rate of change of *f* with respect to time when it is moving at a speed and direction given by **v**.

### Variation using only direction of vector

Some authors define the directional derivative to be with respect to the vector **v** after normalization, thus ignoring its magnitude. In this case, one has

or in case *f* is differentiable at **x**,

This definition has some disadvantages: its applicability is limited to when the norm of a vector is defined and nonzero. It is incompatible with notation used in some other areas of mathematics, physics and engineering, but should be used when what is wanted is the rate of increase in *f* per unit distance.

### Restriction to unit vector

Some authors restrict the definition of the directional derivative to being with respect to a unit vector. With this restriction, the two definitions above become the same.

## Notation

Directional derivatives can be also denoted by:

where *v* is a parameterization of a curve to which **v** is tangent and which determines its magnitude.

## 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**:
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## In differential geometry

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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 (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 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^{[3]} 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.

## In group theory

### Translations

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^{[4]}

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

Proof of the last equation |
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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 f(x+ε): 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 U( We can now plug in our above expression for U( Using the identity we have And since U( 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( 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( 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: and thus for abelian groups, Q.E.D. |

### Rotations

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 =**θ**/θ 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:^{[8]}

## 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. If the normal direction is denoted by , then the directional 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.^{[9]} 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

*Properties:*
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### 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

*Properties:*
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### 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 .

*Properties:*
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### 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

for all second order tensors .

*Properties:*
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## See also

- Fréchet derivative
- Gâteaux derivative
- Derivative (generalizations)
- Lie derivative
- Differential form
- Structure tensor
- Tensor derivative (continuum mechanics)
- Del in cylindrical and spherical coordinates

## Notes

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Technically, the gradient ∇
*f*is a covector, and the "dot product" is the action of this covector on the vector**v**. - ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
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- ↑ J. E. Marsden and T. J. R. Hughes, 2000,
*Mathematical Foundations of Elasticity*, Dover.

## References

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