Directional derivative

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.

Definition[edit]

A contour plot of f(x, y)=x^2 + y^2, showing the gradient vector in blue, and the unit vector \bold{u} scaled by the directional derivative in the direction of \bold{u} in orange. The gradient vector is longer because gradients points in the direction of greatest rate of increase of a function.

Generally applicable definition[edit]

The directional derivative of a scalar function

f(\bold{x}) = f(x_1, x_2, \ldots, x_n)

along a vector

\bold{v} = (v_1, \ldots, v_n)

is the function defined by the limit[1]

\nabla_{\bold{v}}{f}(\bold{x}) = \lim_{h \rightarrow 0}{\frac{f(\bold{x} + h\bold{v}) - f(\bold{x})}{h}}.

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

\nabla_{\bold{v}}{f}(\bold{x}) = \nabla f(\bold{x}) \cdot \bold{v}

where the \nabla on the right denotes the gradient and \cdot is the dot product.[2] At any point x, the directional derivative of f intuitively represents the rate of change of f with respect to time when it is moving at a speed and direction given by v at the point x.

Variation using only direction of vector[edit]

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 the vector v after normalization, thus ignoring its magnitude. In this case, one has

\nabla_{\bold{v}}{f}(\bold{x}) = \lim_{h \rightarrow 0}{\frac{f(\bold{x} + h\bold{v}) - f(\bold{x})}{h|\bold{v}|}},

or in case f is differentiable at x,

\nabla_{\bold{v}}{f}(\bold{x}) = \nabla f(\bold{x}) \cdot \frac{\bold{v}}{|\bold{v}|} .

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[edit]

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[edit]

Directional derivatives can be also denoted by:

\nabla_{\bold{v}}{f}(\bold{x}) \sim \frac{\partial{f(\bold{x})}}{\partial{v}} \sim f'_\mathbf{v}(\bold{x}) \sim D_\bold{v}f(\bold{x}) \sim \mathbf{v}\cdot{\nabla f(\bold{x})} \sim \bold{v}\cdot \frac{\partial f(\bold{x})}{\partial\bold{x}}

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

Properties[edit]

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:

  1. The sum rule:
    \nabla_{\bold{v}} (f + g) = \nabla_{\bold{v}} f + \nabla_{\bold{v}} g
  2. The constant factor rule: For any constant c,
    \nabla_{\bold{v}} (cf) = c\nabla_{\bold{v}} f
  3. The product rule (or Leibniz rule):
    \nabla_{\bold{v}} (fg) = g\nabla_{\bold{v}} f + f\nabla_{\bold{v}} g
  4. The chain rule: If g is differentiable at p and h is differentiable at g(p), then
    \nabla_{\bold{v}}(h\circ g)(\bold{p}) = h'(g(\bold{p})) \nabla_{\bold{v}} g (\bold{p})

In differential geometry[edit]

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 \nabla_{\bold{v}} f(\bold{p}) (see covariant derivative), L_{\bold{v}} f(\bold{p}) (see Lie derivative), or {\bold{v}}_{\bold{p}}(f) (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

\nabla_{\bold{v}} f(\bold{p}) = \left.\frac{d}{d\tau} f\circ\gamma(\tau)\right|_{\tau=0}

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

Normal derivative[edit]

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 \bold{n}, then the directional derivative of a function f is sometimes denoted as \frac{ \partial f}{\partial n}. In other notations

\frac{ \partial f}{\partial n} = \nabla f(\bold{x}) \cdot \bold{n} = \nabla_{\bold{n}}{f}(\bold{x}) = \frac{\partial f}{\partial \bold{x}}\cdot\bold{n} = Df(\bold{x})[\bold{n}]

In the continuum mechanics of solids[edit]

Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors.[3] 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[edit]

Let f(\mathbf{v}) be a real valued function of the vector \mathbf{v}. Then the derivative of f(\mathbf{v}) with respect to \mathbf{v} (or at \mathbf{v}) in the direction \mathbf{u} is defined as


  \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = Df(\mathbf{v})[\mathbf{u}] 
     = \left[\frac{d }{d \alpha}~f(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0}

for all vectors \mathbf{u}.

Properties:

  1. If f(\mathbf{v}) = f_1(\mathbf{v}) + f_2(\mathbf{v}) then 
   \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial f_1}{\partial \mathbf{v}} + \frac{\partial f_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u}
  2. If f(\mathbf{v}) = f_1(\mathbf{v})~ f_2(\mathbf{v}) then 
   \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial f_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)~f_2(\mathbf{v}) + f_1(\mathbf{v})~\left(\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)
  3. If f(\mathbf{v}) = f_1(f_2(\mathbf{v})) then 
   \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \frac{\partial f_1}{\partial f_2}~\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u}

Derivatives of vector valued functions of vectors[edit]

Let \mathbf{f}(\mathbf{v}) be a vector valued function of the vector \mathbf{v}. Then the derivative of \mathbf{f}(\mathbf{v}) with respect to \mathbf{v} (or at \mathbf{v}) in the direction \mathbf{u} is the second-order tensor defined as


  \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = D\mathbf{f}(\mathbf{v})[\mathbf{u}] 
     = \left[\frac{d }{d \alpha}~\mathbf{f}(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0}

for all vectors \mathbf{u}.

Properties:

  1. If \mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v}) + \mathbf{f}_2(\mathbf{v}) then 
   \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}} + \frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u}
  2. If \mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v})\times\mathbf{f}_2(\mathbf{v}) then 
   \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)\times\mathbf{f}_2(\mathbf{v}) + \mathbf{f}_1(\mathbf{v})\times\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)
  3. If \mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{f}_2(\mathbf{v})) then 
   \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \frac{\partial \mathbf{f}_1}{\partial \mathbf{f}_2}\cdot\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)

Derivatives of scalar valued functions of second-order tensors[edit]

Let f(\boldsymbol{S}) be a real valued function of the second order tensor \boldsymbol{S}. Then the derivative of f(\boldsymbol{S}) with respect to \boldsymbol{S} (or at \boldsymbol{S}) in the direction \boldsymbol{T} is the second order tensor defined as


  \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = Df(\boldsymbol{S})[\boldsymbol{T}] 
     = \left[\frac{d }{d \alpha}~f(\boldsymbol{S} + \alpha\boldsymbol{T})\right]_{\alpha = 0}

for all second order tensors \boldsymbol{T}.

Properties:

  1. If f(\boldsymbol{S}) = f_1(\boldsymbol{S}) + f_2(\boldsymbol{S}) then  \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \left(\frac{\partial f_1}{\partial \boldsymbol{S}} + \frac{\partial f_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T}
  2. If f(\boldsymbol{S}) = f_1(\boldsymbol{S})~ f_2(\boldsymbol{S}) then  \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \left(\frac{\partial f_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)~f_2(\boldsymbol{S}) + f_1(\boldsymbol{S})~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)
  3. If f(\boldsymbol{S}) = f_1(f_2(\boldsymbol{S})) then  \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \frac{\partial f_1}{\partial f_2}~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)

Derivatives of tensor valued functions of second-order tensors[edit]

Let \boldsymbol{F}(\boldsymbol{S}) be a second order tensor valued function of the second order tensor \boldsymbol{S}. Then the derivative of \boldsymbol{F}(\boldsymbol{S}) with respect to \boldsymbol{S} (or at \boldsymbol{S}) in the direction \boldsymbol{T} is the fourth order tensor defined as


  \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = D\boldsymbol{F}(\boldsymbol{S})[\boldsymbol{T}] 
     = \left[\frac{d }{d \alpha}~\boldsymbol{F}(\boldsymbol{S} + \alpha\boldsymbol{T})\right]_{\alpha = 0}

for all second order tensors \boldsymbol{T}.

Properties:

  1. If \boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S}) + \boldsymbol{F}_2(\boldsymbol{S}) then  \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}} + \frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T}
  2. If \boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S})\cdot\boldsymbol{F}_2(\boldsymbol{S}) then  \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)\cdot\boldsymbol{F}_2(\boldsymbol{S}) + \boldsymbol{F}_1(\boldsymbol{S})\cdot\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)
  3. If \boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{F}_2(\boldsymbol{S})) then  \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)
  4. If f(\boldsymbol{S}) = f_1(\boldsymbol{F}_2(\boldsymbol{S})) then  \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \frac{\partial f_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)

See also[edit]

Notes[edit]

  1. ^ R. Wrede, M.R. Spiegel (2010). Advanced Calculus (3rd edition ed.). Schaum's Outline Series. ISBN 978-0-07-162366-7. 
  2. ^ Technically, the gradient ∇f is a covector, and the "dot product" is the action of this covector on the vector v.
  3. ^ J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.

References[edit]

  • Hildebrand, F. B. (1976). Advanced Calculus for Applications. Prentice Hall. ISBN 0-13-011189-9. 
  • K.F. Riley, M.P. Hobson, S.J. Bence (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3. 

External links[edit]