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directional-derivative

Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the direction is always taken parallel to one of the coordinate axes.

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

Definition

The directional derivative of a scalar function f(vec{x}) = f(x_1, x_2, ldots, x_n) along a vector vec{v} = (v_1, ldots, v_n) is the function defined by the limit
nabla_{vec{v}}{f}(vec{x}) = lim_{h rightarrow 0}{frac{f(vec{x} + hvec{v}) - f(vec{x})}{h}}.

Sometimes authors write Dv instead of nabla_v. If the function f is differentiable at vec{x}, then the directional derivative exists along any vector vec{v}, and one has

nabla_{vec{v}}{f}(vec{x}) = nabla f(vec{x}) cdot vec{v}

where the nabla on the right denotes the gradient and cdot is the Euclidean inner product. At any point vec{x}, the directional derivative of f intuitively represents the rate of change in f along vec{v} at the point vec{x}. Usually directions are taken to be normalized, so vec{v} is a unit vector, although the definition above works for arbitrary (even zero) vectors.

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:

nabla_v hcirc g (p) = h'(g(p)) nabla_v g (p)

In differential geometry

Let M be a differentiable manifold and p a point of M. Suppose that f is 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_v f(p) (see covariant derivative), L_v f(p) (see Lie derivative), or v_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_v f(p) = left.frac{d}{dtau} fcircgamma(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

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 vec{n}, then the directional derivative of a function ƒ is sometimes denoted as frac{ partial f}{partial n}.

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(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 the vector defined as
frac{partial f}{partial mathbf{v}}cdotmathbf{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}}cdotmathbf{u} = left(frac{partial f_1}{partial mathbf{v}} + frac{partial f_2}{partial mathbf{v}}right)cdotmathbf{u}

2) If f(mathbf{v}) = f_1(mathbf{v})~ f_2(mathbf{v}) then frac{partial f}{partial mathbf{v}}cdotmathbf{u} = left(frac{partial f_1}{partial mathbf{v}}cdotmathbf{u}right)~f_2(mathbf{v}) + f_1(mathbf{v})~left(frac{partial f_2}{partial mathbf{v}}cdotmathbf{u} right)

3) If f(mathbf{v}) = f_1(f_2(mathbf{v})) then frac{partial f}{partial mathbf{v}}cdotmathbf{u} = frac{partial f_1}{partial f_2}~frac{partial f_2}{partial mathbf{v}}cdotmathbf{u}


Derivatives of vector valued functions of vectors

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}}cdotmathbf{u} = Dmathbf{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}}cdotmathbf{u} = left(frac{partial mathbf{f}_1}{partial mathbf{v}} + frac{partial mathbf{f}_2}{partial mathbf{v}}right)cdotmathbf{u}

2) If mathbf{f}(mathbf{v}) = mathbf{f}_1(mathbf{v})timesmathbf{f}_2(mathbf{v}) then frac{partial mathbf{f}}{partial mathbf{v}}cdotmathbf{u} = left(frac{partial mathbf{f}_1}{partial mathbf{v}}cdotmathbf{u}right)timesmathbf{f}_2(mathbf{v}) + mathbf{f}_1(mathbf{v})timesleft(frac{partial mathbf{f}_2}{partial mathbf{v}}cdotmathbf{u} right)

3) If mathbf{f}(mathbf{v}) = mathbf{f}_1(mathbf{f}_2(mathbf{v})) then frac{partial mathbf{f}}{partial mathbf{v}}cdotmathbf{u} = frac{partial mathbf{f}_1}{partial mathbf{f}_2}cdotleft(frac{partial mathbf{f}_2}{partial mathbf{v}}cdotmathbf{u} right)


Derivatives of scalar valued functions of second-order tensors

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

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} = Dboldsymbol{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})cdotboldsymbol{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)cdotboldsymbol{F}_2(boldsymbol{S}) + boldsymbol{F}_1(boldsymbol{S})cdotleft(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)

References

See also

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