方向微分

多変数微積分において、方向微分は、特定の点を通る指定されたベクトルに沿って関数が変化する瞬間的な速度を表します。ベクトルにスカラーが乗算される場合、対応する方向微分も同じスカラーで乗算されます。

初等的な教科書の中には、関数の単位距離あたりの変化率を表すのに「 $v$ の方向における方向微分」という表現を用いるものもある。この慣例では、非ゼロベクトル $v$ はまず単位ベクトルに正規化される。 ${\hat {\mathbf {v} }}=\mathbf {v} /\|\mathbf {v} \|$ 正規化されたベクトルは、サーカムフレックス（ハット）記号で表されます。 $\mathbf {\widehat {}}$ 。

スカラー関数fのベクトルvに関する方向微分は、以下のいずれかの方法で表すことができます。 ${\begin{aligned}\nabla _{\mathbf {v} }{f}(\mathbf {x} )&=f'_{\mathbf {v} }(\mathbf {x} )\\&=D_{\mathbf {v} }f(\mathbf {x} )\\&=Df(\mathbf {x} )(\mathbf {v} )\\&=\partial _{\mathbf {v} }f(\mathbf {x} )\\&={\frac {\partial f(\mathbf {x} )}{\partial \mathbf {v} }}\\&=\mathbf {v} \cdot {\nabla f(\mathbf {x} )}\\&=\mathbf {v} \cdot {\frac {\partial f(\mathbf {x} )}{\partial \mathbf {x} }}.\\\end{aligned}}$

したがって、これは偏微分の概念を一般化したものであり、変化率は曲線座標曲線の1つに沿って取られ、他のすべての座標は一定である。関数解析において、位相ベクトル空間間の関数に対する類似の概念はガトー微分である。

意味

{\displaystyle f(x,y)=x^{2}+y^{2}} — 等高線図 $f(x,y)=x^{2}+y^{2}$ 勾配ベクトルを黒で、単位ベクトルを黒で示しています。 $\mathbf {u}$ 方向微分によってスケーリングされた $\mathbf {u}$ オレンジ色で示されています。勾配ベクトルが長いのは、勾配が関数の増加率が最も高い方向を指しているためです。

スカラー関数の方向微分 $f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})$ ベクトルに沿って $\mathbf {v} =(v_{1},\ldots ,v_{n})$ 関数は $\nabla _{\mathbf {v} }{f}$ 限界によって定義される^{[ 1 ]} $\nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}f(\mathbf {x} +t\mathbf {v} )\right|_{t=0}.$

この定義は、例えばベクトルのノルムが定義されるような幅広い文脈において有効です。有限次元においては、すべてのノルムは等価であるため、ノルムの選択に依存しません。その適用範囲は、計量を持たない有限次元ベクトル空間上の関数や、一般相対性理論における微分可能多様体などにも及びます。

微分可能な関数の場合

関数f がxで微分可能であれば、 xにおける任意のベクトルvに沿って方向微分が存在し、次の式が成り立つ。

$\nabla _{\mathbf {v} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {v}$

どこで $\nabla$ 右側は勾配を示し、 $\cdot$ はドット積です。^{[ 2 ]}

It can be derived by using the property that all directional derivatives at a point make up a single tangent plane which can be defined using partial derivatives. This can be used to find a formula for the gradient vector and an alternative formula for the directional derivative, the latter of which can be rewritten as shown above for convenience.

It also follows from defining a path $\mathbf {h} (t)=\mathbf {x} +t\mathbf {v}$ and using the definition of the derivative as a limit along this path to get ${\begin{aligned}\nabla _{\mathbf {v} }f(\mathbf {x} )-\nabla f(\mathbf {x} )\cdot \mathbf {v} &=\lim _{t\to 0}{\frac {f(\mathbf {x} +t\mathbf {v} )-f(\mathbf {x} )}{t}}-\nabla f(\mathbf {x} )\cdot \mathbf {v} \\&=\lim _{t\to 0}{\frac {f(\mathbf {x} +t\mathbf {v} )-f(\mathbf {x} )-t\nabla f(\mathbf {x} )\cdot \mathbf {v} }{t}}\\&=0,\end{aligned}}$ whence $\nabla f(\mathbf {x} )\cdot \mathbf {v} =\nabla _{\mathbf {v} }f(\mathbf {x} )$ .

Using only direction of vector

In a Euclidean space, some authors^[3]^[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 change of a function $f$ per unit of distance moved in the direction given by $v$ . In this case, one has $\nabla _{\hat {\mathbf {v} }}{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h\|v\|}},$ or in case f is differentiable at x, $\nabla _{\hat {\mathbf {v} }}{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot {\hat {\mathbf {v} }}.$

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:

sum rule: $\nabla _{\mathbf {v} }(f+g)=\nabla _{\mathbf {v} }f+\nabla _{\mathbf {v} }g.$
constant factor rule: For any constant c, $\nabla _{\mathbf {v} }(cf)=c\nabla _{\mathbf {v} }f.$
product rule (or Leibniz's rule): $\nabla _{\mathbf {v} }(fg)=g\nabla _{\mathbf {v} }f+f\nabla _{\mathbf {v} }g.$
chain rule: If g is differentiable at p and h is differentiable at g(p), then $\nabla _{\mathbf {v} }(h\circ g)(\mathbf {p} )=h'(g(\mathbf {p} ))\nabla _{\mathbf {v} }g(\mathbf {p} ).$

In differential geometry

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), $\nabla _{\mathbf {v} }f(\mathbf {p} )$ (see Covariant derivative), $L_{\mathbf {v} }f(\mathbf {p} )$ (see Lie derivative), or ${\mathbf {v} }_{\mathbf {p} }(f)$ (see Tangent space § Definition via derivations), can be defined as follows. Let $γ : [-1, 1] \to M$ be a differentiable curve with $γ (0) = p$ and $γ'(0) = v$ . Then the directional derivative is defined by $\nabla _{\mathbf {v} }f(\mathbf {p} )=\left.{\frac {d}{d\tau }}f\circ \gamma (\tau )\right|_{\tau =0}.$ This definition can be proven independent of the choice of $γ$ , provided $γ$ is selected in the prescribed manner so that $γ (0) = p$ and $γ'(0) = v$ .

The Lie derivative

The Lie derivative of a vector field $W^{\mu }(x)$ along a vector field $V^{\mu }(x)$ is given by the difference of two directional derivatives (with vanishing torsion): ${\mathcal {L}}_{V}W^{\mu }=(V\cdot \nabla )W^{\mu }-(W\cdot \nabla )V^{\mu }.$ In particular, for a scalar field $\phi (x)$ , the Lie derivative reduces to the standard directional derivative: ${\mathcal {L}}_{V}\phi =(V\cdot \nabla )\phi .$

The Riemann tensor

Directional derivatives are often used in introductory derivations of the Riemann curvature tensor. Consider a curved rectangle with an infinitesimal vector $\delta$ along one edge and $\delta '$ along the other. We translate a covector $S$ along $\delta$ then $\delta '$ and then subtract the translation along $\delta '$ and then $\delta$ . Instead of building the directional derivative using partial derivatives, we use the covariant derivative. The translation operator for $\delta$ is thus $1+\sum _{\nu }\delta ^{\nu }D_{\nu }=1+\delta \cdot D,$ and for $\delta '$ , $1+\sum _{\mu }\delta '^{\mu }D_{\mu }=1+\delta '\cdot D.$ The difference between the two paths is then $(1+\delta '\cdot D)(1+\delta \cdot D)S^{\rho }-(1+\delta \cdot D)(1+\delta '\cdot D)S^{\rho }=\sum _{\mu ,\nu }\delta '^{\mu }\delta ^{\nu }[D_{\mu },D_{\nu }]S_{\rho }.$ It can be argued^[6] that the noncommutativity of the covariant derivatives measures the curvature of the manifold: $[D_{\mu },D_{\nu }]S_{\rho }=\pm \sum _{\sigma }R^{\sigma }{}_{\rho \mu \nu }S_{\sigma },$ where $R$ is the Riemann curvature tensor and the sign depends on the sign convention of the author.

In group theory

Translations

In the Poincaré algebra, we can define an infinitesimal translation operator P as $\mathbf {P} =i\nabla .$ (the i ensures that P is a self-adjoint operator) For a finite displacement λ, the unitary Hilbert space representation for translations is^[7] $U({\boldsymbol {\lambda }})=\exp \left(-i{\boldsymbol {\lambda }}\cdot \mathbf {P} \right).$ By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative: $U({\boldsymbol {\lambda }})=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right).$ This is a translation operator in the sense that it acts on multivariable functions f(x) as $U({\boldsymbol {\lambda }})f(\mathbf {x} )=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right)f(\mathbf {x} )=f(\mathbf {x} +{\boldsymbol {\lambda }}).$

Proof of the last equation

In standard single-variable calculus, the derivative of a smooth function f(x) is defined by (for small ε) ${\frac {df}{dx}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.$ This can be rearranged to find f(x+ε): $f(x+\varepsilon )=f(x)+\varepsilon \,{\frac {df}{dx}}=\left(1+\varepsilon \,{\frac {d}{dx}}\right)f(x).$ It follows that $[1+\varepsilon \,(d/dx)]$ is a translation operator. This is instantly generalized^[8] to multivariable functions f(x) $f(\mathbf {x} +{\boldsymbol {\varepsilon }})=\left(1+{\boldsymbol {\varepsilon }}\cdot \nabla \right)f(\mathbf {x} ).$ Here ${\boldsymbol {\varepsilon }}\cdot \nabla$ is the directional derivative along the infinitesimal displacement ε. We have found the infinitesimal version of the translation operator: $U({\boldsymbol {\varepsilon }})=1+{\boldsymbol {\varepsilon }}\cdot \nabla .$ It is evident that the group multiplication law^[9]U(g)U(f)=U(gf) takes the form $U(\mathbf {a} )U(\mathbf {b} )=U(\mathbf {a+b} ).$ So suppose that we take the finite displacement λ and divide it into N parts (N→∞ is implied everywhere), so that λ/N=ε. In other words, ${\boldsymbol {\lambda }}=N{\boldsymbol {\varepsilon }}.$ Then by applying U(ε) N times, we can construct U(λ): $[U({\boldsymbol {\varepsilon }})]^{N}=U(N{\boldsymbol {\varepsilon }})=U({\boldsymbol {\lambda }}).$ We can now plug in our above expression for U(ε): $[U({\boldsymbol {\varepsilon }})]^{N}=\left[1+{\boldsymbol {\varepsilon }}\cdot \nabla \right]^{N}=\left[1+{\frac {{\boldsymbol {\lambda }}\cdot \nabla }{N}}\right]^{N}.$ Using the identity^[10] $\exp(x)=\left[1+{\frac {x}{N}}\right]^{N},$ we have $U({\boldsymbol {\lambda }})=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right).$ And since $U (ε) f (x) = f (x + ε)$ we have $[U({\boldsymbol {\varepsilon }})]^{N}f(\mathbf {x} )=f(\mathbf {x} +N{\boldsymbol {\varepsilon }})=f(\mathbf {x} +{\boldsymbol {\lambda }})=U({\boldsymbol {\lambda }})f(\mathbf {x} )=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right)f(\mathbf {x} ),$ 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 $\xi ^{a}$ . The group multiplication law takes the form $T({\bar {\xi }})T(\xi )=T(f({\bar {\xi }},\xi )).$ Taking $\xi ^{a}=0$ as the coordinates of the identity, we must have $f^{a}(\xi ,0)=f^{a}(0,\xi )=\xi ^{a}.$ 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 $U(T(\xi ))=1+i\sum _{a}\xi ^{a}t_{a}+{\frac {1}{2}}\sum _{b,c}\xi ^{b}\xi ^{c}t_{bc}+\cdots$ is quite good. Suppose that U(T(ξ)) form a non-projective representation, i.e., $U(T({\bar {\xi }}))U(T(\xi ))=U(T(f({\bar {\xi }},\xi ))).$ The expansion of f to second power is $f^{a}({\bar {\xi }},\xi )=\xi ^{a}+{\bar {\xi }}^{a}+\sum _{b,c}f^{abc}{\bar {\xi }}^{b}\xi ^{c}.$ After expanding the representation multiplication equation and equating coefficients, we have the nontrivial condition $t_{bc}=-t_{b}t_{c}-i\sum _{a}f^{abc}t_{a}.$ Since $t_{ab}$ is by definition symmetric in its indices, we have the standard Lie algebra commutator: $[t_{b},t_{c}]=i\sum _{a}(-f^{abc}+f^{acb})t_{a}=i\sum _{a}C^{abc}t_{a},$ with C the structure constant. The generators for translations are partial derivative operators, which commute: $\left[{\frac {\partial }{\partial x^{b}}},{\frac {\partial }{\partial x^{c}}}\right]=0.$ 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: $f_{\text{abelian}}^{a}({\bar {\xi }},\xi )=\xi ^{a}+{\bar {\xi }}^{a},$ and thus for abelian groups, $U(T({\bar {\xi }}))U(T(\xi ))=U(T({\bar {\xi }}+\xi )).$ 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 ${\hat {\theta }}={\boldsymbol {\theta }}/\theta$ is $U(R(\mathbf {\theta } ))=\exp(-i\mathbf {\theta } \cdot \mathbf {L} ).$ Here L is the vector operator that generates SO(3): $\mathbf {L} ={\begin{pmatrix}0&0&0\\0&0&1\\0&-1&0\end{pmatrix}}\mathbf {i} +{\begin{pmatrix}0&0&-1\\0&0&0\\1&0&0\end{pmatrix}}\mathbf {j} +{\begin{pmatrix}0&1&0\\-1&0&0\\0&0&0\end{pmatrix}}\mathbf {k} .$ It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector x by $\mathbf {x} \rightarrow \mathbf {x} -\delta {\boldsymbol {\theta }}\times \mathbf {x} .$ So we would expect under infinitesimal rotation: $U(R(\delta {\boldsymbol {\theta }}))f(\mathbf {x} )=f(\mathbf {x} -\delta {\boldsymbol {\theta }}\times \mathbf {x} )=f(\mathbf {x} )-(\delta {\boldsymbol {\theta }}\times \mathbf {x} )\cdot \nabla f.$ It follows that $U(R(\delta \mathbf {\theta } ))=1-(\delta \mathbf {\theta } \times \mathbf {x} )\cdot \nabla .$ Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:^[11] $U(R(\mathbf {\theta } ))=\exp(-(\mathbf {\theta } \times \mathbf {x} )\cdot \nabla ).$

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 $\mathbf {n}$ , then the normal derivative of a function f is sometimes denoted as ${\textstyle {\frac {\partial f}{\partial \mathbf {n} }}}$ . In other notations, ${\frac {\partial f}{\partial \mathbf {n} }}=\nabla f(\mathbf {x} )\cdot \mathbf {n} =\nabla _{\mathbf {n} }{f}(\mathbf {x} )={\frac {\partial f}{\partial \mathbf {x} }}\cdot \mathbf {n} =Df(\mathbf {x} )[\mathbf {n} ].$

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.^[12] The directional derivative 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

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

for all vectors 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 $f(\mathbf {v} )=f_{1}(\mathbf {v} )+f_{2}(\mathbf {v} )$ then ${\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}+{\frac {\partial f_{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u}$
If $f(\mathbf {v} )=f_{1}(\mathbf {v} )~f_{2}(\mathbf {v} )$ then ${\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)~f_{2}(\mathbf {v} )+f_{1}(\mathbf {v} )~\left({\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)$
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

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

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

for all vectors 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 $\mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )+\mathbf {f} _{2}(\mathbf {v} )$ then ${\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}+{\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u}$
If $\mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )\times \mathbf {f} _{2}(\mathbf {v} )$ then ${\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)\times \mathbf {f} _{2}(\mathbf {v} )+\mathbf {f} _{1}(\mathbf {v} )\times \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)$
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

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:

If $f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})+f_{2}({\boldsymbol {S}})$ then ${\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}$
If $f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})~f_{2}({\boldsymbol {S}})$ then ${\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)~f_{2}({\boldsymbol {S}})+f_{1}({\boldsymbol {S}})~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)$
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 ${\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=D{\boldsymbol {F}}({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~{\boldsymbol {F}}({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}$ for all second order tensors ${\boldsymbol {T}}$ .

Properties:

If ${\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})+{\boldsymbol {F}}_{2}({\boldsymbol {S}})$ then ${\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}$
If ${\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})$ then ${\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})+{\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot \left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)$
If ${\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))$ then ${\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)$
If $f({\boldsymbol {S}})=f_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))$ then ${\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)$

Notes

↑R. Wrede; M.R. Spiegel (2010). Advanced Calculus (3rd ed.). Schaum's Outline Series. ISBN 978-0-07-162366-7.
↑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.
↑Stewart, James (2015). "14.6 Directional Derivates and the Gradient Vector". Calculus (8th ed.). Cengage Learning. ISBN 978-1-285-74062-1.
↑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.
↑Zee, A. (2013). Einstein Gravity in a Nutshell. Princeton: Princeton University Press. p. 341. ISBN 9780691145587.
↑Weinberg, Steven (1999). The quantum theory of fields (Reprinted (with corr.). ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521550017.
↑Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. ISBN 9780691145587.
↑Cahill, Kevin Cahill (2013). Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press. ISBN 978-1107005211.
↑Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (9th ed.). Belmont: Brooks/Cole. ISBN 9780547209982.
↑Shankar, R. (1994). Principles of quantum mechanics (2nd ed.). New York: Kluwer Academic / Plenum. p. 318. ISBN 9780306447907.
↑J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.

References

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.
Shapiro, A. (1990). "On concepts of directional differentiability". Journal of Optimization Theory and Applications. 66 (3): 477–487. doi:10.1007/BF00940933. S2CID 120253580.

External links

Media related to Directional derivative at Wikimedia Commons

Directional derivatives at MathWorld.
Directional derivative at PlanetMath.

[1] R. Wrede; M.R. Spiegel (2010). Advanced Calculus (3rd ed.). Schaum's Outline Series. ISBN 978-0-07-162366-7.

[2] 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.

[3] Thomas, George B. Jr.; and Finney, Ross L. (1979) Calculus and Analytic Geometry, Addison-Wesley Publ. Co., fifth edition, p. 593.

[4] Stewart, James (2015). "14.6 Directional Derivates and the Gradient Vector". Calculus (8th ed.). Cengage Learning. ISBN 978-1-285-74062-1.

[5] 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.

[6] Zee, A. (2013). Einstein Gravity in a Nutshell. Princeton: Princeton University Press. p. 341. ISBN 9780691145587.

[7] Weinberg, Steven (1999). The quantum theory of fields (Reprinted (with corr.). ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521550017.

[8] Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. ISBN 9780691145587.

[9] Cahill, Kevin Cahill (2013). Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press. ISBN 978-1107005211.

[10] Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (9th ed.). Belmont: Brooks/Cole. ISBN 9780547209982.

[11] Shankar, R. (1994). Principles of quantum mechanics (2nd ed.). New York: Kluwer Academic / Plenum. p. 318. ISBN 9780306447907.

[Marsden00-12] J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.

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方向微分

意味

微分可能な関数の場合

Using only direction of vector

Properties

In differential geometry

The Lie derivative

The Riemann tensor

In group theory

Translations

Rotations

Normal derivative

In the continuum mechanics of solids

Derivatives of scalar valued functions of vectors

Derivatives of vector valued functions of vectors

Derivatives of scalar valued functions of second-order tensors

Derivatives of tensor valued functions of second-order tensors

See also

Notes

References

External links

関連するトピック

多変数微積分

ベクトル

サーカムフレックス

スカラー関数