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多変数ベクトル関数の連鎖律 📂多変数ベクトル解析

多変数ベクトル関数の連鎖律

定理

二つの関数 g:D⊂Rm→Rk\mathbf{g} : D \subset \mathbb{R}^{m} \to \mathbb{R}^{k}、f:g(Rk)⊂Rk→Rn\mathbf{f} : \mathbf{g}(\mathbb{R}^{k}) \subset \mathbb{R}^{k} \to \mathbb{R}^{n}が微分可能だとしよう。すると、これら二つの関数の合成 F=f∘g:Rm→Rn\mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{n}も微分可能であり、F\mathbf{F}の(全)導関数は次を満たす。

F′(x)=f′(g(x))g′(x) \mathbf{F}^{\prime}(\mathbf{x}) = \mathbf{f}^{\prime}\left( \mathbf{g}(\mathbf{x}) \right) \mathbf{g}^{\prime}(\mathbf{x})

解説

これを連鎖律と呼ぶ。

x=(x1,…,xm)\mathbf{x} = (x_{1}, \dots, x_{m})、g(x)=(g1,…,gk)\mathbf{g}(\mathbf{x}) = (g_{1}, \dots, g_{k})、f(g1,…,gk)=(f1,…,fn)\mathbf{f}(g_{1}, \dots, g_{k}) = (f_{1}, \dots, f_{n})とした場合、公式の具体的な形は全導関数の定義から次のようなn×mn \times m行列である。

F′(x)= [∂f1(g(x))∂g1∂f1∂g2⋯∂f1∂gk∂f2∂g1∂f2∂g2⋯∂f2∂gk⋮⋮⋱⋮∂fn∂g1∂fn∂g2⋯∂fn∂gk][∂g1(x)∂x1∂g1∂x2⋯∂g1∂xm∂g2∂x1∂g2∂x2⋯∂g2∂xm⋮⋮⋱⋮∂gk∂x1∂gk∂x2⋯∂gk∂xm]= [∂f1∂g1∂g1∂x1+∂f1∂g2∂g2∂x1+⋯+∂f1∂gk∂gk∂x1…∂f1∂g1∂g1∂x1+∂f1∂g2∂g2∂xm+⋯+∂f1∂gk∂gk∂xm⋮⋱⋮∂fn∂g1∂g1∂x1+∂fn∂g2∂g2∂x1+⋯+∂fn∂gk∂gk∂x1⋯∂fn∂g1∂g1∂x1+∂fn∂g2∂g2∂xm+⋯+∂fn∂gm∂gk∂xm]= [∑ℓ=1k∂f1∂gℓ∂gℓ∂x1…∑ℓ=1k∂f1∂gℓ∂gℓ∂xm⋮⋱⋮∑ℓ=1k∂fn∂gℓ∂gℓ∂x1…∑ℓ=1k∂fn∂gℓ∂gℓ∂xm] \begin{align*} \mathbf{F}^{\prime} (\mathbf{x}) =&\ \begin{bmatrix} \dfrac{\partial f_{1}(\mathbf{g}(\mathbf{x}))}{\partial g_{1}} & \dfrac{\partial f_{1}}{\partial g_{2}} & \cdots & \dfrac{\partial f_{1}}{\partial g_{k}} \\[1em] \dfrac{\partial f_{2}}{\partial g_{1}} & \dfrac{\partial f_{2}}{\partial g_{2}} & \cdots & \dfrac{\partial f_{2}}{\partial g_{k}} \\[1em] \vdots & \vdots & \ddots & \vdots \\[1em] \dfrac{\partial f_{n}}{\partial g_{1}} & \dfrac{\partial f_{n}}{\partial g_{2}} & \cdots & \dfrac{\partial f_{n}}{\partial g_{k}} \end{bmatrix} \begin{bmatrix} \dfrac{\partial g_{1}(\mathbf{x})}{\partial x_{1}} & \dfrac{\partial g_{1}}{\partial x_{2}} & \cdots & \dfrac{\partial g_{1}}{\partial x_{m}} \\[1em] \dfrac{\partial g_{2}}{\partial x_{1}} & \dfrac{\partial g_{2}}{\partial x_{2}} & \cdots & \dfrac{\partial g_{2}}{\partial x_{m}} \\[1em] \vdots & \vdots & \ddots & \vdots \\[1em] \dfrac{\partial g_{k}}{\partial x_{1}} & \dfrac{\partial g_{k}}{\partial x_{2}} & \cdots & \dfrac{\partial g_{k}}{\partial x_{m}} \end{bmatrix} \\[1em] =&\ \begin{bmatrix} \dfrac{\partial f_{1}}{\partial g_{1}} \dfrac{\partial g_{1}}{\partial x_{1}}+\dfrac{\partial f_{1}}{\partial g_{2}}\dfrac{\partial g_{2}}{\partial x_{1}} + \cdots + \dfrac{\partial f_{1}}{\partial g_{k}} \dfrac{\partial g_{k}}{\partial x_{1}} & \dots & \dfrac{\partial f_{1}}{\partial g_{1}} \dfrac{\partial g_{1}}{\partial x_{1}}+\dfrac{\partial f_{1}}{\partial g_{2}}\dfrac{\partial g_{2}}{\partial x_{m}} + \cdots + \dfrac{\partial f_{1}}{\partial g_{k}} \dfrac{\partial g_{k}}{\partial x_{m}} \\[1em] \vdots & \ddots & \vdots \\[1em] \dfrac{\partial f_{n}}{\partial g_{1}} \dfrac{\partial g_{1}}{\partial x_{1}}+\dfrac{\partial f_{n}}{\partial g_{2}}\dfrac{\partial g_{2}}{\partial x_{1}} + \cdots + \dfrac{\partial f_{n}}{\partial g_{k}} \dfrac{\partial g_{k}}{\partial x_{1}} & \cdots & \dfrac{\partial f_{n}}{\partial g_{1}} \dfrac{\partial g_{1}}{\partial x_{1}}+\dfrac{\partial f_{n}}{\partial g_{2}}\dfrac{\partial g_{2}}{\partial x_{m}} + \cdots + \dfrac{\partial f_{n}}{\partial g_{m}} \dfrac{\partial g_{k}}{\partial x_{m}} \end{bmatrix} \\[1em] =&\ \begin{bmatrix} \displaystyle \sum\limits_{\ell =1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \displaystyle \sum\limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \\[1em] \vdots & \ddots & \vdots \\[1em] \displaystyle \sum\limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \displaystyle \sum\limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \end{bmatrix} \end{align*}

アインシュタインの記法で簡単に表すと、1≤i≤n1 \le i \le n、1≤j≤m1 \le j \le mに対して

F′=[Fij′]=[∂f1∂gℓ∂gℓ∂x1…∂f1∂gℓ∂gℓ∂xm⋮⋱⋮∂fn∂gℓ∂gℓ∂x1…∂fn∂gℓ∂gℓ∂xm] \mathbf{F}^{\prime} = \left[ F_{ij}^{\prime} \right] = \begin{bmatrix} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \\[1em] \vdots & \ddots & \vdots \\[1em] \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \end{bmatrix}

Fij′=∂fi∂gℓ∂gℓ∂xj F_{ij}^{\prime} = \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}}

これは最も一般化された形なので、k,m,nk, m, nに従って、さまざまな具体的な公式を得ることができる。

公式

  • ケース 1. g:R→Rg : \mathbb{R} \to \mathbb{R}、f:R→Rf : \mathbb{R} \to \mathbb{R}、F=f∘g:R→RF = f \circ g : \mathbb{R} \to \mathbb{R}

    x∈Rx \in \mathbb{R}、g=g(x)g = g(x)、f=f(g(x))f = f(g(x))のとき、

    F′=dFdx=dfdgdgdx F^{\prime} = \dfrac{d F}{d x} = \dfrac{d f}{d g} \dfrac{d g}{d x}

    証明

  • ケース 2. g:R→Rk\mathbf{g} : \mathbb{R} \to \mathbb{R}^{k}、f:Rk→Rf : \mathbb{R}^{k} \to \mathbb{R}、F=f∘g:R→RF = f \circ \mathbf{g} : \mathbb{R} \to \mathbb{R}

    x∈Rx \in \mathbb{R}、g(x)=(g1,…,gk)\mathbf{g}(x) = (g_{1}, \dots, g_{k})、f=f(g1,…,gk)f = f(g_{1}, \dots ,g_{k})のとき、

    F′=dFdx=∑ℓ=1k∂f∂gℓdgℓdx F^{\prime} = \dfrac{d F}{d x} = \sum \limits_{\ell=1}^{k}\dfrac{\partial f}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x}

  • ケース 3. g:Rm→Rg : \mathbb{R}^{m} \to \mathbb{R}、f:R→Rf : \mathbb{R} \to \mathbb{R}、F=f∘g:Rm→RF = f \circ g : \mathbb{R}^{m} \to \mathbb{R}

    x=(x1,…,xn)∈Rn\mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n}、g=g(x)g = g(\mathbf{x})、f=f(g(x))f = f(g(\mathbf{x}))のとき、

    F′=dFdx=[dfdg∂g∂x1…dfdg∂g∂xm] F^{\prime} = \dfrac{d F}{d \mathbf{x}} = \begin{bmatrix} \dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{m}} \end{bmatrix}

    Fj′=dfdg∂g∂xj,1≤j≤m F_{j}^{\prime} = \dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{j}},\quad 1 \le j \le m

  • ケース 4. g:Rm→Rk\mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{k}、f:Rk→Rf : \mathbb{R}^{k} \to \mathbb{R}、F=f∘g:Rm→RF = f \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}

    x=(x1,…,xn)∈Rn\mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n}、g(x)=(g1,…,gk)\mathbf{g}(x) = (g_{1}, \dots, g_{k})、f=f(g1,…,gk)f = f(g_{1}, \dots, g_{k})のとき、

    F′=dFdx=[∑ℓ=1k∂f∂gℓ∂gℓ∂x1…∑ℓ=1k∂f∂gℓ∂gℓ∂xm] F^{\prime} = \dfrac{d F}{d \mathbf{x}} = \begin{bmatrix} \sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \end{bmatrix}

    Fj′=∑ℓ=1k∂f∂gℓ∂gℓ∂xj,1≤j≤m F_{j}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}},\quad 1 \le j \le m

  • ケース 5. g:R→Rg : \mathbb{R} \to \mathbb{R}、f:R→Rn\mathbf{f} : \mathbb{R} \to \mathbb{R}^{n}、F=f∘g:R→Rn\mathbf{F} = \mathbf{f} \circ g : \mathbb{R} \to \mathbb{R}^{n}

    x∈Rx \in \mathbb{R}、g=g(x)g = g(x)、f(g(x))=(f1,…,fn)\mathbf{f}(g(x)) = (f_{1}, \dots, f_{n})のとき、

    F′=dFdx=[df1dgdgdx⋮dfndgdgdx] \mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d x} = \begin{bmatrix} \dfrac{d f_{1}}{d g} \dfrac{d g}{d x} \\[1em] \vdots \\[1em] \dfrac{d f_{n}}{d g} \dfrac{d g}{d x} \end{bmatrix}

    Fi′=dfidgdgdx,1≤i≤n F_{i}^{\prime} = \dfrac{d f_{i}}{d g} \dfrac{d g}{d x},\quad 1\le i \le n

  • ケース 6. g:R→Rk\mathbf{g} : \mathbb{R} \to \mathbb{R}^{k}、f:Rk→Rn\mathbf{f} : \mathbb{R}^{k} \to \mathbb{R}^{n}、F=f∘g:R→Rn\mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R} \to \mathbb{R}^{n}

    x∈Rx \in \mathbb{R}、g(x)=(g1,…,gk)\mathbf{g}(x) = (g_{1}, \dots, g_{k})、f(g1,…,gk)=(f1,…,fn)\mathbf{f}(g_{1}, \dots ,g_{k}) = (f_{1}, \dots, f_{n})のとき、

    F′=dFdx=[∑ℓ=1k∂f1∂gℓdgℓdx⋮∑ℓ=1k∂fn∂gℓdgℓdx] \mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d x} = \begin{bmatrix} \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x} \\[1em] \vdots \\[1em] \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x} \end{bmatrix}

    Fi′=∑ℓ=1k∂fi∂gℓdgℓdx,1≤i≤n F_{i}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x},\quad 1\le i \le n

  • ケース 7. g:Rm→Rg : \mathbb{R}^{m} \to \mathbb{R}、f:R→Rn\mathbf{f} : \mathbb{R} \to \mathbb{R}^{n}、F=f∘g:Rm→Rn\mathbf{F} = \mathbf{f} \circ g : \mathbb{R}^{m} \to \mathbb{R}^{n}

    x=(x1,…,xn)∈Rn\mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n}、g=g(x)g = g(\mathbf{x})、f(g(x))=(f1,…,fn)\mathbf{f}(g(\mathbf{x})) = (f_{1}, \dots, f_{n})のとき、

    F′=dFdx=[df1dg∂g∂x1…df1dg∂g∂xm⋮⋱⋮dfndg∂g∂x1…dfndg∂g∂xm] \mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d \mathbf{x}} = \begin{bmatrix} \dfrac{d f_{1}}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f_{1}}{d g} \dfrac{\partial g}{\partial x_{m}} \\[1em] \vdots & \ddots & \vdots \\[1em] \dfrac{d f_{n}}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f_{n}}{d g} \dfrac{\partial g}{\partial x_{m}} \end{bmatrix}

    Fij′=dfidg∂g∂xj,1≤i≤n,1≤j≤m F_{ij}^{\prime} = \dfrac{d f_{i}}{d g} \dfrac{\partial g}{\partial x_{j}},\quad 1\le i \le n, 1 \le j \le m

  • ケース 8. g:Rm→Rk\mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{k}、f:Rk→Rn\mathbf{f} : \mathbb{R}^{k} \to \mathbb{R}^{n}、F=f∘g:Rm→Rn\mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{n}

    x=(x1,…,xn)∈Rn\mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n}、g(x)=(g1,…,gk)g(\mathbf{x}) = (g_{1}, \dots, g_{k})、f(g1,…,gk)=(f1,…,fn)\mathbf{f}(g_{1}, \dots, g_{k}) = (f_{1}, \dots, f_{n})のとき、

    F′=dFdx=[∑ℓ=1k∂f1∂gℓ∂gℓ∂x1…∑ℓ=1k∂f1∂gℓ∂gℓ∂xm⋮⋱⋮∑ℓ=1k∂fn∂gℓ∂gℓ∂x1…∑ℓ=1k∂fn∂gℓ∂gℓ∂xm] \mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d \mathbf{x}} = \begin{bmatrix} \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \\[1em] \vdots & \ddots & \vdots \\[1em] \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \end{bmatrix}

    Fij′=∑ℓ=1k∂fi∂gℓ∂gℓ∂xj,1≤i≤n,1≤j≤m F_{ij}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}},\quad 1\le i \le n, 1 \le j \le m

証明

一般化された証明を参照。

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