logo

合成関数のヤコビアン 📂多変数ベクトル解析

合成関数のヤコビアン

概要

関数f:Rn→Rmf : \mathbb{R}^{n} \to \mathbb{R}^{m}とg:Rm→Rkg : \mathbb{R}^{m} \to \mathbb{R}^{k}が与えられたとする。ffのヤコビアンをJ(f)J(f)と表記しよう。すると、以下が成り立つ。

J(g∘f)=J(g)J(f) J(g \circ f) = J(g) J(f)

説明

ヤコビアンは最も一般化された導関数であるため、上記の定理は連鎖律の一般化である。

証明

ヤコビアンの定義により、

J(g∘f)=[∂(g∘f)1∂x1⋯∂(g∘f)1∂xn⋮⋱⋮∂(g∘f)k∂x1⋯∂(g∘f)k∂xn]=[∂g1∂x1⋯∂g1∂xn⋮⋱⋮∂gk∂x1⋯∂gk∂xn] J(g \circ f) = \begin{bmatrix} \dfrac{\partial (g \circ f)_{1}}{\partial x_{1}} & \cdots & \dfrac{\partial (g \circ f)_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial (g \circ f)_{k}}{\partial x_{1}} & \cdots & \dfrac{\partial (g \circ f)_{k}}{\partial x_{n}} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial g_{1}}{\partial x_{1}} & \cdots & \dfrac{\partial g_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial g_{k}}{\partial x_{1}} & \cdots & \dfrac{\partial g_{k}}{\partial x_{n}} \end{bmatrix}

この時gi=gi(f1(x),…,fm(x)))g_{i} = g_{i}(f_{1}(\mathbf{x}), \dots, f_{m}(\mathbf{x})))であるため、

∂gi∂xj=∑ℓ=1m∂gi∂fℓ∂fℓ∂xj \dfrac{\partial g_{i}}{\partial x_{j}} = \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{i}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{j}}

したがって、

J(g∘f)= [∑ℓ=1m∂g1∂fℓ∂fℓ∂x1⋯∑ℓ=1m∂g1∂fℓ∂fℓ∂xn⋮⋱⋮∑ℓ=1m∂gk∂fℓ∂fℓ∂x1⋯∑ℓ=1m∂gk∂fℓ∂fℓ∂xm]= [∂g1∂f1⋯∂g1∂fm⋮⋱⋮∂gk∂f1⋯∂gk∂fm][∂f1∂x1⋯∂f1∂xn⋮⋱⋮∂fm∂x1⋯∂fm∂xn]= J(g)J(f) \begin{align*} J(g \circ f) =&\ \begin{bmatrix} \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{1}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{1}} & \cdots & \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{1}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{k}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{1}} & \cdots & \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{k}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{m}} \end{bmatrix} \\ =&\ \begin{bmatrix} \dfrac{\partial g_{1}}{\partial f_{1}} & \cdots & \dfrac{\partial g_{1}}{\partial f_{m}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial g_{k}}{\partial f_{1}}& \cdots & \dfrac{\partial g_{k}}{\partial f_{m}} \end{bmatrix} \begin{bmatrix} \dfrac{\partial f_{1}}{\partial x_{1}} & \cdots & \dfrac{\partial f_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_{m}}{\partial x_{1}} & \cdots & \dfrac{\partial f_{m}}{\partial x_{n}} \end{bmatrix} \\ =&\ J(g) J(f) \end{align*}

■