合成写像の微分
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2022/01/02 17:04 UTC 版)
次に、合成写像の微分について説明する。 E {\displaystyle {\textbf {E}}} を R m {\displaystyle {\mathbb {R} ^{m}}} の開集合とし、 E {\displaystyle {\textbf {E}}} は、 f {\displaystyle {\textbf {f}}} の値域を含む（つまり、 f ( D ) ⊂ E {\displaystyle {\textbf {f}}({\textbf {D}})\subset {\textbf {E}}} 、特に f ( p ) ∈ E {\displaystyle {\textbf {f}}({\textbf {p}})\in {\textbf {E}}} とする）とする。多変数ベクトル値関数 g ( y ) = ( g 1 ( y ) ⋮ g m ( y ) ) {\displaystyle \mathbf {g} (\mathbf {y} )=\left({\begin{matrix}{{g}_{1}}(\mathbf {y} )\\\vdots \\{{g}_{m}}(\mathbf {y} )\\\end{matrix}}\right)} (3-6) は、 E {\displaystyle {\textbf {E}}} で定義され、 R l {\displaystyle {{\mathbb {R} }^{l}}} に値をとるとする。このとき、 g {\displaystyle \mathbf {g} } と f {\displaystyle {\textbf {f}}} との合成写像 g ∘ f {\displaystyle \mathbf {g} \circ {\textbf {f}}} は、 D {\displaystyle {\textbf {D}}} で定義され、 R l {\displaystyle {{\mathbb {R} }^{l}}} に値をとる多変数ベクトル値関数である。 f {\displaystyle {\textbf {f}}} が点 p {\displaystyle {\textbf {p}}} で微分可能で、 g {\displaystyle \mathbf {g} } が、点 f ( p ) {\displaystyle \mathbf {f} (\mathbf {p} )} で微分可能であるとき、 g ∘ f {\displaystyle \mathbf {g} \circ {\textbf {f}}} も p {\displaystyle {\textbf {p}}} で微分可能で、 ( J ( g ∘ f ) ) [ p ] {\displaystyle {{\left(J\left({\textbf {g}}\circ \mathbf {f} \right)\right)}_{[{\textbf {p}}]}}} = ( J g ) [ f ( p ) ] {\displaystyle {{\left(J\mathbf {g} \right)}_{[{\textbf {f}}(\mathbf {p} )]}}} ⋅ ( J f ) p {\displaystyle \cdot {(J{\textbf {f}})}_{\textbf {p}}} (3-7) ここで“ ⋅ {\displaystyle \cdot } ”とは、行列としての積である。 ■証明 f {\displaystyle {\textbf {f}}} を点 p {\displaystyle {\textbf {p}}} で一次展開し、 g {\displaystyle {\textbf {g}}} を点 f ( p ) {\displaystyle \mathbf {f} (\mathbf {p} )} で(2-16)同様に一次展開すると、 f ( x ) {\displaystyle \mathbf {f} (\mathbf {x} )} = ( J f ) [ p ] ⋅ ( x − p ) + f ( p ) + o [ f , p ] ( x ) {\displaystyle ={{(J\mathbf {f} )}_{[\mathbf {p} ]}}\cdot (\mathbf {x} -\mathbf {p} )+\mathbf {f} (\mathbf {p} )+{{\mathbf {o} }_{[\mathbf {f} ,\mathbf {p} ]}}(\mathbf {x} )} (3-8) g ( y ) {\displaystyle \mathbf {g} (\mathbf {y} )} = ( J g ) [ f ( p ) ] ⋅ ( y − f ( p ) ) + g ( f ( p ) ) + o [ g , f ( p ) ] ( f ( p ) ) {\displaystyle ={{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot (\mathbf {y} -\mathbf {f} (\mathbf {p} ))+\mathbf {g} (\mathbf {f} (\mathbf {p} ))+{{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}({\textbf {f}}(\mathbf {p} ))} (3-9) となるので、 g ∘ f ( x ) {\displaystyle \mathbf {g} \circ {\textbf {f}}({\textbf {x}})} = g ( f ( x ) ) {\displaystyle =\mathbf {g} ({\textbf {f}}({\textbf {x}}))} = ( J g ) [ f ( p ) ] ⋅ ( f ( x ) − f ( p ) ) + g ( f ( p ) ) + o [ g , f ( p ) ] ( f ( p ) ) {\displaystyle ={{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot (\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} ))+\mathbf {g} (\mathbf {f} (\mathbf {p} ))+{{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}({\textbf {f}}(\mathbf {p} ))} = ( J g ) [ f ( p ) ] ⋅ ( ( J f ) [ p ] ⋅ ( x − p ) + o [ f , p ] ( x ) ) {\displaystyle ={{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot ({{(J\mathbf {f} )}_{[\mathbf {p} ]}}\cdot (\mathbf {x} -\mathbf {p} )+{{\mathbf {o} }_{[\mathbf {f} ,\mathbf {p} ]}}(\mathbf {x} ))} + g ( f ( p ) ) + o [ g , f ( p ) ] ( f ( p ) ) {\displaystyle +\mathbf {g} (\mathbf {f} (\mathbf {p} ))+{{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))} = ( J g ) [ f ( p ) ] ⋅ ( J f ) [ p ] ⋅ ( x − p ) + ( J g ) [ f ( p ) ] ⋅ o [ f , p ] ( x ) {\displaystyle ={{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot {{(J\mathbf {f} )}_{[\mathbf {p} ]}}\cdot (\mathbf {x} -\mathbf {p} )+{{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot {{\mathbf {o} }_{[\mathbf {f} ,\mathbf {p} ]}}(\mathbf {x} )} + g ( f ( p ) ) + o [ g , f ( p ) ] ( f ( p ) ) {\displaystyle +\mathbf {g} (\mathbf {f} (\mathbf {p} ))+{{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))} = ( J g ) [ f ( p ) ] ⋅ ( J f ) [ p ] ⋅ ( x − p ) + g ( f ( p ) ) {\displaystyle ={{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot {{(J\mathbf {f} )}_{[\mathbf {p} ]}}\cdot (\mathbf {x} -\mathbf {p} )+\mathbf {g} (\mathbf {f} (\mathbf {p} ))} + o [ g , f ( p ) ] ( f ( p ) ) + ( J g ) [ f ( p ) ] ⋅ o [ f , p ] ( x ) {\displaystyle +{{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))+{{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot {{\mathbf {o} }_{[\mathbf {f} ,\mathbf {p} ]}}(\mathbf {x} )} (3-10) である。従って lim x → p ( o [ g , f ( p ) ] ( f ( p ) ) + ( J g ) [ f ( p ) ] ⋅ o [ f , p ] ( x ) ) ‖ x − p ‖ = 0 {\displaystyle {\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,\,\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))+{{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot {{\mathbf {o} }_{[\mathbf {f} ,\mathbf {p} ]}}(\mathbf {x} )\right)}{\left\|\mathbf {x} -\mathbf {p} \right\|}}\,=0} (3-11) を示すを示せば終証である。 以下(3-11)を示す。 ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ x − p ‖ {\displaystyle {\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {x} -\mathbf {p} \right\|}}} = ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ x − p ‖ ( ‖ f ( x ) − f ( p ) ‖ ‖ f ( x ) − f ( p ) ‖ ) {\displaystyle =\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {x} -\mathbf {p} \right\|}}\left({\frac {\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}{\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}}\right)} = ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ f ( x ) − f ( p ) ‖ ( ‖ f ( x ) − f ( p ) ‖ ‖ x − p ‖ ) {\displaystyle \,={\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}}\left({\frac {\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}{\left\|\mathbf {x} -\mathbf {p} \right\|}}\right)\,} (3-12) より、 lim x → p ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ x − p ‖ {\displaystyle {\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {x} -\mathbf {p} \right\|}}} = lim x → p ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ f ( x ) − f ( p ) ‖ ( ‖ f ( x ) − f ( p ) ‖ ‖ x − p ‖ ) {\displaystyle \,={\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}}\left({\frac {\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}{\left\|\mathbf {x} -\mathbf {p} \right\|}}\right)\,} (3-13) 一方、 lim x → p ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ f ( x ) − f ( p ) ‖ {\displaystyle {\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}}} = lim f ( x ) → f ( p ) ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ f ( x ) − f ( p ) ‖ {\displaystyle {\underset {\mathbf {f} (\mathbf {x} )\to \mathbf {f} (\mathbf {p} )}{\mathop {\lim } }}\,\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}}} (3-14) は、 lim y → f ( p ) ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ y − f ( p ) ‖ {\displaystyle {\underset {\mathbf {y} \to \mathbf {f} (\mathbf {p} )}{\mathop {\lim } }}\,\,\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {y} -\mathbf {f} (\mathbf {p} )\right\|}}} (3-15) の特殊なケースに過ぎないので、 lim x → p ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ f ( x ) − f ( p ) ‖ = 0 {\displaystyle {\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}}=0} (3-16) さらに、 lim x → p ( ‖ f ( x ) − f ( p ) ‖ ‖ x − p ‖ ) {\displaystyle {\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,\left({\frac {\left\|\mathbf {f} (\mathbf {x} )-\mathbf {f} (\mathbf {p} )\right\|}{\left\|\mathbf {x} -\mathbf {p} \right\|}}\right)\,} (3-17) は有限の値であることから、 lim x → p ( o [ g , f ( p ) ] ( f ( p ) ) ) ‖ x − p ‖ = 0 {\displaystyle {\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,\,{\frac {\left({{\mathbf {o} }_{[\mathbf {g} ,\mathbf {f} (\mathbf {p} )]}}(\mathbf {f} (\mathbf {p} ))\right)}{\left\|\mathbf {x} -\mathbf {p} \right\|}}=0} (3-18) また、 lim x → p ( ( J g ) [ f ( p ) ] ⋅ o [ f , p ] ( x ) ) ‖ x − p ‖ = 0 {\displaystyle {\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,\,\,\,{\frac {\left({{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot {{\mathbf {o} }_{[\mathbf {f} ,\mathbf {p} ]}}(\mathbf {x} )\right)}{\left\|\mathbf {x} -\mathbf {p} \right\|}}=0} (3-19) は、 lim x → p ( ( J g ) [ f ( p ) ] ⋅ o [ f , p ] ( x ) ) ‖ x − p ‖ {\displaystyle {\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,\,\,{\frac {\left({{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\cdot {{\mathbf {o} }_{[\mathbf {f} ,\mathbf {p} ]}}(\mathbf {x} )\right)}{\left\|\mathbf {x} -\mathbf {p} \right\|}}} = lim x → p ( ( J g ) [ f ( p ) ] ( o [ f , p ] ( x ) ‖ x − p ‖ ) ) {\displaystyle ={\underset {\mathbf {x} \to \mathbf {p} }{\mathop {\lim } }}\,\,\,\left({{(J\mathbf {g} )}_{[\mathbf {f} (\mathbf {p} )]}}\left({\frac {{{\mathbf {o} }_{[\mathbf {f} ,\mathbf {p} ]}}(\mathbf {x} )}{\left\|\mathbf {x} -\mathbf {p} \right\|}}\right)\,\right)} (3-20) であることと、線形写像の連続性から明らかである。 ■ (3-7)を行列として具体的に表記すると ( J ( g ∘ f ) ) [ p ] {\displaystyle {{\left(J\left({\textbf {g}}\circ \mathbf {f} \right)\right)}_{[{\textbf {p}}]}}} = ( ∂ g 1 ∂ x 1 | [ f ( p ) ] ⋯ ∂ g 1 ∂ x m | [ f ( p ) ] ⋮ ⋮ ⋮ ∂ g l ∂ x 1 | [ f ( p ) ] ⋯ ∂ g l ∂ x m | [ f ( p ) ] ) {\displaystyle \left({\begin{matrix}{{\left.{\frac {\partial {{g}_{1}}}{\partial {{x}_{1}}}}\right|}_{[\mathbf {f} (\mathbf {p} )]}}&\cdots &{{\left.{\frac {\partial {{g}_{1}}}{\partial {{x}_{m}}}}\right|}_{[\mathbf {f} (\mathbf {p} )]}}\\\vdots &\vdots &\vdots \\{{\left.{\frac {\partial {{g}_{l}}}{\partial {{x}_{1}}}}\right|}_{[\mathbf {f} (\mathbf {p} )]}}&\cdots &{{\left.{\frac {\partial {{g}_{l}}}{\partial {{x}_{m}}}}\right|}_{[\mathbf {f} (\mathbf {p} )]}}\\\end{matrix}}\right)} ( ∂ f 1 ∂ x 1 | [ p ] ⋯ ∂ f 1 ∂ x n | [ p ] ⋮ ⋮ ⋮ ∂ f m ∂ x 1 | [ p ] ⋯ ∂ f m ∂ x n | [ p ] ) {\displaystyle \left({\begin{matrix}{{\left.{\frac {\partial {{f}_{1}}}{\partial {{x}_{1}}}}\right|}_{[\mathbf {p} ]}}&\cdots &{{\left.{\frac {\partial {{f}_{1}}}{\partial {{x}_{n}}}}\right|}_{[\mathbf {p} ]}}\\\vdots &\vdots &\vdots \\{{\left.{\frac {\partial {{f}_{m}}}{\partial {{x}_{1}}}}\right|}_{[\mathbf {p} ]}}&\cdots &{{\left.{\frac {\partial {{f}_{m}}}{\partial {{x}_{n}}}}\right|}_{[\mathbf {p} ]}}\\\end{matrix}}\right)} (3-21) となる。これから、 ∂ ( f ∘ g ) i ∂ x j | [ p ] = ∑ k = 1 m ∂ f i ∂ x k | [ f ( p ) ] ∂ g k ∂ x j | [ p ] {\displaystyle {{\left.{\frac {\partial {{(\mathbf {f} \circ \mathbf {g} )}_{i}}}{\partial {{x}_{j}}}}\right|}_{[{\textbf {p}}]}}={{\sum \limits _{k=1}^{m}{\left.{\frac {\partial {{f}_{i}}}{\partial {{x}_{k}}}}\right|}}_{[{\textbf {f}}({\textbf {p}})]}}{{\left.{\frac {\partial {{g}_{k}}}{\partial {{x}_{j}}}}\right|}_{[{\textbf {p}}]}}} (3-22) が分かる。
※この「合成写像の微分」の解説は、「多変数の微分」の解説の一部です。
「合成写像の微分」を含む「多変数の微分」の記事については、「多変数の微分」の概要を参照ください。
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