多変数の関数高次導関数、テイラーの定理合成関数、内積、等式

学習環境

解析入門(中) (松坂和夫数学入門シリーズ 5) (松坂和夫 (著)、岩波書店)の第14章(多変数の関数)、14.2(高次導関数、テイラーの定理)、問題3の解答を求めてみる。

\frac{\partial^{2} z}{\partial x^{2}} = \frac{\partial z}{\partial x} (f (a x + b y) + x f' (a x + b y) a + y g' (a x + b y) a)

= f' (a x + b y) a + f' (a x + b y) a + f'' (a x + b y) a^{2} + y g'' (a x + b) a^{2}

= 2 a f' (a x + b y) + a^{2} x f'' (a x + b y) + a^{2} y g'' (a x + b)

\frac{\partial^{2} z}{\partial x \partial y} = \frac{\partial}{\partial x} (b x f' (a x + b y) + g (a x + b y) + b y g' (a x + b y))

= b f' (a x + b y) + a b x f'' (a x + b y) + a g' (a x + b y) + a b y g'' (a x + b y)

\frac{\partial^{2} z}{\partial y^{2}} = b^{2} x f'' (a x + b y) + b g' (a x + b y) + b g' (a x + b y) + b^{2} y g' (a x + b y)

= b^{2} x f'' (a x + b y) + 2 b g' (a x + b y) + b^{2} y g'' (a x + b y)

よって、

b^{2} \frac{\partial^{2} z}{\partial x^{2}} - 2 a b \frac{\partial^{2} z}{\partial x \partial y} + a^{2} \frac{\partial^{2} z}{\partial y^{2}}

\begin{array}{l} = (2 a b^{2} - 2 a b^{2}) f' (a x + b y) + (a^{2} b^{2} x - 2 a^{2} b^{2} x + a^{2} b^{2} x) f'' (a x + b y) \\ + (- 2 a^{2} b + 2 a^{2} b) g' (a x + b y) + (a^{2} b^{2} y - 2 a^{2} b^{2} y + a^{2} b^{2} y) g'' (a x + b y) \end{array}

= 0

上記において、

\begin{array}{l} a x + b y = t \\ f' = \frac{\partial f}{\partial t}, g' = \frac{\partial y}{\partial t} \end{array}

\frac{\partial^{2} z}{\partial u \partial v} = \frac{\partial}{d u} ((\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) \cdot (1, 1)) = \frac{\partial}{\partial u} (\frac{\partial z}{\partial x} - \frac{\partial z}{\partial y}) = \frac{\partial^{2} z}{\partial x^{2}} - \frac{\partial^{2} z}{\partial y^{2}}

\frac{\partial^{2} z}{\partial u^{2}} = \frac{\partial}{\partial u} ((\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) \cdot (e^{u}, - e^{- u}))

= \frac{\partial}{\partial u} (\frac{\partial z}{\partial x} e^{u} - \frac{\partial z}{\partial y} e^{- u})

= \frac{\partial}{\partial x} (\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) \cdot (e^{u}, - e^{- u}) e^{u} + \frac{\partial z}{\partial x} e^{u} - \frac{\partial}{\partial y} (\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) \cdot (e^{u}, - e^{- u}) e^{- u} + \frac{\partial z}{\partial y} e^{- u}

= \frac{\partial^{2} z}{\partial x^{2}} e^{2 u} - \frac{\partial^{2} z}{\partial x \partial y} + \frac{\partial z}{\partial x} e^{u} - \frac{\partial^{2} z}{\partial x \partial y} + \frac{\partial^{2} z}{\partial y^{2}} e^{- 2 u} + \frac{\partial z}{\partial y} e^{- u}

= \frac{\partial^{2} z}{\partial x^{2}} e^{2 u} - 2 \frac{\partial^{2} z}{\partial x \partial y} + \frac{\partial z}{\partial x} e^{u} + \frac{\partial^{2} z}{\partial y^{2}} e^{- 2 u} + \frac{\partial z}{\partial y} e^{- u}

\frac{\partial^{2} z}{\partial u \partial v} = \frac{\partial}{\partial u} ((\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) \cdot (e^{v}, - e^{- v}))

= \frac{\partial}{\partial u} (\frac{\partial z}{\partial x} e^{v} - \frac{\partial z}{\partial y} e^{- v})

\begin{array}{l} = \frac{\partial}{\partial x} ((\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) \cdot (e^{u}, - e^{- u}) e^{v}) \\ - \frac{\partial}{\partial y} ((\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) \cdot (e^{u}, - e^{- u}) e^{- v}) \end{array}

= \frac{\partial^{2} z}{\partial x^{2}} e^{u + v} - \frac{\partial^{2} z}{\partial x \partial y} e^{- u + v} - \frac{\partial^{2} z}{\partial x \partial y} e^{u - v} + \frac{\partial^{2} z}{\partial y^{2}} e^{- u - v}

\frac{\partial^{2} z}{\partial v^{2}} = \frac{\partial^{2} z}{\partial x^{2}} e^{2 v} - 2 \frac{\partial^{2} z}{\partial x \partial y} + \frac{\partial z}{\partial x} e^{v} + \frac{\partial^{2} z}{\partial y^{2}} e^{- 2 v} + \frac{\partial z}{\partial y} e^{- v}

よって、

\frac{\partial^{2} z}{\partial u^{2}} + 2 \frac{\partial^{2} z}{\partial u \partial v} + \frac{\partial^{2} z}{\partial v^{2}}

\begin{array}{l} = (e^{2 u} + e^{u + u} + e^{2 v}) \frac{\partial^{2} z}{\partial x^{2}} - 2 (2 + e^{- u + v} + e^{u - v}) \frac{\partial^{2} z}{\partial x \partial y} \\ + (e^{- 2 u} + e^{- u - v} + e^{- 2 v}) \frac{\partial^{2} z}{\partial y^{2}} + (e^{u} + e^{v}) \frac{\partial z}{\partial x} + (e^{- u} + e^{- v}) \frac{\partial z}{\partial y} \end{array}

= x^{2} \frac{\partial^{2} z}{\partial x^{2}} - 2 x y \frac{\partial^{2} z}{\partial x \partial y} + y^{2} \frac{\partial^{2} z}{\partial y^{2}} + x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y}