数学 - 解析学 - 多変数の関数 - 微分可能性と勾配ベクトル(極座標、微分鎖律)

解析入門〈3〉

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• 参考書籍
  • Pythonからはじめる数学入門
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  • インタラクティブ・データビジュアライゼーション

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.1(微分可能性と勾配ベクトル)、問題14.を取り組んでみる。

14. 
    
    $\begin{array}{}x=r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\\ y=r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\end{array}$
    $\begin{array}{}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->r}\\ =grad\left(f\left(x,y\right)\right)·\; <!--\; middle\; dot\; -->\left(\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ =\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x},\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)·\; <!--\; middle\; dot\; -->\left(\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ =\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\end{array}$
    $\begin{array}{}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}\\ =\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x},\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)·\; <!--\; middle\; dot\; -->\left(-r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ =r\left(-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\end{array}$
    ${\left(\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->r}\right)}^2={\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)}^2{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)}^2{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+2\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}$
    $\frac{1}{r^2}{\left(\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}\right)}^2={\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)}^2{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)}^2{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-2\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}$
    $\begin{array}{}{\left(\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->r}\right)}^2+\frac{1}{r^2}{\left(\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}\right)}^2={\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)}^2\left({\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)}^2\left({\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ {\left(\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->r}\right)}^2+\frac{1}{r^2}{\left(\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}\right)}^2={\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)}^2+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)}^2\end{array}$