n 次元極座標のラプラシアンを直接求める

直交座標$(x_1,x_2,\dots,x_n)$と極座標$(r,\theta_1,\dots,\theta_{n-1})$の関係を以下のように定義します．

直交座標におけるラプラシアン$\displaystyle\Delta = \sum_{i=1}^n \pdv[2]{x_i} = \pdv[2]{x_1} + \pdv[2]{x_2} + \dots + \pdv[2]{x_n}$を極座標に変換した形を求めるとき，$2$次元の場合の変換を繰り返す導出が一般的ですが，ここでは$1$回の変換でまとめて計算します．

$1$階微分

連鎖律より，$\displaystyle\pdv{r}, \pdv{\theta_1}, \dots, \pdv{\theta_{n-1}}$と$\displaystyle\pdv{x_1}, \pdv{x_2}, \dots, \pdv{x_n}$の間には以下のような関係が成り立ちます．

行列$\begin{pmatrix} \pdv{x_1}{r} & \pdv{x_2}{r} & \cdots & \pdv{x_n}{r} \\ \pdv{x_1}{\theta_1} & \pdv{x_2}{\theta_1} & \dots & \pdv{x_n}{\theta_1} \\ \vdots & \vdots & & \vdots \\ \pdv{x_1}{\theta_{n-1}} & \pdv{x_2}{\theta_{n-1}} & \cdots & \pdv{x_n}{\theta_{n-1}} \end{pmatrix}$を$A$とおきます．

$\displaystyle\pdv{x_1}, \pdv{x_2}, \dots, \pdv{x_n}$を$\displaystyle\pdv{r}, \pdv{\theta_1}, \dots, \pdv{\theta_{n-1}}$で表したいので，左から$A$の逆行列をかけることを考えます．そのため，$A$の成分を具体的に表し，より簡単な行列の積として書くことを考えます．

\begin{align*} A &= \begin{pmatrix} \pdv{x_1}{r} & \pdv{x_2}{r} & \cdots & \pdv{x_n}{r} \\ \pdv{x_1}{\theta_1} & \pdv{x_2}{\theta_1} & \dots & \pdv{x_n}{\theta_1} \\ \vdots & \vdots & & \vdots \\ \pdv{x_1}{\theta_{n-1}} & \pdv{x_2}{\theta_{n-1}} & \cdots & \pdv{x_n}{\theta_{n-1}} \end{pmatrix} \\ &= \begin{pmatrix} \cos\theta_1 & \sin\theta_1\cos\theta_2 & \sin\theta_1\sin\theta_2\cos\theta_3 & \cdots & \sin\theta_1\sin\theta_2\dots\sin\theta_{n-3}\cos\theta_{n-2} & \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} \\ -r\sin\theta_1 & r\cos\theta_1\cos\theta_2 & r\cos\theta_1\sin\theta_2\cos\theta_3 & \cdots & r\cos\theta_1\sin\theta_2\dots\sin\theta_{n-3}\cos\theta_{n-2} & r\cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & r\cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} \\ 0 & -r\sin\theta_1\sin\theta_2 & r\sin\theta_1\cos\theta_2\cos\theta_3 & \cdots & r\sin\theta_1\cos\theta_2\cdots\sin\theta_{n-3}\cos\theta_{n-2} & r\sin\theta_1\cos\theta_2\cdots\sin\theta_{n-2}\cos\theta_{n-1} & r\sin\theta_1\cos\theta_2\cdots\sin\theta_{n-2}\sin\theta_{n-1} \\ \vdots & 0 & -r\sin\theta_1\sin\theta_2\sin\theta_3 & & \vdots & \vdots & \vdots \\ \vdots & \vdots & 0 & \ddots & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & -r\sin\theta_1\sin\theta_2\dots\sin\theta_{n-3}\sin\theta_{n-2} & r\sin\theta_1\sin\theta_2\dots\cos\theta_{n-2}\cos\theta_{n-1} & r\sin\theta_1\sin\theta_2\dots\cos\theta_{n-2}\sin\theta_{n-1}\\ 0 & 0 & 0 & \cdots & 0 & -r\sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} & r\sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1}\end{pmatrix} \end{align*}
第$1,2,3,\dots,n$行から，それぞれ$1,r,r\sin\theta_1,\dots,r\sin\theta_1\dots\sin\theta_{n-2}$をくくり出します．行列の第$1,2,\dots,n$行の各成分をそれぞれ$a_1,a_2,\dots,a_n$倍する操作は，左から$\begin{pmatrix}a_1 & 0 & \dots & \dots & 0 \\ 0 & a_2 & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & a_{n-1} & 0 \\ 0 & \cdots & \cdots & 0 & a_n \end{pmatrix}$を掛ける操作と言い換えられます．
\begin{align} A &= \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & r & \ddots & & & \vdots \\ \vdots & \ddots & r\sin\theta_1 & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & r\sin\theta_1\dots\sin\theta_{n-3} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & r\sin\theta_1\dots\sin\theta_{n-2}\end{pmatrix} \begin{pmatrix} \cos\theta_1 & \sin\theta_1\cos\theta_2 & \sin\theta_1\sin\theta_2\cos\theta_3 & \cdots & \sin\theta_1\sin\theta_2\dots\sin\theta_{n-3}\cos\theta_{n-2} & \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} \\ -\sin\theta_1 & \cos\theta_1\cos\theta_2 & \cos\theta_1\sin\theta_2\cos\theta_3 & \cdots & \cos\theta_1\sin\theta_2\dots\sin\theta_{n-3}\cos\theta_{n-2} & \cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} \\ 0 & -\sin\theta_2 & \cos\theta_2\cos\theta_3 & \cdots & \cos\theta_2\cdots\sin\theta_{n-3}\cos\theta_{n-2} & \cos\theta_2\cdots\sin\theta_{n-2}\cos\theta_{n-1} & \cos\theta_2\cdots\sin\theta_{n-2}\sin\theta_{n-1} \\ \vdots & \ddots & -\sin\theta_3 & \ddots & \vdots & \vdots & \vdots \\ \vdots & & \ddots & \ddots & \cos\theta_{n-3}\cos\theta_{n-2} & \vdots & \vdots \\ \vdots & & & \ddots & -\sin\theta_{n-2} & \cos\theta_{n-2}\cos\theta_{n-1} &\cos\theta_{n-2}\sin\theta_{n-1}\\ 0 & \cdots & \cdots & \cdots & 0 & -\sin\theta_{n-1} & \cos\theta_{n-1}\end{pmatrix} \end{align}
実は右側の行列は直交行列になっているのですが，ここではそれは用いずに，さらに分解を進めます．以下，簡単のため$\sin\theta_i,\cos\theta_i \neq 0$を仮定します．
第$n-1$列の$-\tan\theta_{n-1}$倍を第$n$列に加え，$i=n-2,n-3,\dots,2,1$の順に第$i$列の$-\tan\theta_i\cos\theta_{i-1}$倍を第$i+1$列に加えます．列基本変形は，右から基本行列をかけることで表されます．$(i,i) \ (2 \leq i \leq n-1)$成分は$\displaystyle\cos\theta_{i-1}\cos\theta_i - \tan\theta_{i-1}\cos\theta_i(-\sin\theta_{i-1}) = \left(\frac{\cos^2\theta_{i-1}}{\cos\theta_{i-1}}+\frac{\sin^2\theta_{i-1}}{\cos\theta_{i-1}}\right)\cos\theta_i = \frac{\cos\theta_i}{\cos\theta_{i-1}}$，$(n,n)$成分は$\displaystyle\cos\theta_{n-1}-\tan\theta_{n-1}(-\sin\theta_{n-1}) = \frac{\cos^2\theta_{n-1}}{\cos\theta_{n-1}}+\frac{\sin^2\theta_{n-1}}{\cos\theta_{n-1}} = \frac{1}{\cos\theta_{n-1}}$となります．

\begin{align} A &= \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & r & \ddots & & & \vdots \\ \vdots & \ddots & r\sin\theta_1 & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & r\sin\theta_1\dots\sin\theta_{n-3} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & r\sin\theta_1\dots\sin\theta_{n-2}\end{pmatrix} \begin{pmatrix} \cos\theta_1 & 0 & \cdots & \cdots & \cdots & 0 \\ -\sin\theta_1 & \frac{\cos\theta_2}{\cos\theta_1} & \ddots & & & \vdots \\ 0 & -\sin\theta_2 & \frac{\cos\theta_3}{\cos\theta_2} & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & -\sin\theta_{n-2} & \frac{\cos\theta_{n-1}}{\cos\theta_{n-2}} & 0 \\ 0 & \cdots & \cdots & 0 & -\sin\theta_{n-1} & \frac{1}{\cos\theta_{n-1}} \end{pmatrix} \begin{pmatrix} 1 & \tan\theta_1\cos\theta_2 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 1 & 0 \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix} \dots \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 1 & \ddots & \vdots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 & 0 \\ \vdots & & \ddots & 1 & \tan\theta_{n-2}\cos\theta_{n-1} & 0 \\ \vdots & & & \ddots & 1 & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 \\ \vdots & & \ddots & 1 & \tan\theta_{n-1} \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix} \end{align}

第$1$行の$\tan\theta_1$倍を第$2$行に加え，$i=2,3,\dots,n-1$の順に第$i$行の$\cos\theta_{i-1}\tan\theta_i$倍を第$i+1$行に加えます．行基本変形は，左から基本行列をかけることで表されます．

\begin{align} A &= \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & r & \ddots & & & \vdots \\ \vdots & \ddots & r\sin\theta_1 & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & r\sin\theta_1\dots\sin\theta_{n-3} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & r\sin\theta_1\dots\sin\theta_{n-2}\end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ -\tan\theta_1 & 1 & \ddots & & \vdots \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & & \vdots \\ 0 & -\cos\theta_1\tan\theta_2 & 1 & \ddots & & \vdots \\ 0 & 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \dots \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \\ 0 & \cdots & 0 & -\cos\theta_{n-2}\tan\theta_{n-1} & 1\end{pmatrix} \begin{pmatrix} \cos\theta_1 & 0 & \cdots & \cdots & 0 \\ 0 & \frac{\cos\theta_2}{\cos\theta_1} & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \frac{\cos\theta_{n-1}}{\cos\theta_{n-2}} & 0 \\ 0 & \cdots & \cdots & 0 & \frac{1}{\cos\theta_{n-1}}\end{pmatrix} \begin{pmatrix} 1 & \tan\theta_1\cos\theta_2 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 1 & 0 \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix} \dots \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 1 & \ddots & \vdots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 & 0 \\ \vdots & & \ddots & 1 & \tan\theta_{n-2}\cos\theta_{n-1} & 0 \\ \vdots & & & \ddots & 1 & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 \\ \vdots & & \ddots & 1 & \tan\theta_{n-1} \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix} \end{align}

$r,\sin\theta_i,\cos\theta_i \neq 0$のとき，上の積に現れるそれぞれの行列は正則なので，その積である$A$も正則となります．逆行列は以下のようになります．
\begin{align} A^{-1} &= \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 \\ \vdots & & \ddots & 1 & -\tan\theta_{n-1} \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 1 & \ddots & \vdots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 & 0 \\ \vdots & & \ddots & 1 & -\tan\theta_{n-2}\cos\theta_{n-1} & 0 \\ \vdots & & & \ddots & 1 & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & 1 \end{pmatrix} \dots \begin{pmatrix} 1 & -\tan\theta_1\cos\theta_2 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 1 & 0 \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} \frac{1}{\cos\theta_1} & 0 & \cdots & \cdots & 0 \\ 0 & \frac{\cos\theta_1}{\cos\theta_2} & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \frac{\cos\theta_{n-2}}{\cos\theta_{n-1}} & 0 \\ 0 & \cdots & \cdots & 0 & \cos\theta_{n-1}\end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \\ 0 & \cdots & 0 & \cos\theta_{n-2}\tan\theta_{n-1} & 1\end{pmatrix} \dots \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & & \vdots \\ 0 & \cos\theta_1\tan\theta_2 & 1 & \ddots & & \vdots \\ 0 & 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ \tan\theta_1 & 1 & \ddots & & \vdots \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & \frac{1}{r} & \ddots & & & \vdots \\ \vdots & \ddots & \frac{1}{r\sin\theta_1} & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-3}} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-2}}\end{pmatrix} \end{align}

$\begin{pmatrix} \frac{1}{\cos\theta_1} & 0 & \cdots & \cdots & 0 \\ 0 & \frac{\cos\theta_1}{\cos\theta_2} & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \frac{\cos\theta_{n-2}}{\cos\theta_{n-1}} & 0 \\ 0 & \cdots & \cdots & 0 & \cos\theta_{n-1}\end{pmatrix}$に対し，$i=2,3,\dots,n-1$の順に第$i$行の$-\tan\theta_{i-1}\cos\theta_i$倍を第$i-1$行に加え，第$n$行の$-\tan\theta_{n-1}$倍を第$n-1$行に加えます．

\begin{align} A^{-1} &= \begin{pmatrix} \frac{1}{\cos\theta_1} & -\sin\theta_1 & 0 & \cdots & \cdots & 0 \\ 0 & \frac{\cos\theta_1}{\cos\theta_2} & -\sin\theta_2 & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & -\sin\theta_{n-2} & 0 \\ \vdots & & & \ddots & \frac{\cos\theta_{n-2}}{\cos\theta_{n-1}} & -\sin\theta_{n-1} \\ 0 & \cdots & \cdots & \cdots & 0 & \cos\theta_{n-1} \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \\ 0 & \cdots & 0 & \cos\theta_{n-2}\tan\theta_{n-1} & 1\end{pmatrix} \dots \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & & \vdots \\ 0 & \cos\theta_1\tan\theta_2 & 1 & \ddots & & \vdots \\ 0 & 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ \tan\theta_1 & 1 & \ddots & & \vdots \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & \frac{1}{r} & \ddots & & & \vdots \\ \vdots & \ddots & \frac{1}{r\sin\theta_1} & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-3}} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-2}}\end{pmatrix} \end{align}

第$n-1$列に第$n$列の$\cos\theta_{n-2}\tan\theta_{n-1}$倍を加えます．$(n-1,n-1)$成分は$\displaystyle\frac{\cos\theta_{n-2}}{\cos\theta_{n-1}}+\cos\theta_{n-2}\tan\theta_{n-1}(-\sin\theta_{n-1}) = \cos\theta_{n-2}\left(\frac{1}{\cos\theta_{n-1}}-\frac{\sin^2\theta_{n-1}}{\cos\theta_{n-1}}\right) = \cos\theta_{n-2}\cos\theta_{n-1}$となります．

\begin{align} A^{-1} &= \begin{pmatrix} \frac{1}{\cos\theta_1} & -\sin\theta_1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & \frac{\cos\theta_1}{\cos\theta_2} & -\sin\theta_2 & \ddots & & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & -\sin\theta_{n-3} & \ddots & \vdots \\ \vdots & & & \ddots & \frac{\cos\theta_{n-3}}{\cos\theta_{n-2}} & -\sin\theta_{n-2} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & \cos\theta_{n-2}\cos\theta_{n-1} & -\sin\theta_{n-1} \\ 0 & \cdots & \cdots & \cdots & 0 & \cos\theta_{n-2}\sin\theta_{n-1} & \cos\theta_{n-1} \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & & \vdots \\ \vdots & \ddots & \ddots & \ddots & & \vdots \\ 0 & \cdots & 0 & 1 & \ddots & \vdots \\ 0 & \cdots & 0 & \cos\theta_{n-3}\tan\theta_{n-2} & 1 & 0 \\ 0 & \cdots & 0 & 0 & 0 & 1 \end{pmatrix} \dots \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & & \vdots \\ 0 & \cos\theta_1\tan\theta_2 & 1 & \ddots & & \vdots \\ 0 & 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ \tan\theta_1 & 1 & \ddots & & \vdots \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & \frac{1}{r} & \ddots & & & \vdots \\ \vdots & \ddots & \frac{1}{r\sin\theta_1} & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-3}} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-2}}\end{pmatrix} \end{align}

第$n-2$列に第$n-1$列の$\cos\theta_{n-3}\tan\theta_{n-2}$倍を加えます．$(n-2,n-2)$成分は$\displaystyle\frac{\cos\theta_{n-3}}{\cos\theta_{n-2}}+\cos\theta_{n-3}\tan\theta_{n-2}(-\sin\theta_{n-2}) = \cos\theta_{n-3}\left(\frac{1}{\cos\theta_{n-2}}-\frac{\sin^2\theta_{n-2}}{\cos\theta_{n-2}}\right) = \cos\theta_{n-3}\cos\theta_{n-2}$となります．

\begin{align} A^{-1} &= \begin{pmatrix} \frac{1}{\cos\theta_1} & -\sin\theta_1 & 0 & \cdots & \cdots & \cdots & \cdots & 0 \\ 0 & \frac{\cos\theta_1}{\cos\theta_2} & -\sin\theta_2 & \ddots & & & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & & & \vdots \\ \vdots & & \ddots & \ddots & -\sin\theta_{n-4} & \ddots & & \vdots \\ \vdots & & & \ddots & \frac{\cos\theta_{n-4}}{\cos\theta_{n-3}} & -\sin\theta_{n-3} & \ddots & \vdots \\ 0 & \cdots & \cdots & \cdots & 0 & \cos\theta_{n-3}\cos\theta_{n-2} & -\sin\theta_{n-2} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & \cos\theta_{n-3}\sin\theta_{n-2}\cos\theta_{n-1} & \cos\theta_{n-2}\cos\theta_{n-1} & -\sin\theta_{n-1} \\ 0 & \cdots & \cdots & \cdots & 0 & \cos\theta_{n-3}\sin\theta_{n-2}\sin\theta_{n-1} & \cos\theta_{n-2}\sin\theta_{n-1} & \cos\theta_{n-1} \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & & & \vdots \\ \vdots & \ddots & \ddots & \ddots & & & \vdots \\ 0 & \cdots & 0 & 1 & \ddots & & \vdots \\ 0 & \cdots & 0 & \cos\theta_{n-4}\tan\theta_{n-3} & 1 & \ddots & \vdots \\ 0 & \cdots & 0 & 0 & 0 & 1 & 0 \\ 0 & \cdots & 0 & 0 & 0 & 0 & 1 \end{pmatrix} \dots \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & 1 & \ddots & & & \vdots \\ 0 & \cos\theta_1\tan\theta_2 & 1 & \ddots & & \vdots \\ 0 & 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ \tan\theta_1 & 1 & \ddots & & \vdots \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & \frac{1}{r} & \ddots & & & \vdots \\ \vdots & \ddots & \frac{1}{r\sin\theta_1} & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-3}} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-2}}\end{pmatrix} \end{align}

これを繰り返し，第$2$列に第$3$列の$\cos\theta_1\tan\theta_2$倍を加えるまで進めます．

\begin{align} A^{-1} &= \begin{pmatrix} \frac{1}{\cos\theta_1} & -\sin\theta_1 & 0 & \cdots & \cdots & 0 \\ 0 & \cos\theta_1\cos\theta_2 & -\sin\theta_2 & \ddots & & \vdots \\ \vdots & \cos\theta_1\sin\theta_2\cos\theta_3 & \cos\theta_2\cos\theta_3 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & -\sin\theta_{n-2} & 0 \\ 0 & \cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \cos\theta_2\sin\theta_3\dots\sin\theta_{n-2}\cos\theta_{n-1} & \cdots & \cos\theta_{n-2}\cos\theta_{n-1} & -\sin\theta_{n-1} \\ 0 & \cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} & \cos\theta_2\sin\theta_3\dots\sin\theta_{n-2}\sin\theta_{n-1} & \cdots & \cos\theta_{n-2}\sin\theta_{n-1} & \cos\theta_{n-1} \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ \tan\theta_1 & 1 & \ddots & & \vdots \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & \frac{1}{r} & \ddots & & & \vdots \\ \vdots & \ddots & \frac{1}{r\sin\theta_1} & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-3}} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-2}}\end{pmatrix} \end{align}

第$1$列に第$2$列の$\tan\theta_1$倍を加えます．$(1,1)$成分は$\displaystyle\frac{1}{\cos\theta_1}+\tan\theta_1(-\sin\theta_1) = \frac{1}{\cos\theta_1} - \frac{\sin^2\theta_1}{\cos\theta_1} = \cos\theta_1$となります．

\begin{align} A^{-1} &= \begin{pmatrix} \cos\theta_1 & -\sin\theta_1 & 0 & \cdots & \cdots & 0 \\ \sin\theta_1\cos\theta_2 & \cos\theta_1\cos\theta_2 & -\sin\theta_2 & \ddots & & \vdots \\ \sin\theta_1\sin\theta_2\cos\theta_3 & \cos\theta_1\sin\theta_2\cos\theta_3 & \cos\theta_2\cos\theta_3 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & -\sin\theta_{n-2} & 0 \\ \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \cos\theta_2\sin\theta_3\dots\sin\theta_{n-2}\cos\theta_{n-1} & \cdots & \cos\theta_{n-2}\cos\theta_{n-1} & -\sin\theta_{n-1} \\ \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} & \cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} & \cos\theta_2\sin\theta_3\dots\sin\theta_{n-2}\sin\theta_{n-1} & \cdots & \cos\theta_{n-2}\sin\theta_{n-1} & \cos\theta_{n-1} \end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & \frac{1}{r} & \ddots & & & \vdots \\ \vdots & \ddots & \frac{1}{r\sin\theta_1} & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ \vdots & & & \ddots & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-3}} & 0 \\ 0 & \cdots & \cdots & \cdots & 0 & \frac{1}{r\sin\theta_1\dots\sin\theta_{n-2}}\end{pmatrix} \end{align}

第$1,2,3,\dots,n$列の各成分をそれぞれ$\displaystyle 1,\frac{1}{r},\frac{1}{r\sin\theta_1},\dots,\frac{1}{r\sin\theta_1\dots\sin\theta_{n-2}}$ 倍します．

\begin{align} A^{-1} &= \begin{pmatrix} \cos\theta_1 & -\frac{1}{r}\sin\theta_1 & 0 & \cdots & \cdots & 0 \\ \sin\theta_1\cos\theta_2 & \frac{1}{r}\cos\theta_1\cos\theta_2 & -\frac{1}{r}\frac{1}{\sin\theta_1}\sin\theta_2 & \ddots & & \vdots \\ \sin\theta_1\sin\theta_2\cos\theta_3 & \frac{1}{r}\cos\theta_1\sin\theta_2\cos\theta_3 & \frac{1}{r}\frac{1}{\sin\theta_1}\cos\theta_2\cos\theta_3 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & -\frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-3}}\sin\theta_{n-2} & 0 \\ \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \frac{1}{r}\cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \frac{1}{r}\frac{1}{\sin\theta_1}\cos\theta_2\sin\theta_3\dots\sin\theta_{n-2}\cos\theta_{n-1} & \cdots & \frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-3}}\cos\theta_{n-2}\cos\theta_{n-1} & -\frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-2}}\sin\theta_{n-1} \\ \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} & \frac{1}{r}\cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} & \frac{1}{r}\frac{1}{\sin\theta_1}\cos\theta_2\sin\theta_3\dots\sin\theta_{n-2}\sin\theta_{n-1} & \cdots & \frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-3}}\cos\theta_{n-2}\sin\theta_{n-1} & \frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-2}}\cos\theta_{n-1} \end{pmatrix} \end{align}

よって，以下が成り立ちます．
\begin{align} \begin{pmatrix} \pdv{x_1} \\ \pdv{x_2} \\ \vdots \\ \pdv{x_n} \end{pmatrix} = \begin{pmatrix} \cos\theta_1 & -\frac{1}{r}\sin\theta_1 & 0 & \cdots & \cdots & 0 \\ \sin\theta_1\cos\theta_2 & \frac{1}{r}\cos\theta_1\cos\theta_2 & -\frac{1}{r}\frac{1}{\sin\theta_1}\sin\theta_2 & \ddots & & \vdots \\ \sin\theta_1\sin\theta_2\cos\theta_3 & \frac{1}{r}\cos\theta_1\sin\theta_2\cos\theta_3 & \frac{1}{r}\frac{1}{\sin\theta_1}\cos\theta_2\cos\theta_3 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & -\frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-3}}\sin\theta_{n-2} & 0 \\ \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \frac{1}{r}\cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1} & \frac{1}{r}\frac{1}{\sin\theta_1}\cos\theta_2\sin\theta_3\dots\sin\theta_{n-2}\cos\theta_{n-1} & \cdots & \frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-3}}\cos\theta_{n-2}\cos\theta_{n-1} & -\frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-2}}\sin\theta_{n-1} \\ \sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} & \frac{1}{r}\cos\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1} & \frac{1}{r}\frac{1}{\sin\theta_1}\cos\theta_2\sin\theta_3\dots\sin\theta_{n-2}\sin\theta_{n-1} & \cdots & \frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-3}}\cos\theta_{n-2}\sin\theta_{n-1} & \frac{1}{r}\frac{1}{\sin\theta_1}\frac{1}{\sin\theta_2}\dots\frac{1}{\sin\theta_{n-2}}\cos\theta_{n-1} \end{pmatrix} \begin{pmatrix} \pdv{r} \\ \pdv{\theta_1} \\ \vdots \\ \pdv{\theta_{n-1}}\end{pmatrix} \end{align}
これをそれぞれの成分の式に書き直すと，以下のようになります．

$2$階微分$(x_1,x_2,\dots,x_{n-1})$

$1 \leq i \leq n-1$として，$\displaystyle\pdv[2]{x_i}$を計算します．
\begin{align} \pdv[2]{x_i} &= \left(\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\pdv{r}+\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\pdv{\theta_j}-\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i\pdv{\theta_i}\right)\left(\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\pdv{r}+\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\pdv{\theta_j}-\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i\pdv{\theta_i}\right) \\ \end{align}
左側の項を$\displaystyle\pdv{r}, \pdv{\theta_j}, \pdv{\theta_i}$に分けて展開します．$\displaystyle\pdv{\theta_j}$については，右側の項で添字がかぶらないように，$j,k$をそれぞれ$k,l$に置き換えます．
\begin{align} \pdv[2]{x_i} &= \prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\pdv{r}\left(\underbrace{\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\pdv{r}}_{(1)} + \underbrace{\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\pdv{\theta_j}}_{(2)} + \underbrace{\left(-\frac{1}{r}\right)\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i\pdv{\theta_i}}_{(3)}\right) \\ &\phantom{{}={}}+\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\pdv{\theta_j}\left(\underbrace{\prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i\pdv{r}}_{(4)} + \underbrace{\sum_{k=1}^{i-1}\left(\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i\right)\pdv{\theta_k}}_{(5)} - \underbrace{\frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i\pdv{\theta_i}}_{(6)}\right) \\ &\phantom{{}={}}-\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i\pdv{\theta_i}\left(\underbrace{\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\pdv{r}}_{(7)} + \underbrace{\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\pdv{\theta_j}}_{(8)} - \underbrace{\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i\pdv{\theta_i}}_{(9)}\right) \end{align}

積の微分に注意して，$(1)$から$(9)$をそれぞれ微分します．

• $(1)$: 係数に$r$が含まれないので，$1$階偏微分は現れず，$\displaystyle\pdv{r}\left(\pdv{r}\right)=\pdv[2]{r}$となります．
• $(2)$: 係数に含まれる$r$に依存する項は$\displaystyle\frac{1}{r}$なので，$\displaystyle\pdv{r}\left(\frac{1}{r}\pdv{\theta_j}\right)=\frac{1}{r}\pdv[2]{}{r}{\theta_j}-\frac{1}{r^2}\pdv{\theta_j}$となります．
• $(3)$: 係数に含まれる$r$に依存する項は$\displaystyle-\frac{1}{r}$なので，$\displaystyle\pdv{r}\left(-\frac{1}{r}\pdv{\theta_j}\right)=-\frac{1}{r}\pdv[2]{}{r}{\theta_j}+\frac{1}{r^2}\pdv{\theta_j}$となります．
• $(4)$: $1 \leq j \leq i-1$より，係数に含まれる$\theta_j$に依存する項は$\sin\theta_j$です．$\displaystyle\pdv{\theta_j}\left(\sin\theta_j\pdv{r}\right)=\sin\theta_j\pdv[2]{}{\theta_j}{r}+\cos\theta_j\pdv{r}=\sin\theta_j\left(\pdv[2]{}{\theta_j}{r}+\frac{\cos\theta_j}{\sin\theta_j}\pdv{r}\right)$なので，$\displaystyle\pdv{r}$を$\displaystyle\pdv[2]{}{\theta_j}{r}+\frac{\cos\theta_j}{\sin\theta_j}\pdv{r}$で置き換えます．
• $(5)$: 係数に含まれる$\theta_j$に依存する項の形は，$j$と$k$の大小関係により異なります．
  • $1 \leq k \leq j-1$のとき，$k+1 \leq j \leq i-1$なので，係数に含まれる$\theta_j$に依存する項は$\sin\theta_j$です．$\displaystyle\pdv{\theta_j}\left(\sin\theta_j\pdv{\theta_k}\right)=\sin\theta_j\pdv[2]{}{\theta_j}{\theta_k}+\cos\theta_j\pdv{\theta_k}=\sin\theta_j\left(\pdv[2]{}{\theta_j}{\theta_k}+\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_k}\right)$なので，$\displaystyle\pdv{\theta_k}$を$\displaystyle\pdv[2]{}{\theta_j}{\theta_k}+\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_k}$で置き換えます．
  • $k=j$のとき，係数に含まれる$\theta_j$に依存する項は$\cos\theta_j$なので，$\displaystyle\pdv{\theta_j}\left(\cos\theta_j\pdv{\theta_j}\right)=\cos\theta_j\pdv[2]{\theta_j}-\sin\theta_j\pdv{\theta_j}$となります．
  • $j+1 \leq k \leq i-1$のとき，$1 \leq j \leq k-1$なので，係数に含まれる$\theta_j$に依存する項は$\displaystyle\frac{1}{\sin\theta_j}$です．$\displaystyle\pdv{\theta_j}\left(\frac{1}{\sin\theta_j}\pdv{\theta_k}\right)=\frac{1}{\sin\theta_j}\pdv[2]{}{\theta_j}{\theta_k}-\frac{\cos\theta_j}{\sin^2\theta_j}\pdv{\theta_k}=\frac{1}{\sin\theta_j}\left(\pdv[2]{}{\theta_j}{\theta_k}-\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_k}\right)$なので，$\displaystyle\pdv{\theta_k}$を$\displaystyle\pdv[2]{}{\theta_j}{\theta_k}-\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_k}$で置き換えます．
• $(6)$: $1 \leq j \leq i-1$より，係数に含まれる$\theta_j$に依存する項は$\displaystyle\frac{1}{\sin\theta_j}$です．$\displaystyle\pdv{\theta_j}\left(\frac{1}{\sin\theta_j}\pdv{\theta_i}\right)=\frac{1}{\sin\theta_j}\pdv[2]{}{\theta_j}{\theta_i}-\frac{\cos\theta_j}{\sin^2\theta_j}\pdv{\theta_i}=\frac{1}{\sin\theta_j}\left(\pdv[2]{}{\theta_j}{\theta_i}-\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_i}\right)$なので，$\displaystyle\pdv{\theta_i}$を$\displaystyle\pdv[2]{}{\theta_j}{\theta_i}-\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_i}$で置き換えます．
• $(7)$: 係数に含まれる$\theta_i$に依存する項は$\cos\theta_i$なので，$\displaystyle\pdv{\theta_i}\left(\cos\theta_i\pdv{r}\right)=\cos\theta_i\pdv[2]{}{\theta_i}{r}-\sin\theta_i\pdv{r}$となります．
• $(8)$: 係数に含まれる$\theta_i$に依存する項は$\cos\theta_i$なので，$\displaystyle\pdv{\theta_i}\left(\cos\theta_i\pdv{\theta_j}\right)=\cos\theta_i\pdv[2]{}{\theta_i}{\theta_j}-\sin\theta_i\pdv{\theta_j}$となります．
• $(9)$: 係数に含まれる$\theta_i$に依存する項は$\sin\theta_i$なので，$\displaystyle\pdv{\theta_i}\left(\sin\theta_i\pdv{\theta_i}\right)=\sin\theta_i\pdv[2]{\theta_i}+\cos\theta_i\pdv{\theta_i}$となります．

\begin{align} \pdv[2]{x_i} &= \prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\left(\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\pdv[2]{r}+\sum_{j=1}^{i-1}\left(\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\left(\frac{1}{r}\pdv[2]{}{r}{\theta_j}-\frac{1}{r^2}\pdv{\theta_j}\right)+\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i\left(-\frac{1}{r}\pdv[2]{}{r}{\theta_i}+\frac{1}{r^2}\pdv{\theta_i}\right)\right) \\ &\phantom{{}={}} +\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\left(\prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i\left(\pdv[2]{}{\theta_j}{r}+\frac{\cos\theta_j}{\sin\theta_j}\pdv{r}\right)+\sum_{k=1}^{j-1}\left(\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i\right)\left(\pdv[2]{}{\theta_j}{\theta_k}+\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_k}\right)+\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\prod_{k=j+1}^{i-1}\sin\theta_l\cos\theta_i\left(\cos\theta_j\pdv[2]{\theta_j}-\sin\theta_j\pdv{\theta_j}\right)+\sum_{k=j+1}^{i-1}\left(\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i\right)\left(\pdv[2]{}{\theta_j}{\theta_k}-\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_k}\right)-\frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i\left(\pdv[2]{}{\theta_j}{\theta_i}-\frac{\cos\theta_j}{\sin\theta_j}\pdv{\theta_i}\right)\right) \\ &\phantom{{}={}} -\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i\left(\prod_{j=1}^{i-1}\sin\theta_j\left(\cos\theta_i\pdv[2]{}{\theta_i}{r}-\sin\theta_i\pdv{r}\right)+\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\right)\left(\cos\theta_i\pdv[2]{}{\theta_i}{\theta_j}-\sin\theta_i\pdv{\theta_j}\right)-\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\left(\sin\theta_i\pdv[2]{\theta_i}+\cos\theta_i\pdv{\theta_i}\right)\right) \end{align}

偏微分の種類ごとにまとめます．異なる変数に関する偏微分は交換してもよいとします．
\begin{align} \pdv[2]{x_i} &= \left(\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\cdot\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\right)\pdv[2]{r} \\ &\phantom{{}={}} +\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\pdv[2]{\theta_j} \\ &\phantom{{}={}} +\left(-\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \left(-\frac{1}{r}\right)\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i\right)\pdv[2]{\theta_i} \\ &\phantom{{}={}} +\sum_{j=1}^{i-1}\left(\prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i \cdot \prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{1}{r} + \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i\right)\pdv[2]{}{r}{\theta_j} \\ &\phantom{{}={}} + \left(\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \left(-\frac{1}{r}\right) - \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i\right)\pdv[2]{}{r}{\theta_i} \\ &\phantom{{}={}} +\sum_{j=1}^{i-1}\sum_{k=j+1}^{i-1}\left(\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\cos\theta_j\prod_{l=j+1}^{i-1}\sin\theta_l\cos\theta_i + \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\cos\theta_j\prod_{l=j+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i\right)\pdv[2]{}{\theta_j}{\theta_k} \\ &\phantom{{}={}} +\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \left(-\frac{1}{r}\right)\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i - \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i \cdot \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i\right)\pdv[2]{}{\theta_j}{\theta_i} \\ &\phantom{{}={}} +\left(\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{\cos\theta_j}{\sin\theta_j}\right)-\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \prod_{j=1}^{i-1}\sin\theta_j \cdot (-\sin\theta_i)\right)\pdv{r} \\ &\phantom{{}={}}+\sum_{j=1}^{i-1}\Bigg(\prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i \cdot \prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \left(-\frac{1}{r^2}\right) + \sum_{k=j+1}^{i-1}\left(\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\cos\theta_j\prod_{l=j+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \frac{\cos\theta_k}{\sin\theta_k}\right) + \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot (-\sin\theta_j) \\ &\phantom{{}={}+\sum_{j=1}^{i-1}\Bigg(}+\sum_{k=1}^{i-1}\left(\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\cos\theta_j\prod_{l=j+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \left(-\frac{\cos\theta_k}{\sin\theta_k}\right)\right) - \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i \cdot \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k \cdot (-\sin\theta_i)\Biggr)\pdv{\theta_j} \\ &\phantom{{}={}}+\left(\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \frac{1}{r^2} + \sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \left(-\frac{1}{r}\right)\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i \cdot \left(-\frac{\cos\theta_j}{\sin\theta_j}\right)\right) - \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \left(-\frac{1}{r}\right)\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j} \cdot \cos\theta_i\right)\pdv{\theta_i} \end{align}

偏微分の種類ごとに，係数を計算します．

$\displaystyle\pdv[2]{r}$

$\displaystyle\pdv[2]{r}$の係数は$\displaystyle\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i$となります．

$\displaystyle\pdv[2]{\theta_j} \ (1 \leq j \leq i-1)$

$\displaystyle\pdv[2]{\theta_j}$の係数は$\displaystyle\frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\cos^2\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i$となります．

$\displaystyle\pdv[2]{\theta_i}$

$\displaystyle\pdv[2]{\theta_i}$の係数は$\displaystyle\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin^2\theta_i$となります．

$\displaystyle\pdv[2]{}{r}{\theta_j} \ (1 \leq j \leq i-1)$

$\displaystyle\pdv[2]{}{r}{\theta_j}$の係数は$2$つの同じ項の和となっており，そのうち$1$つは
\begin{align} &\phantom{{}={}}\prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i \cdot \prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{1}{r} \\ &= \prod_{k=1}^{j-1}\sin\theta_k \sin\theta_j \prod_{k=j+1}^{i-1}\sin\theta_k \cos\theta_i \cdot \prod_{k=1}^{j-1}\frac{1}{\sin\theta_k} \cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{1}{r} \\ &= \frac{1}{r}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i \end{align}
なので，$\displaystyle\pdv[2]{}{r}{\theta_j}$の係数は$\displaystyle\frac{2}{r}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i$です．

$\displaystyle\pdv[2]{}{r}{\theta_i}$

$\displaystyle\pdv[2]{}{r}{\theta_i}$の係数は$2$つの同じ項の和となっており，そのうち$1$つは$\displaystyle\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \left(-\frac{1}{r}\right)=-\frac{1}{r}\sin\theta_i\cos\theta_i$なので，$\displaystyle\pdv[2]{}{r}{\theta_i}$の係数は$\displaystyle-\frac{2}{r}\sin\theta_i\cos\theta_i$です．

$\displaystyle\pdv[2]{}{\theta_j}{\theta_k} \ (1 \leq j < k \leq i-1)$

$\displaystyle\pdv[2]{}{\theta_j}{\theta_k}$の係数は$2$つの同じ項の和となっており，そのうち$1$つは
\begin{align} &\phantom{{}={}}\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\cos\theta_j\prod_{l=j+1}^{i-1}\sin\theta_l\cos\theta_i \\ &= \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\frac{1}{\sin\theta_j}\prod_{l=j+1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\cos\theta_j\prod_{l=j+1}^{k-1}\sin\theta_l\sin\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i \\ &= \frac{1}{r^2}\prod_{l=1}^{j-1}\frac{1}{\sin^2\theta_l}\frac{\cos\theta_j}{\sin\theta_j}\sin\theta_k\cos\theta_k\prod_{l=k+1}^{i-1}\sin^2\theta_l\cos^2\theta_l \end{align}
なので，$\displaystyle\pdv[2]{}{\theta_j}{\theta_k}$の係数は$\displaystyle\frac{2}{r^2}\prod_{l=1}^{j-1}\frac{1}{\sin^2\theta_l}\frac{\cos\theta_j}{\sin\theta_j}\sin\theta_k\cos\theta_k\prod_{l=k+1}^{i-1}\sin^2\theta_l\cos^2\theta_i$です．

$\displaystyle\pdv[2]{}{\theta_j}{\theta_i} \ (1 \leq j \leq i-1)$

$\displaystyle\pdv[2]{}{\theta_j}{\theta_i}$の係数は$2$つの同じ項の和となっており，そのうち$1$つは
\begin{align} &\phantom{{}={}}\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \left(-\frac{1}{r}\right)\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i \\ &= \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \left(-\frac{1}{r}\right)\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\frac{1}{\sin\theta_j}\prod_{k=j+1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i \\ &= -\frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sin\theta_i\cos\theta_i \end{align}
なので，$\displaystyle\pdv[2]{}{\theta_j}{\theta_i}$の係数は$\displaystyle-\frac{2}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sin\theta_i\cos\theta_i$です．

$\displaystyle\pdv{r}$

$\displaystyle\pdv{r}$の係数は
\begin{align} &\phantom{{}={}}\sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{\cos\theta_j}{\sin\theta_j}\right)-\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \prod_{j=1}^{i-1}\sin\theta_j \cdot (-\sin\theta_i) \\ &= \sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \prod_{k=1}^{j-1}\sin\theta_k\sin\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{\cos\theta_j}{\sin\theta_j}\right)+ \frac{1}{r}\sin^2\theta_i \\ &= \sum_{j=1}^{i-1}\left(\frac{1}{r}\cos^2\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i\right)+\frac{1}{r}\sin^2\theta_i \\ &= \frac{1}{r}\sum_{j=1}^{i-1}\left((1-\sin^2\theta_j)\prod_{k=j+1}^{i-1}\sin^2\theta_k\right)\cos^2\theta_i+\frac{1}{r}\sin^2\theta_i \\ &= \frac{1}{r}\sum_{j=1}^{i-1}\left(\prod_{k=j+1}^{i-1}\sin^2\theta_k-\prod_{k=j}^{i-1}\sin^2\theta_k\right)\cos^2\theta_i+\frac{1}{r}\sin^2\theta_i \\ &= \frac{1}{r}\left(1-\prod_{j=1}^{i-1}\sin^2\theta_j\right)\cos^2\theta_i+\frac{1}{r}\sin^2\theta_i \\ &= \frac{1}{r}-\frac{1}{r}\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i \end{align}

$\displaystyle\pdv{\theta_j} \ (1 \leq j \leq i-1)$

$\displaystyle\pdv{\theta_j}$の係数は
\begin{align} &\phantom{{}={}}\prod_{k=1}^{i-1}\sin\theta_k\cos\theta_i \cdot \prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \left(-\frac{1}{r^2}\right) + \sum_{k=j+1}^{i-1}\left(\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=j+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \frac{\cos\theta_k}{\sin\theta_k}\right) + \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot (-\sin\theta_j)+\sum_{k=1}^{i-1}\left(\frac{1}{r}\prod_{l=1}^{k-1}\frac{1}{\sin\theta_l}\cos\theta_k\prod_{l=k+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \frac{1}{r}\prod_{l=1}^{j-1}\frac{1}{\sin\theta_l}\cos\theta_j\prod_{l=j+1}^{i-1}\sin\theta_l\cos\theta_i \cdot \left(-\frac{\cos\theta_k}{\sin\theta_k}\right)\right) - \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i \cdot \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k \cdot (-\sin\theta_i) \\ &= -\frac{1}{r^2}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \sum_{k=j+1}^{i-1}\left(\frac{1}{r^2}\prod_{l=1}^{j-1}\frac{1}{\sin^2\theta_l}\frac{\cos\theta_j}{\sin\theta_j}\cos^2\theta_k\prod_{l=k+1}^{i-1}\sin^2\theta_l\cos^2\theta_i\right) - \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \sum_{k=1}^{j-1}\left(-\frac{1}{r^2}\prod_{l=1}^{k-1}\frac{1}{\sin^2\theta_l}\frac{\cos^2\theta_k}{\sin^2\theta_k}\sin\theta_j\cos\theta_j\prod_{l=j-1}^{i-1}\sin^2\theta_l\cos^2\theta_i\right) + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sin^2\theta_i \\ &= -\frac{1}{r^2}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sum_{k=j+1}^{i-1}\left(\cos^2\theta_k\prod_{l=k+1}^{i-1}\sin^2\theta_l\right)\cos^2\theta_i - \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i -\frac{1}{r^2}\sum_{k=1}^{j-1}\left(\prod_{l=1}^{k-1}\frac{1}{\sin^2\theta_l}\frac{\cos^2\theta_k}{\sin^2\theta_k}\right)\sin\theta_j\cos\theta_j\prod_{k=j-1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sin^2\theta_i \\ &= -\frac{1}{r^2}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sum_{k=j+1}^{i-1}\left((1-\sin^2\theta_k)\prod_{l=k+1}^{i-1}\sin^2\theta_l\right)\cos^2\theta_i - \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i -\frac{1}{r^2}\sum_{k=1}^{j-1}\left(\prod_{l=1}^{k-1}\frac{1}{\sin^2\theta_l}\left(\frac{1}{\sin^2\theta_k}-1\right)\right)\sin\theta_j\cos\theta_j\prod_{k=j-1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sin^2\theta_i \\ &= -\frac{1}{r^2}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sum_{k=j+1}^{i-1}\left(\prod_{l=k+1}^{i-1}\sin^2\theta_l - \prod_{l=k}^{i-1}\sin^2\theta_l\right)\cos^2\theta_i - \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i -\frac{1}{r^2}\sum_{k=1}^{j-1}\left(\prod_{l=1}^k\frac{1}{\sin^2\theta_l}-\prod_{l=1}^{k-1}\frac{1}{\sin^2\theta_l}\right)\sin\theta_j\cos\theta_j\prod_{k=j-1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sin^2\theta_i \\ &= -\frac{1}{r^2}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\left(1-\prod_{k=j+1}^{i-1}\sin^2\theta_k\right)\cos^2\theta_i - \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i -\frac{1}{r^2}\left(\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}-1\right)\sin\theta_j\cos\theta_j\prod_{k=j-1}^{i-1}\sin^2\theta_k\cos^2\theta_i + \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sin^2\theta_i \\ &= \left(\frac{1}{r^2}\frac{\cos\theta_j}{\sin\theta_j}\cos^2\theta_i + \frac{1}{r^2}\frac{\cos\theta_j}{\sin\theta_j}\sin^2\theta_i\right)\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k} + \left(-\frac{1}{r^2}\sin\theta_j\cos\theta_j + \frac{1}{r^2}\sin\theta_j\cos\theta_j\right)\prod_{k=j-1}^{i-1}\sin^2\theta_k + \left(-\frac{1}{r^2}\frac{\cos\theta_j}{\sin\theta_j}\cos^2\theta_i - \frac{1}{r^2}\sin\theta_j\cos\theta_j\cos^2\theta_i - \frac{1}{r^2}\sin\theta_j\cos\theta_j\cos^2\theta_i\right)\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\prod_{k=j+1}^{i-1}\sin^2\theta_k \\ &= \frac{1}{r^2}\frac{\cos\theta_j}{\sin\theta_j}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k} -\frac{1}{r^2}\left(\frac{\cos\theta_j}{\sin\theta_j} + 2\sin\theta_j\cos\theta_j\right)\cos^2\theta_i\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\prod_{k=j+1}^{i-1}\sin^2\theta_k \\ &= \frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}-\frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\left(\frac{\cos\theta_j}{\sin\theta_j}+2\sin\theta_j\cos\theta_j\right)\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i \end{align}

$\displaystyle\pdv{\theta_i}$

$\displaystyle\pdv{\theta_i}$の係数は
\begin{align} &\phantom{{}={}}\prod_{j=1}^{i-1}\sin\theta_j\cos\theta_i \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \frac{1}{r^2} + \sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{i-1}\sin\theta_k\cos\theta_i \cdot \left(-\frac{1}{r}\right)\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\sin\theta_i \cdot \left(-\frac{\cos\theta_j}{\sin\theta_j}\right)\right) - \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\sin\theta_i \cdot \left(-\frac{1}{r}\right)\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j} \cdot \cos\theta_i \\ &= \frac{1}{r^2}\sin\theta_i\cos\theta_i + \sum_{j=1}^{i-1}\left(\frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos^2\theta_j}{\sin^2\theta_j}\sin\theta_i\cos\theta_i\right)+\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i \\ &= \frac{1}{r^2}\sin\theta_i\cos\theta_i + \frac{1}{r^2}\sum_{j=1}^{i-1}\left(\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos^2\theta_j}{\sin^2\theta_j}\right)\sin\theta_i\cos\theta_i + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i \\ &= \frac{1}{r^2}\sin\theta_i\cos\theta_i + \frac{1}{r^2}\sum_{j=1}^{i-1}\left(\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\left(\frac{1}{\sin^2\theta_j}-1\right)\right)\sin\theta_i\cos\theta_i + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i \\ &= \frac{1}{r^2}\sin\theta_i\cos\theta_i + \frac{1}{r^2}\sum_{j=1}^{i-1}\left(\prod_{k=1}^j\frac{1}{\sin^2\theta_k}-\prod_{k-1}^{j-1}\frac{1}{\sin^2\theta_k}\right)\sin\theta_i\cos\theta_i + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i \\ &= \frac{1}{r^2}\sin\theta_i\cos\theta_i + \frac{1}{r^2}\left(\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j} - 1\right)\sin\theta_i\cos\theta_i + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i \\ &= \frac{2}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i \end{align}

よって，以下の式を得ます．

$2$階微分$(x_n)$

$\displaystyle\pdv[2]{x_n}$を計算します．
\begin{align} \pdv[2]{x_n} &= \left(\prod_{i=1}^{n-1}\sin\theta_i\pdv{r} + \sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j\right)\pdv{\theta_i} \right)\left(\prod_{i=1}^{n-1}\sin\theta_i\pdv{r} + \sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j\right)\pdv{\theta_i} \right) \\ \end{align}
左側の項を$\displaystyle\pdv{r}, \pdv{\theta_i}$に分けて展開します．$\displaystyle\pdv{\theta_i}$については，右側の項で添字がかぶらないように，$i,j$をそれぞれ$j,k$に置き換えます．
\begin{align} \pdv[2]{x_n} &= \prod_{i=1}^{n-1}\sin\theta_i\pdv{r}\left(\prod_{i=1}^{n-1}\sin\theta_i\pdv{r} + \sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j\right)\pdv{\theta_i}\right) \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j\right)\pdv{\theta_i}\left(\prod_{j=1}^{n-1}\sin\theta_j\pdv{r} + \sum_{j=1}^{n-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k\right)\pdv{\theta_j}\right) \\ &= \prod_{i=1}^{n-1}\sin\theta_i\left(\prod_{i=1}^{n-1}\sin\theta_i\pdv[2]{r} + \sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j\right)\left(\frac{1}{r}\pdv[2]{}{r}{\theta_i}-\frac{1}{r^2}\pdv{\theta_i}\right)\right) \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j\right)\left(\prod_{j=1}^{n-1}\sin\theta_j\left(\pdv[2]{}{\theta_i}{r}+\frac{\cos\theta_i}{\sin\theta_i}\pdv{r}\right) + \sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k\right)\left(\pdv[2]{}{\theta_i}{\theta_j}+\frac{\cos\theta_i}{\sin\theta_i}\pdv{\theta_j}\right) + \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\prod_{j=i+1}^{n-1}\sin\theta_j\left(\cos\theta_i\pdv[2]{\theta_i} - \sin\theta_i\pdv{\theta_i}\right) + \sum_{j=i+1}^{n-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k\right)\left(\pdv[2]{}{\theta_i}{\theta_j} - \frac{\cos\theta_i}{\sin\theta_i}\pdv{\theta_j}\right)\right) \\ &= \left(\prod_{i=1}^{n-1}\sin\theta_i \cdot \prod_{i=1}^{n-1}\sin\theta_i\right)\pdv[2]{r} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \cos\theta_i\right)\pdv[2]{\theta_i} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\prod_{j=1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \frac{1}{r} + \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{n-1}\sin\theta_j\right) \pdv[2]{}{r}{\theta_i} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\cos\theta_i\prod_{k=i+1}^{n-1}\sin\theta_k + \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\cos\theta_i\prod_{k=i+1}^{n-1}\sin\theta_k \cdot \frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k\right)\pdv[2]{}{\theta_i}{\theta_j} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{n-1}\sin\theta_j \cdot \frac{\cos\theta_i}{\sin\theta_i}\right)\pdv{r} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\prod_{j=1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \left(-\frac{1}{r^2}\right) + \sum_{j=i+1}^{n-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\cos\theta_i\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{\cos\theta_j}{\sin\theta_j}\right) + \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\prod_{j=i+1}^{n-1}\sin\theta_j \cdot (-\sin\theta_i) + \sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\cos\theta_i\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \left(-\frac{\cos\theta_j}{\sin\theta_j}\right)\right)\right)\pdv{\theta_i} \end{align}

偏微分の種類ごとに，係数を計算します．

$\displaystyle\pdv[2]{r}$

$\displaystyle\pdv[2]{r}$の係数は$\displaystyle\prod_{i=1}^{n-1}\sin^2\theta_i$となります．

$\displaystyle\pdv[2]{\theta_i} \ (1 \leq i \leq n-1)$

$\displaystyle\pdv[2]{\theta_j}$の係数は$\displaystyle\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j$となります．

$\displaystyle\pdv[2]{}{r}{\theta_i} \ (1 \leq i \leq n-1)$

$\displaystyle\pdv[2]{}{r}{\theta_i}$の係数は$2$つの同じ項の和となっており，そのうち$1$つは
\begin{align} &\phantom{{}={}}\prod_{j=1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \frac{1}{r} \\ &= \prod_{j=1}^{i-1}\sin\theta_j\sin\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \frac{1}{r} \\ &= \frac{1}{r}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{i-1}\sin^2\theta_j \end{align}
なので，$\displaystyle\pdv[2]{}{r}{\theta_i}$の係数は$\displaystyle\frac{2}{r}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{i-1}\sin^2\theta_j$です．

$\displaystyle\pdv[2]{}{\theta_i}{\theta_j} \ (1 \leq i < j \leq n-1)$

$\displaystyle\pdv[2]{}{\theta_i}{\theta_j}$の係数は$2$つの同じ項の和となっており，そのうち$1$つは
\begin{align} &\phantom{{}={}}\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\cos\theta_i\prod_{k=i+1}^{n-1}\sin\theta_k \\ &= \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\frac{1}{\sin\theta_i}\prod_{k=i+1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\cos\theta_i\prod_{k=i+1}^{j-1}\sin\theta_k\sin\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k \\ &= \frac{1}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\cos\theta_j\sin\theta_j\prod_{k=j+1}^{n-1}\sin^2\theta_k \end{align}
なので，$\displaystyle\pdv[2]{}{\theta_j}{\theta_k}$の係数は$\displaystyle\frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\cos\theta_j\sin\theta_j\prod_{k=j+1}^{n-1}\sin^2\theta_k$です．

$\displaystyle\pdv{r}$

$\displaystyle\pdv{r}$の係数は
\begin{align} &\phantom{{}={}}\sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{n-1}\sin\theta_j \cdot \frac{\cos\theta_i}{\sin\theta_i}\right) \\ &= \sum_{i=1}^{n-1}\left(\frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{i-1}\sin\theta_j\sin\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \frac{\cos\theta_i}{\sin\theta_i}\right) \\ &= \sum_{i=1}^{n-1}\left(\frac{1}{r}\cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &= \frac{1}{r}\sum_{i=1}^{n-1}\left((1-\cos^2\theta_i)\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &= \frac{1}{r}\sum_{i=1}^{n-1}\left(\prod_{j=i+1}^{n-1}\sin^2\theta_j - \prod_{j=i}^{n-1}\sin^2\theta_j\right) \\ &= \frac{1}{r}\left(1-\prod_{i=1}^{n-1}\sin^2\theta_i\right) \\ &= \frac{1}{r} - \frac{1}{r}\prod_{i=1}^{n-1}\sin^2\theta_i \end{align}

$\displaystyle\pdv{\theta_i} \ (1 \leq i \leq n-1)$

$\displaystyle\pdv{\theta_i}$の係数は
\begin{align} &\phantom{{}={}}\prod_{j=1}^{n-1}\sin\theta_j \cdot \prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \left(-\frac{1}{r^2}\right) + \sum_{j=i+1}^{n-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\cos\theta_i\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{\cos\theta_j}{\sin\theta_j}\right) + \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j \cdot \frac{1}{r}\prod_{j=1}^{i-1}\frac{1}{\sin\theta_j}\prod_{j=i+1}^{n-1}\sin\theta_j \cdot (-\sin\theta_i) + \sum_{j=1}^{i-1}\left(\frac{1}{r}\prod_{k=1}^{j-1}\frac{1}{\sin\theta_k}\cos\theta_j\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \frac{1}{r}\prod_{k=1}^{i-1}\frac{1}{\sin\theta_k}\cos\theta_i\prod_{k=j+1}^{n-1}\sin\theta_k \cdot \left(-\frac{\cos\theta_j}{\sin\theta_j}\right)\right) \\ &= -\frac{1}{r^2}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j + \sum_{j=i+1}^{n-1}\left(\frac{1}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\cos^2\theta_j\prod_{k=j+1}^{n-1}\sin^2\theta_k\right) - \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j + \sum_{j=1}^{i-1}\left(-\frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos^2\theta_j}{\sin^2\theta_j}\sin\theta_i\cos\theta_i\prod_{k=i+1}^{n-1}\sin^2\theta_k\right) \\ &= -\frac{1}{r^2}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\frac{\cos\theta_i}{\sin\theta_i}\sum_{j=i+1}^{n-1}\left(\cos^2\theta_j\prod_{k=j+1}^{n-1}\sin^2\theta_k\right) - \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j -\frac{1}{r^2} \sum_{j=1}^{i-1}\left(\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos^2\theta_j}{\sin^2\theta_j}\right)\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &= -\frac{1}{r^2}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\frac{\cos\theta_i}{\sin\theta_i}\sum_{j=i+1}^{n-1}\left((1-\sin^2\theta_j)\prod_{k=j+1}^{n-1}\sin^2\theta_k\right) - \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j - \frac{1}{r^2}\sum_{j=1}^{i-1}\left(\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\left(\frac{1}{\sin^2\theta_j}-1\right)\right)\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &= -\frac{1}{r^2}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\frac{\cos\theta_i}{\sin\theta_i}\sum_{j=i+1}^{n-1}\left(\prod_{k=j+1}^{n-1}\sin^2\theta_k-\prod_{k=j}^{n-1}\sin^2\theta_k\right) - \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j - \frac{1}{r^2}\sum_{j=1}^{i-1}\left(\prod_{k=1}^j\frac{1}{\sin^2\theta_k}-\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\right)\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &= -\frac{1}{r^2}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin\theta_j + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\frac{\cos\theta_i}{\sin\theta_i}\left(1-\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) - \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j - \frac{1}{r^2}\left(\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}-1\right)\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &= \left(\frac{1}{r^2}\frac{\cos\theta_i}{\sin\theta_i}\right)\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j} + \left(-\frac{1}{r^2}\sin\theta_i\cos\theta_i + \frac{1}{r^2}\sin\theta_i\cos\theta_i\right)\prod_{j=i+1}^{n-1}\sin^2\theta_j + \left(-\frac{1}{r^2}\frac{\cos\theta_i}{\sin\theta_i}-\frac{1}{r^2}\sin\theta_i\cos\theta_i-\frac{1}{r^2}\sin\theta_i\cos\theta_i\right)\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &= \frac{1}{r^2}\frac{\cos\theta_i}{\sin\theta_i}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}-\frac{1}{r^2}\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\frac{\cos\theta_i}{\sin\theta_i}-\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{j=i+1}^{n-1}\sin^2\theta_j \end{align}

よって，以下の式を得ます．

ラプラシアン

\begin{align} \Delta &= \sum_{i=1}^{n-1}\pdv[2]{x_i} + \pdv[2]{x_n} \\ &= \sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i\pdv[2]{r} + \sum_{j=1}^{i-1}\left(\frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\cos^2\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i\right)\pdv[2]{\theta_j} + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin^2\theta_i\pdv[2]{\theta_i} + \sum_{j=1}^{i-1}\left(\frac{2}{r}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i\right)\pdv[2]{}{r}{\theta_j} - \frac{2}{r}\sin\theta_i\cos\theta_i\pdv[2]{}{r}{\theta_i} + \sum_{j=1}^{i-1}\sum_{k=j+1}^{i-1}\left(\frac{2}{r^2}\prod_{l=1}^{j-1}\frac{1}{\sin^2\theta_l}\frac{\cos\theta_j}{\sin\theta_j}\sin\theta_k\cos\theta_k\prod_{l=k+1}^{i-1}\sin^2\theta_l\cos^2\theta_i\right)\pdv[2]{}{\theta_j}{\theta_k} + \sum_{j=1}^{i-1}\left(-\frac{2}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}\sin\theta_i\cos\theta_i\right)\pdv[2]{}{\theta_j}{\theta_i} + \left(\frac{1}{r}-\frac{1}{r}\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i\right)\pdv{r} + \sum_{j=1}^{i-1}\left(\frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_j}{\sin\theta_j}-\frac{1}{r^2}\prod_{k=1}^{j-1}\frac{1}{\sin^2\theta_k}\left(\frac{\cos\theta_j}{\sin\theta_j}+2\sin\theta_j\cos\theta_j\right)\prod_{k=j+1}^{i-1}\sin^2\theta_k\cos^2\theta_i\right)\pdv{\theta_j}+\frac{2}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i\pdv{\theta_i}\right) \\ &\phantom{{}={}}+\left(\prod_{i=1}^{n-1}\sin^2\theta_i\pdv[2]{r} + \sum_{i=1}^{n-1}\left(\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right)\pdv[2]{\theta_i} + \sum_{i=1}^{n-1}\left(\frac{2}{r}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right)\pdv[2]{}{r}{\theta_i} + \sum_{i=1}^{n-1}\sum_{j=i+1}^{n-1}\left(\frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\cos\theta_j\sin\theta_j\prod_{k=j+1}^{n-1}\sin^2\theta_k\right)\pdv[2]{}{\theta_i}{\theta_j} + \left(\frac{1}{r}-\frac{1}{r}\prod_{i=1}^{n-1}\sin^2\theta_i\right)\pdv{r} + \sum_{i=1}^{n-1}\left(\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\frac{\cos\theta_i}{\sin\theta_i} - \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{j=i+1}^{n-1}\sin^2\theta_j\right)\pdv{\theta_i}\right) \\ &= \left(\sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i\right)+\prod_{i=1}^{n-1}\sin^2\theta_i\right)\pdv[2]{r} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\sum_{j=i+1}^{n-1}\left(\frac{1}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\cos^2\theta_i\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)+\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin^2\theta_i + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right)\pdv[2]{\theta_i} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\sum_{j=i+1}^{n-1}\left(\frac{2}{r}\sin\theta_i\cos\theta_i\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)-\frac{2}{r}\sin\theta_i\cos\theta_i+\frac{2}{r}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right)\pdv[2]{}{r}{\theta_i} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n-1}\left(\sum_{k=j+1}^{n-1}\left(\frac{2}{r^2}\prod_{l=1}^{i-1}\frac{1}{\sin^2\theta_l}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j\prod_{l=j+1}^{k-1}\sin^2\theta_l\cos^2\theta_k\right)-\frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j+\frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{n-1}\sin^2\theta_k\right)\pdv[2]{}{\theta_i}{\theta_j} \\ &\phantom{{}={}}+\left(\sum_{i=1}^{n-1}\left(\frac{1}{r}-\frac{1}{r}\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i\right)+\left(\frac{1}{r}-\frac{1}{r}\prod_{i=1}^{n-1}\sin^2\theta_i\right)\right)\pdv{r} \\ &\phantom{{}={}}+\sum_{i=1}^{n-1}\left(\sum_{j=i+1}^{n-1}\left(\frac{1}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}-\frac{1}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)+\frac{2}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i+\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\frac{\cos\theta_i}{\sin\theta_i}-\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{j=i+1}^{n-1}\sin^2\theta_j\right)\pdv{\theta_i} \end{align}

偏微分の種類ごとに，係数を計算します．

$\displaystyle\pdv[2]{r}$

$\displaystyle\pdv[2]{r}$の係数は
\begin{align} &\phantom{{}={}}\sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i\right)+\prod_{i=1}^{n-1}\sin^2\theta_i \\ &= \sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\sin^2\theta_j(1-\sin^2\theta_i)\right)+\prod_{i=1}^{n-1}\sin^2\theta_i \\ &= \sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\sin^2\theta_j - \prod_{j=1}^i\sin^2\theta_j\right) + \prod_{i=1}^{n-1}\sin^2\theta_i \\ &= \left(1-\prod_{i=1}^{n-1}\sin^2\theta_i\right) + \prod_{i=1}^{n-1}\sin^2\theta_i \\ &= 1 \end{align}

$\displaystyle\pdv[2]{\theta_i} \ (1 \leq i \leq n-1)$

$\displaystyle\pdv[2]{\theta_i}$の係数は
\begin{align} &\phantom{{}={}}\sum_{j=i+1}^{n-1}\left(\frac{1}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\cos^2\theta_i\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)+\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin^2\theta_i + \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\sum_{j=i+1}^{n-1}\left(\cos^2\theta_i\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right) + \sin^2\theta_i + \cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\cos^2\theta_i\sum_{j=i+1}^{n-1}\left(\prod_{k=i+1}^{j-1}\sin^2\theta_k(1-\sin^2\theta_j)\right) + \sin^2\theta_i + \cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\cos^2\theta_i\sum_{j=i+1}^{n-1}\left(\prod_{k=i+1}^{j-1}\sin^2\theta_k-\prod_{k=i+1}^j\sin^2\theta_k\right) + \sin^2\theta_i + \cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\cos^2\theta_i\left(1-\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) + \sin^2\theta_i + \cos^2\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\cos^2\theta_i + \sin^2\theta_i\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j} \end{align}

$\displaystyle\pdv[2]{}{r}{\theta_i} \ (1 \leq i \leq n-1)$

$\displaystyle\pdv[2]{}{r}{\theta_i}$の係数は
\begin{align} &\phantom{{}={}}\sum_{j=i+1}^{n-1}\left(\frac{2}{r}\sin\theta_i\cos\theta_i\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)-\frac{2}{r}\sin\theta_i\cos\theta_i+\frac{2}{r}\sin\theta_i\cos\theta_i\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &=\frac{2}{r}\sin\theta_i\cos\theta_i\left(\sum_{j=i+1}^{n-1}\left(\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)-1+\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &=\frac{2}{r}\sin\theta_i\cos\theta_i\left(\sum_{j=i+1}^{n-1}\left(\prod_{k=i+1}^{j-1}\sin^2\theta_k(1-\sin^2\theta_j)\right)-1+\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &=\frac{2}{r}\sin\theta_i\cos\theta_i\left(\sum_{j=i+1}^{n-1}\left(\prod_{k=i+1}^{j-1}\sin^2\theta_k-\prod_{k=i+1}^j\sin^2\theta_k\right)-1+\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &=\frac{2}{r}\sin\theta_i\cos\theta_i\left(1-\prod_{j=i+1}^{n-1}\sin^2\theta_j-1+\prod_{j=i+1}^{n-1}\sin^2\theta_j\right) \\ &= 0 \end{align}

$\displaystyle\pdv[2]{}{\theta_i}{\theta_j} \ (1 \leq i < j \leq n-1)$

$\displaystyle\pdv[2]{}{\theta_i}{\theta_j}$の係数は
\begin{align} &\phantom{{}={}}\sum_{k=j+1}^{n-1}\left(\frac{2}{r^2}\prod_{l=1}^{i-1}\frac{1}{\sin^2\theta_l}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j\prod_{l=j+1}^{k-1}\sin^2\theta_l\cos^2\theta_k\right)-\frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j+\frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j\prod_{k=j+1}^{n-1}\sin^2\theta_k \\ &= \frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j\left(\sum_{k=j+1}^{n-1}\left(\prod_{l=j+1}^{k-1}\sin^2\theta_l\cos^2\theta_k\right)-1+\prod_{k=j+1}^{n-1}\sin^2\theta_k\right) \\ &= \frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j\left(\sum_{k=j+1}^{n-1}\left(\prod_{l=j+1}^{k-1}\sin^2\theta_l(1-\sin^2\theta_k)\right)-1+\prod_{k=j+1}^{n-1}\sin^2\theta_k\right) \\ &= \frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j\left(\sum_{k=j+1}^{n-1}\left(\prod_{l=j+1}^{k-1}\sin^2\theta_l-\prod_{l=j+1}^k\sin^2\theta_l\right)-1+\prod_{k=j+1}^{n-1}\sin^2\theta_k\right) \\ &= \frac{2}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}\sin\theta_j\cos\theta_j\left(1-\prod_{k=j+1}^{n-1}\sin^2\theta_k-1+\prod_{k=j+1}^{n-1}\sin^2\theta_k\right) \\ &= 0 \end{align}

$\displaystyle\pdv{r}$

$\displaystyle\pdv{r}$の係数は
\begin{align} &\phantom{{}={}}\sum_{i=1}^{n-1}\left(\frac{1}{r}-\frac{1}{r}\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i\right)+\left(\frac{1}{r}-\frac{1}{r}\prod_{i=1}^{n-1}\sin^2\theta_i\right) \\ &= \frac{n}{r}-\frac{1}{r}\sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\sin^2\theta_j\cos^2\theta_i\right)-\frac{1}{r}\prod_{i=1}^{n-1}\sin^2\theta_i \\ &= \frac{n}{r}-\frac{1}{r}\sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\sin^2\theta_j(1-\sin^2\theta_i)\right)-\frac{1}{r}\prod_{i=1}^{n-1}\sin^2\theta_i \\ &= \frac{n}{r}-\frac{1}{r}\sum_{i=1}^{n-1}\left(\prod_{j=1}^{i-1}\sin^2\theta_j-\prod_{j=1}^i\sin^2\theta_j\right)-\frac{1}{r}\prod_{i=1}^{n-1}\sin^2\theta_i \\ &= \frac{n}{r}-\frac{1}{r}\left(1-\prod_{i=1}^{n-1}\sin^2\theta_i\right)-\frac{1}{r}\prod_{i=1}^{n-1}\sin^2\theta_i \\ &= \frac{n}{r}-\frac{1}{r} \\ &= \frac{n-1}{r} \\ \end{align}

$\displaystyle\pdv{\theta_i} \ (1 \leq i \leq n-1)$

$\displaystyle\pdv{\theta_i}$の係数は
\begin{align} &\phantom{{}={}}\sum_{j=i+1}^{n-1}\left(\frac{1}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\frac{\cos\theta_i}{\sin\theta_i}-\frac{1}{r^2}\prod_{k=1}^{i-1}\frac{1}{\sin^2\theta_k}\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)+\frac{2}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\sin\theta_i\cos\theta_i+\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\frac{\cos\theta_i}{\sin\theta_i}-\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{j=i+1}^{n-1}\sin^2\theta_j \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\sum_{j=i+1}^{n-1}\left(\frac{\cos\theta_i}{\sin\theta_i}-\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)+2\sin\theta_i\cos\theta_i+\frac{\cos\theta_i}{\sin\theta_i}-\left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\prod_{j=i+1}^{n-1}\sin^2\theta_k\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left((n-i-1)\frac{\cos\theta_i}{\sin\theta_i} + \left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\left(-\sum_{j=i+1}^{n-1}\left(\prod_{k=i+1}^{j-1}\sin^2\theta_k\cos^2\theta_j\right)+1-\prod_{j=i+1}^{n-1}\sin^2\theta_k\right)\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left((n-i-1)\frac{\cos\theta_i}{\sin\theta_i} + \left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\left(-\sum_{j=i+1}^{n-1}\left(\prod_{k=i+1}^{j-1}\sin^2\theta_k(1-\sin^2\theta_j)\right)+1-\prod_{j=i+1}^{n-1}\sin^2\theta_k\right)\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left((n-i-1)\frac{\cos\theta_i}{\sin\theta_i} + \left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\left(-\sum_{j=i+1}^{n-1}\left(\prod_{k=i+1}^{j-1}\sin^2\theta_k-\prod_{k=i+1}^j\sin^2\theta_k\right)+1-\prod_{j=i+1}^{n-1}\sin^2\theta_k\right)\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left((n-i-1)\frac{\cos\theta_i}{\sin\theta_i} + \left(\frac{\cos\theta_i}{\sin\theta_i}+2\sin\theta_i\cos\theta_i\right)\left(-1+\prod_{j=i+1}^{n-1}\sin^2\theta_j+1-\prod_{j=i+1}^{n-1}\sin^2\theta_k\right)\right) \\ &= \frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}(n-i-1)\frac{\cos\theta_i}{\sin\theta_i} \\ \end{align}

よって，以下の式を得ます．
\begin{align} \Delta &= \pdv[2]{r} + \sum_{i=1}^{n-1}\left(\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\right)\pdv[2]{\theta_i} + \frac{n-1}{r}\pdv{r} + \sum_{i=1}^{n-1}\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}(n-i-1)\frac{\cos\theta_i}{\sin\theta_i}\pdv{\theta_i} \\ &= \pdv[2]{r} + \frac{n-1}{r}\pdv{r} + \sum_{i=1}^{n-1}\frac{1}{r^2}\prod_{j=1}^{i-1}\frac{1}{\sin^2\theta_j}\left(\pdv[2]{\theta_i}+(n-i-1)\frac{\cos\theta_i}{\sin\theta_i}\pdv{\theta_i}\right) \end{align}