直線座標の上では，つぎの形式が計量テンソルとなった (  「直線座標の計量テンソル」)：
\[
s^2 = \sum_{i,j=1}^n d_{ij}\, {x^i}{x^j}
\]
これから一般座標 \( u^i \) での計量テンソルを導いてみる。
直線座標 \( x^i \) と一般座標 \( u^i \) 間の変換は：
\[
\left(
\begin{array}{c}
x^1 \\
\vdots \\
x^n \\
\end{array}
\right)
\begin{array}{c}
\left(\,\frac{\partial x^i}{\partial u^j}\,\right) \\
\longleftarrow \\
\longrightarrow \\
\left(\,\frac{\partial u^i}{\partial x^j}\,\right) \\
\end{array}
\left(
\begin{array}{c}
u^1 \\
\vdots \\
u^n \\
\end{array}
\right)
\\
x^i = \sum_{j=1}^n \frac{\partial x^i}{\partial u^j} u^j \ \ \ \ (\, i = 1, \cdots, n \,)
\]
よって，
\[
ds^2 = \sum_{k,l=1}^n d_{kl}\, {dx^k}{dx^l} \\
= \sum_{k,l=1}^n d_{kl}
\left( \sum_{i=1}^n \frac{\partial x^k}{\partial u^i} du^i \right)
\left( \sum_{j=1}^n \frac{\partial x^l}{\partial u^j} du^j\right) \\
= \sum_{i,j=1}^n
\left( \sum_{k,l=1}^n d_{kl}\, \frac{\partial x^k}{\partial u^i} \frac{\partial x^l}{\partial u^j} \right)
du^i \, du^j \\
\]
形式
\[
\sum_{i,j=1}^n
\left( \sum_{k,l=1}^n d_{kl}\, \frac{\partial x^k}{\partial u^i} \frac{\partial x^l}{\partial u^j} \right) du^i \, du^j
\]
は，\( x^i,\, u^i \) の座標変換
\[
\left(
\begin{array}{c}
x^1 \\
\vdots \\
x^n \\
\end{array}
\right)
\begin{array}{c}
\left(\,a^i_j\,\right) \\
\longleftarrow \\
\longrightarrow \\
\left(\,b^i_j\,\right) \\
\end{array}
\left(
\begin{array}{c}
{x^{'}}^1 \\
\vdots \\
{x^{'}}^n \\
\end{array}
\right)
\ \ \ \ \ \ \ \ \
\left(
\begin{array}{c}
u^1 \\
\vdots \\
u^n \\
\end{array}
\right)
\begin{array}{c}
\left(\,\frac{\partial u^i}{\partial {u^{'}}^j}\,\right) \\
\longleftarrow \\
\longrightarrow \\
\left(\,\frac{\partial {u^{'}}^i}{\partial u^j}\,\right) \\
\end{array}
\left(
\begin{array}{c}
{u^{'}}^1 \\
\vdots \\
{u^{'}}^n \\
\end{array}
\right)
\]
で保たれる：
\[
\sum_{i,j=1}^n
\left(
\sum_{k,l=1}^n d_{kl}\,
\frac{\partial x^k}{\partial u^i}
\frac{\partial x^l}{\partial u^j}
\right)
du^i \, du^j
\\ =
\sum_{i,j=1}^n
\left(
\sum_{k,l=1}^n d_{kl}
\left(
\frac{\partial x^k}{\partial {x^{'}}^k}
\frac{\partial {x^{'}}^k}{\partial {u^{'}}^i}
\frac{\partial {u^{'}}^i}{\partial u^i}
\right)
\left(
\frac{\partial x^l}{\partial {x^{'}}^l}
\frac{\partial {x^{'}}^l}{\partial {u^{'}}^j}
\frac{\partial {u^{'}}^j}{\partial u^j}
\right)
\right)
du^i \, du^j
\\ =
\sum_{i,j=1}^n
\left(
\sum_{k,l=1}^n d_{kl}
\left(
\frac{\partial x^k}{\partial {x^{'}}^k}
\frac{\partial x^l}{\partial {x^{'}}^l}
\right)
\left(
\frac{\partial {x^{'}}^l}{\partial {u^{'}}^i}
\frac{\partial {x^{'}}^l}{\partial {u^{'}}^j}
\right)
\right)
\frac{\partial {u^{'}}^i}{\partial u^i} du^i
\frac{\partial {u^{'}}^j}{\partial u^j} du^j
\\ =
\sum_{i,j=1}^n
\left(
\sum_{k,l=1}^n d_{kl}
\left( a^k_k\, a^l_l \right)
\frac{\partial {x^{'}}^k}{\partial {u^{'}}^i}
\frac{\partial {x^{'}}^l}{\partial {u^{'}}^j}
\right)
d{u^{'}}^i d{u^{'}}^j
\\ =
\sum_{i,j=1}^n
\left(
\sum_{k,l=1}^n d^{'}_{kl}
\frac{\partial {x^{'}}^k}{\partial {u^{'}}^i}
\frac{\partial {x^{'}}^l}{\partial {u^{'}}^j}
\right)
d{u^{'}}^i d{u^{'}}^j
\\
\ \ \ \ \ \ \ \ \ d^{'}_{kl} = d_{kl} \, a^k_k \, a^l_l
\]
そこで，形式
\[
ds^2 = \sum_{i,j=1}^n g_{ij}\, du^i \, du^j \\
g_{ij} = \sum_{k,l=1}^n d_{kl} \frac{\partial x^k}{\partial u^i} \frac{\partial x^l}{\partial u^j}
\]
を「直線座標 \( x^i \) に対する座標 \( u^i \) の上の計量テンソル」と定める。
\( g_{ij} \) は， (0, 1) テンソル \( \frac{\partial x^k}{\partial u^i},\, \frac{\partial x^l}{\partial u^j}\) の積の和ということで， (0, 2) テンソルである。
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