Editing First fundamental form (section)

==Further notation==
When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.
<math display="block">\mathrm{I}(v)= \langle v,v \rangle = |v|^2</math>

The first fundamental form is often written in the modern notation of the [[metric tensor]].  The coefficients may then be written as {{mvar|g<sub>ij</sub>}}: 
<math display="block"> \left(g_{ij}\right) = \begin{pmatrix}
g_{11} & g_{12} \\
g_{21} & g_{22}
\end{pmatrix} =\begin{pmatrix}
E & F \\
F & G
\end{pmatrix}</math>

The components of this tensor are calculated as the scalar product of tangent vectors {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}}:
<math display="block">g_{ij} = \langle X_i, X_j \rangle  </math>
for {{math|1=''i'', ''j'' = 1, 2}}. See example below.