The first fundamental form is very famous in differential geometry, but the definition is not very consistent in many articles. There are two versions of definitions and they are not exactly the same things.
- it is a dot from $T_p\times T_p$ to $R$
- it is the expression ds^2=Edu^2+2Fdudv+Gdv^2.
In fact, the second one comes from the first one at as special case. If we set X=X(u,v), and set $dw=X_udu+X_vdv$ then $dw\cdot dw = X_u\cdot X_udu^2+2X_u\cdot X_vdudv + X_v\cdot X_vdv^2$, and if we set $E=X_u\cdot X_u, F=X_u\cdot X_v, G=X_v\cdot X_v, dw\cdot dw= ds^2$, then we got 2.
But although the two definitions have relationship, it seems that they are different things or at least different by definition. So can I say that the first fundamental form lacks a strict and universal definition?
You can define an euclidian structure on a vector space (in this case of dimension 2) either by mean of a symmetric bilinear form $<x,y>$, positive and definite, or by a quadratic form $q(x)$ (i.e. a degree 2 homogenous polynomial in the coordinates $(u,v)$ of the vector $x$ positive and definite .
The link is $q(x)= <x,x>$, $2<x,y>=q(x+y)-q(x)-q(y)$
In coordinates if $q(x)= Eu^2+2Fu v+Gv^2$, then $<x,x'>= E u u'+F(uu'+vv')+Gvv'$