# Gauss fundamental form in differential geometry : use of dot products

In textbooks on differential geometry, the first fundamental form looks like $$E^2+2FG+H^2$$, and its length is calculated through the help of the dot product. However, the inner product did not exist. The analytic formula was known that $$x_1x_2+y_1y_2+z_1z_2$$ . Nor did the vector exist. So how did he prove it? Same story with the unit normal: in textbooks they use the cross product $$r(u) \times r(v)$$ But this did not exist either. Again, how did he manage to calculate it?

• Try formatting it with MathJax – Euler_Salter Dec 26 '16 at 20:36
• Who is "he" supposed to be? – Danu Dec 29 '16 at 23:31