Fundamental forms of surface: The two fundamental forms called the first fundamental form(metric) and second fundamental form are of great importance in the study of differential geometry. Now we shall discuss them in detail.
First fundamental form or metric form of surface: Let x = x(u, v) be the equation of surface. Then the quadratic differential form
I = ds2 = E du2 + 2F du dv + G dv2 ——————————(i)
Where E = xu . xu, F = xu . xv, G = xv . xv is called the first fundamental form or metric and the quantities E, F, G are called first fundamental coefficients.
Geometrical interpretation of first fundamental form or metric: Consider a curve u = u(t), v = v(t) on the surface x = x(u, v). Let x and x + dx be the position vectors of the two neighboring points P and Q corresponding to u , v and u + du, v + dv respectively on the surface.
We have x = x (u, v)
= x u du + xv dv ——————(ii)
Since P and Q are two neighboring points, we have
ds = | dx|
⟹ ds2 = |dx|2
= (xu du+ xv dv) . (xu du + xv dv)
= xu . xu du2 + xu . xv du dv + xv . xu dv du + xv . xv dv2
= xu . xu du2 + 2xu . xv du dv + xv . xv dv2
= E du2 + 2F du dv + G dv2
∴ ds2 = E du2 + 2F du dv + G dv2
Where ds is the infinitesimal distance from the point P(u,v) to the point Q(u + du, v + dv). (Proved)