**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 = d__s__^{2} = E d__u__^{2} + 2F d__u__ d__v__ + G d__v__^{2} ——————————(i)

Where E = __x___{u} . __x___{u}, F = __x___{u} . __x___{v}, G = __x___{v} . __x___{v}* 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__ + d__x__ be the position vectors of the two neighboring points P and Q corresponding to __u__ , __v__ and __u__ + d__u__, __v__ + d__v__ respectively on the surface.

We have __x__ = __x__ (u, v)

= __x__ _{u }d__u__ + __x___{v} d__v__ ——————(ii)

Since P and Q are two neighboring points, we have

ds = | d__x__|

⟹ d__s__^{2} = |d__x__|^{2}

= (__x___{u} d__u__+ __x___{v} d__v__) . (__x___{u} d__u__ + __x___{v} d__v__)

= __x___{u} . __x___{u} d__u__^{2} + __x___{u} . __x___{v} d__u__ d__v__ + __x___{v} . __x___{u} d__v__ d__u__ + __x___{v} . __x___{v} d__v__^{2}

= __x___{u} . __x___{u} d__u__^{2} + 2__x___{u} . __x___{v} d__u__ d__v__ + x_{v} . __x___{v} d__v__^{2}

= E d__u__^{2} + 2F d__u__ d__v__ + G d__v__^{2}

∴ d__s__^{2} = E d__u__^{2} + 2F d__u__ d__v__ + G d__v__^{2}

## Where ds is the infinitesimal distance from the point P(u,v) to the point Q(u + du, v + dv). **(Proved)**