# Property: First fundamental form (or metric) is a positive definite quadratic form in du, dv.

**Proof:** The first fundamental form is given by

I = ds^{2} = E(du)^{2} + 2 F du dv + G (dv)^{2}

= 1 / E [E^{2}(du)^{2} + 2 EF du dv + EG (dv)^{2}] ** [∵ E>0]**

= 1 /E [(E du + F dv)^{2} – F^{2} (dv)^{2} + EG(dv)^{2}]

= 1 /E [(E du + F dv)^{2} + (EG – F^{2}) (dv)^{2}] —————————- (i)

## Since EG – F^{2}>0, E>0 and for all real values of du and dv, we have form (i)

I = ds^{2} = E du^{2} + 2 F du dv + G dv^{2 }≥ 0.

### Again, if I = 0 i.e., E (du)^{2} + 2F du dv + G (dv)^{2} = 0 then from (i) we have

1 /E [(E du + F dv)^{2} + (EG – F^{2}) (dv)^{2}] = 0

⟹ (E du + F dv)^{2} + (EG – F^{2}) (dv)^{2} = 0

⟹ (E du + F dv)^{2} = 0 and (EG – F^{2}) (dv)^{2} = 0

⟹ (E du + F dv)^{2} = 0 and dv = 0 **[∵ EG – F ^{2}>0]**

⟹ E du = 0 and dv = 0 **[ ∵ dv = 0]**

⟹ du = 0 and dv = 0 **[∵ E>0]**

**But both du and dv cannot vanish together. Thus 1 ≥ 0. That is the first fundamental form is a positive definite quadratic form in du, dv.** *(Proved)*