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First fundamental property of surface

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

Proof: The first fundamental form is given by

I = ds2 = E(du)2 + 2 F du dv + G (dv)2

= 1 / E [E2(du)2 + 2 EF du dv + EG (dv)2]       [∵ E>0]

= 1 /E [(E du + F dv)2 – F2 (dv)2 + EG(dv)2]

= 1 /E [(E du + F dv)2 + (EG – F2) (dv)2] —————————- (i)

Since EG – F2>0, E>0 and for all real values of du and dv, we have form (i)

I = ds2 = E du2 + 2 F du dv + G dv2 ≥ 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 – F2) (dv)2] = 0

⟹ (E du + F dv)2 + (EG – F2) (dv)2 = 0

⟹ (E du + F dv)2 = 0 and (EG – F2) (dv)2 = 0

⟹ (E du + F dv)2 = 0 and dv = 0   [∵ EG – F2>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)

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