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FIRST FUNDAMENTAL OR METRIC FORM OF SURFACE

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.

Screenshot_18

 

 

We have x = x (u, v)

Screenshot_17

= 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)

 

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First fundamental property of surface

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

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