Given a surface S: r (u, v), the differential form

with coefficients (in standard notation, unrelated to F, G elsewhere in this chapter)

is called the first fundamental form of S. This form is basic because it permits us to calculate lengths, angles, and areas on S. To show this prove (a)€“(c):

(a) For a curve C: u = u(t), v = v(t), α ‰¤ t ‰¤ b, on S, formulas (10), Sec. 9.5, and (14) give the length

(b) The angle γ between two intersecting curves C₁: u = g(t), v = h(t) and C₂: u = p(t), v = q(t) on S: r(u, v) is obtained from

where a = r_ug' + r_vh' and b = r_up' + r_vq' are tangent vectors of C₁ and C₂.

(c) The square of the length of the normal vector N can be written

so that formula (8) for the area A(S) of S becomes

(d) For polar coordinates u (= r) and v (= θ) defined by x = u cos v, y = u sin v we have E = 1, F = 0, G = u², so that

Calculate from this and (18) the area of a disk of radius α.

(e) Find the first fundamental form of the torus in Example 5. Use it to calculate the area A of the torus. Show that A can also be obtained by the theorem of Pappus, which states that the area of a surface of revolution equals the product of the length of a meridian C and the length of the path of the center of gravity of C when C is rotated through the angle 2Ï€.

ds2 = E du? + 2Fdu dv + G dv2 (13) |(14) E = r, ry, F= r rg, G = r, r, F = r, r, G= r,r,