1.3. Geodesics 11 We will see, later, that the shortest paths on a sphere are arcs of great circles. Unfortunately, we cannot expect every interesting surface to be the level set for some common coordinate. We may, however, hope to represent our surface parametrically. We may prescribe the x, y, and z coordinates of points on the surface using the parameters u and v and write our surface in the vector form r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k . (1.23) We can now specify a curve on this surface by prescribing u and v in terms of a single parameter — call it t — so that u = u(t) , v = v(t) . (1.24) The vector ˙ ≡ dr dt = ∂r ∂u ˙ + ∂r ∂v ˙ (1.25) is tangent to both the curve and the surface. We find the square of the distance between two points on a curve by integrating ds2 = dr · dr = ∂r ∂u du + ∂r ∂v dv · ∂r ∂u du + ∂r ∂v dv (1.26) along the curve. Equation (1.26) is often written ds2 = E du2 + 2 F du dv + G dv2 , (1.27) where E = ∂r ∂u · ∂r ∂u , F = ∂r ∂u · ∂r ∂v , G = ∂r ∂v · ∂r ∂v . (1.28) The right-hand side of equation (1.27) is called the first funda- mental form of the surface. The coeﬃcients E(u, v), F (u, v), and G(u, v) have many names. They are sometimes called first-order fun- damental magnitudes or quantities. Other times, they are simply called the coeﬃcients of the first fundamental form. The distance between the two points A = (ua,va) and B = (ub,vb) on the curve u = u(t), v = v(t) may now be written s = tb ta E du dt 2 + 2 F du dt dv dt + G dv dt 2 dt , (1.29)

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