Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 21 3of Problem 6 following Section 2.13 apply. We write r,, r,, r,,, r,,, r,, for partialderivatives, all assumed to be continuous.a) Show that the unit tangent vector T = drlds satisfies the equation T = (du/ds)r, +(dv/ds)r,, whereb) Show that the acceleration vector a = dTlds satisfies the equationc) Show that the curvature K satisfies the equationwhere L = r,, . n, M = r,, - n, N = r,, . n. The quadratic differential formL du2 + 2M du dv + N dv2 is called the second fundamental form of S.d) Write Lll = L, LI2 = L21 = M, L22 = N and show that Lij is a tensor asindicated. [Hint: Fix a point Po: (uo, vo) on S and choose a special basis il, jl, klfor vectors in space with kl = n at Po. Then write r = ro + til + r]jl + Jkl, with(6, r], J) = (0,0,O) at Po. Show that at Po r, .n = 0 and r, .n = 0, so that at (uo, vo)J, = 0 and 5; = 0 and L = J,,, M =