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Question 1. Using the transformation law patterns for upper and lower indices, predict a transfor- mation law giving Laß in terms of Lij in the presence of a coordinate transformation f : U U, and then show that your guess is correct. (This shows that the Lij functions do indeed form tensor coefficients.) Question 2. Using the transformation law patterns for upper and lower indices, predict a transfor- mation law giving T? aB in terms of The in the presence of a coordinate transformation f : U U, and then show that your guess is INCORRECT by proving the correct transformation law below: FI aB = [ T* ij 0² fk Oh^ + dua duB auk dua duB duk ijk k (This shows that the functions This do NOT assemble to form the coefficients of a tensor.) HINT: Use Gauss's intrinsic formula for the The and/or the Taß. Question 3. Suppose is a unit speed curve on the unit sphere S2. Prove that Kin is constant. Question 4. Consider the simple surface defined below u U u2 + v2 and the surface curve = x o r U defined by t2 U (t) == . t (Here we are taking U = R².) Find the normal curvature Kin of when t = 1. Be careful! as written may not be unit speed but we do not need to reparametrize the entire curve by arc length. Instead, it is enough to find / = Ci(ru) X, i at t = 1 and renormalize just this single tangent vector to be unit size before using our formula for Kin.

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