FINAL SAMPLE QUESTION
MTH 420 1. CURVATURE OF A CURVE Let C be a curve in (Euclidean) R3 . Either using the heuristic argument ds2 = dx2 + dy 2 + dz 2 = ds = dx dy dz dx + dy + dz ds ds ds
where s is arc length, or by noting that the unit tangent vector T to C satisfies T dr = T T ds = ds we see that it is natural to define T = ds to be the unit 1-form tangent to the curve C. In practice, it is common to parameterize the curve using an arbitrary parameter t, not necessarily arc length. In this case, the s derivatives are replaced using chain and rule, we have in general dx dx + dy dy + dz dz dt dt T = dt 2 2 2 dx + dy + dz ds dt dt FACT: If dT = 0, there is a unique positive function and a unique unit 1-form N such that N is orthogonal to T and dT = T N (If dT = 0 we define = 0 and N is not defined.) We call the curvature of C. (a) Find the curvature of a...