Anthropology homework help

Problem 1. (a) State the inverse function theorem for functions of a single variable.(You don’t need to prove it.)(b) Let I R be an open interval, and let g : I ! R be a smooth function. Let Jbe the range of g, i.e. J = g(I). Prove that if g0(t) 6= 0 for all t 2 I, then thereexists a smooth function f : J ! I which is the inverse to g, i.e. g f = 1J , andf g = 1I . Hint: Use part (a).(c) Prove that the function f has the property that, for all t 2 J, we havef0(t) =1g0(f(t)): (1)Problem 2. Let   : I ! R3 be a regular smooth curve, (not necessarily parametrizedby arc length). Let g : I ! R be the arc length of  . Set J = g(I), and let fbe the inverse of g. For the existence of g, we refer to Problem 1. Recall that thecurve   =   f : J ! R3 is parametrized by arc length, (you don’t need to provethis). By de nition, the curvature of   at s 2 I, is the curvature of   at f 1, i.e.k (s) = k (f 1(s)). In this problem, we’re going to  nd a formula for the curvature of .(a) Prove that, for all s 2 I we have 00(f 1(s)) =  00(s) (f0(f 1(s)))2 +  0(s) f00(f 1(s)):1(b) Use Eq. 1 to express f0(g(s)) in terms of k 0(s)k, and then use the result to provethat, for all s 2 If00(g(s)) = 00(s)  0(s)k 0(s)k4 :HINT: First prove thatddsk 0(s)k = 00(s)  0(s)k 0(s)k:(c) Use parts (a) and (b) to show thatk 00(f 1(s))k2 =1k 0(s)k6k 00(s)k2k 0(s)k2 ( 00(s)  0(s))2 ;and, then take the square root of both sides of this equation, and use an appropriateformula for the cross product, to provek (s) =k 00(s)  0(s)kk 0(s)k3 : (2)Problem 3. Consider the smooth curve   : R ! R2, de ned as (t) = e tp 2 + 1(cos(t); sin(t)):(a) Determine its velocity  0(t), and show that   is not parametrized by arc length.(b) Determine the curvature k (t) of  . HINT: use Equation 2.(c) Find a reparametrization of   so that it is parametrized by arc length. That is,  nda smooth bijective function f : I ! R with smooth inverse f 1 : R ! I, such thatthe smooth curve   =   f : I ! R2 is parametrized by arc length. (HINT: Startby determining the arc length of  .)(d) Determine the curvature of  , and compare with part (b).Problem 4. Consider the smooth curve   : (0;1) ! R3, de ned as (t) = (e t8 cos(t); e t8 sin(t); t):(a) Sketch the trace of  , and in the same  gure, draw the tangent line of   at t = =2.(b) Compute the curvature k  and the torsion  of  .(c) Set a(t) =  00(t), and de nea?(t) = a(t) a(t)  0(t)k 0(t)k:2Recall that the unit normal is then de ned byn (t) =a?(t)ka?(t)k:Calculate n (t).(d) The osculating circle of   at t0 2 (0;1) is the trace of the curve(s) =1k (t0)cos(s) 0(t0)k 0(t0)k+ sin(s)n (t0)+  (t0) +1k (t0)n (t0):Set t0 = and draw the corresponding osculating circle Draw the trace of   in thesame picture.