10 Find the Frenet (TNB) frame (c+) = (t sint, & cosct), t for any A)....
and B alt) =(+², t, t3) find frenet frame i that (a) torsion and K find curvature (6)
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a E R3 with R3 be smooth with = 1 and curvature k and torsion r, both Assume there exists a unit Ta constant = COS a. circular helix is an example of such curve a) Show that b) Show that N -a 0. c) Show that k/T =constant ttan a 2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a...
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field on a, consider the Darboux frame field T, V, U—where T is the unit tangent of a, U is the surface normal restricted to a, and V = U * T (Fig. 5.34). (a) Show that T' = gV + kU V' =-gT + tU, U' = -KT - tv, 263/518 where k = S(T) · T is the normal curvature k(T) of M...
. (sint Cost Find v(t) a) . (sint Cost Find v(t) a)
please answer all the 4 parts of this question 2. Consider the circular helix r(t)- (a cos t, a sin t, bt) where a > 0,b > 0. Let P(0, a, T) be a point on the helix (a) Find the Frenet frame (T, N, B) at the point P (b) Find equations for the tangent and normal line at P (c) Find equations for the normal plane and the osculating plane at P (d) What is the curvature at...
Given the path C: x(t) = (cost, sint, t), 0<t<2n. Let f(t, y, z) = x2 + y2 + 22. Evaluate (12 pts) f(,y,z)ds.
Suppose C is a curve parametrized by r(t)=<cost,sint,1> and S is the portion of z=x^2+y^2 enclosed by C, located in the vector field F=<z,-x,y>. 25. Suppose C is the curve parametrized by F(t) = (cost, sint, 1) and S is the portion of z = x2 + y2 enclosed by C, located in the vector field F = (2, -,y). Verify Stokes' theorem. That is, find show they are, in fact, the same. fe dr and SIC (curl ) ñds...
(1 point) Calculate the integral of f(x, y, z)-3x2 + 3уг + z6 over the curve c(t) (cost, sint, t) for 0 < t < π (1 point) Calculate the integral of f(x, y, z)-3x2 + 3уг + z6 over the curve c(t) (cost, sint, t) for 0
Find f. f'(t) = 2t - 4 sint, (0)= 5 Select one: a. f(t) = 2t - 3 sint +5 O b. f(t)= +2 +4 cost +1 c. f(t)=12 - 3 cost-5 d. f(t) = x2 +3 cost e. None of these
Given ř(t) =< 2 cost, t, 2 sint > as a trace of a moving object. (a) Find the curvature of K(t). (b) Find the arc length when 0<t <31. (c) Find the unit normal and binormal vectors of F(t).