Sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(...
' cos(3t), t<n/2, 2. Let f(t) = sin(2t), 7/2<t< , Write f(t) in terms of the unit step e3 St. function. Then find c{f(t)}.
15. Let γ be the unit circle. Prove that sin z
Let V be a vector space, let S, T L(V), and assume that ST = TS. Prove that if ˇ V is an eigenvector for T with eigenvalue λ, then λ is also an eigenvalue for S Find an eigenvector for λ with respect to S, and prove your answer is correct. Let V be a vector space, let S, T L(V), and assume that ST = TS. Prove that if ˇ V is an eigenvector for T with eigenvalue...
Assume sin S = {}, cos S = 4, sin T = 1, cos T = Find the given quantities without using a calculator. Give answer as a fraction (for instance on half would be written 1/2) Sin (S +T)= Cos (S+T)=
Problem 2. Let be the quarter torus with outward normal. Use the parameterization r(u, v) = (4 + 2 cos(v)) cos(u)i + (4 + 2 cos(u)) sin(u)j + 2 sin(v)k, for 0 Susand 0 <0527 (a) Find a parameterization for each of the curves forming the boundary of E. Make sure the orientation of the curves match the orientation induced by S. (b) Let F(x, y, z) = xyi+yzj+rzk. Evaluate S/.( VF) ds.
(f) If a and b are unit vectors and θ is the angle between them. prove that sin -b Prove that v2·“2 + 2,6 , where the symbol's h) have their usual meanings 12. Find the following dx dx tan'xdx (i) 1+cos x 13 Find the area enclosed between the circle 2y25 and the straight line x +y- s 14. Solve the following equations dy dy dy dx (f) If a and b are unit vectors and θ is the...
Question 7 Let r(t) = ( 11t, cos 5t, sin 5t> Find the unit tangent vector and the unit normal vector of r(t) at + = (Round to 2 decimal places) TE == NG) = < bic rocnonse
Question 9 Let r(t)={cos 2t, sin 2t, V5t) a) Find the unit tangent vector and the unit normal vector of r(t) at += TI (Round to 2 decimal places) TE)= NG) = < b) Find the binormal vector of r(t) at t = TT 2 (Round to 2 decimal places) BC) =< A Moving to another question will save this response.
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the unit sphere S2. Let and show that a new parametrization of the coordinate neighborhood x(U) = V can be given by y(u, (sech u cos e, sech u sin e, tanh u Prove that in the parametrization y the coefficients of the first fundamental form are Thus, y-1: V : S2 → R2 is a conformal...
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0 4. (a Let (sin( x cos( ) dr...