sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(s) Let w zdz Λ dy. Find w. sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(s) Let w zdz Λ dy. Find w.
Given a positive integer n and a real number θ E (0,7), prove that sin n θ 2 sin θ where γ is the circle of radius 2 centered at the origin, oriented counterclockwise. Given a positive integer n and a real number θ E (0,7), prove that sin n θ 2 sin θ where γ is the circle of radius 2 centered at the origin, oriented counterclockwise.
Vector / Complex Calculus 6. Calculate the integrals of cos(z)/z" and sin(x)/2" over the unit circle, where n is a positive integer.
(sin(π/z) -1dd 2. Compute the integral: sin(π/s)-.--d γ is the cl γ is the curve shown in the 2. where 721-1 following figure: arked points on the coordinate axes correspond to T,-T, 2, 2. (sin(π/z) -1dd 2. Compute the integral: sin(π/s)-.--d γ is the cl γ is the curve shown in the 2. where 721-1 following figure: arked points on the coordinate axes correspond to T,-T, 2, 2.
mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism. mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism.
(proof) n all 26. Let P(z) = 0 stand for an the zeros of which are in the unit circle |z| < 1. Replacing each coefficient of P() by its conjugate we obtain the polynomial P(2). We define p*()=P( The roots of the equation P(z) + P*(2) = 0 are all on the unit circle |z| = 1 algebraic equation of degree n all 26. Let P(z) = 0 stand for an the zeros of which are in the unit...
Let S = {(x, y) = RP.22 + y2 = 1} denote the unit circle in R2 with the subspace topology. Define the function F: (0,1) + S via th (cos(2), sin(24t)) Prove that F is one-to-one, onto, and continuous, but not a homeomorphism.
Let f(x)=( sin(1/z), ifrj0 if x = 0. Prove that f(x) has the Intermediate Value Property, although f(r) is not continuous
Problem 4.9 (e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
Problem 5. (i) Prove that sin (5) if 0 < If z = 0 £1 f(z) = 1。 is Riemann integrable on 0, (ii) Prove that if z if z E {0, π, 2r) g(z) = 0 is Riemann integrable on [0,2