Exercise 7.6 Use the angel sum formulas for sin(θ+y) and cos(θ+p) to demonstrate that, in general...
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
Time series analysis 1. (a) Use Euler's identity e¡θ-cos θ + i sin θ to prove that sin θ=-(eiO , 2i (b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer and j E 1,2,... ,n} then TL TL sin-(2Ttj/n)- n/2 so long as J关[m/2, where Laj is the greatest integer that is smaller than or equal to x (c) Show that when j 0 we have...
Essential_Mathematical_Meth_Arfken_Weber_ 1.3.7 Using the vectors P = ˆx cos θ + ˆy sin θ, Q = ˆx cos ϕ − ˆy sin ϕ, R = ˆx cos ϕ + ˆy sin ϕ, prove the familiar trigonometric identities sin(θ + ϕ) = sin θ cos ϕ + cos θ sin ϕ, cos(θ + ϕ) = cos θ cos ϕ − sin θ sin ϕ.
Use the sum-to-product formulas to write the sum as a product. sin 7θ − sin 3θ cos 2θ cos 4θ Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles. sin4(2x)
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...
Verify that Prn (cos θ) solves sin θΟθ (sin ea, Θ) + (E(1 + 1) sin2 θ-m2) Θ 0. Use that pr(z)-(1-z2 )T (4), Pr(r) with Pr(z) a Legendre polynomi 1 Verify that Prn (cos θ) solves sin θΟθ (sin ea, Θ) + (E(1 + 1) sin2 θ-m2) Θ 0. Use that pr(z)-(1-z2 )T (4), Pr(r) with Pr(z) a Legendre polynomi 1
14. (6 points) Use sin(– ) = sin r cos y – cos sin y to evaluate sin ( - 7). BONUS. (8 points) Find all r values in (0,27 that satisfy the following equation. sin cos? Hint: sin?. + cos2 = 1.
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
Consider a point P on the x − y plane. Show that ur = cos(θ) ˆi + sin(θ) ˆj, uθ = − sin(θ) ˆi + cos(θ) ˆj. Q5) Relationship between the Polar and Cartesian Unit Axis Vec- tors] Consider a point P on the x - y plane. Show that u.-cos(0)İ + sin(θ)j. u.--sin(θ)i + oos@j
Find all points (if any) of horizontal and vertical tangency to the portion of the curve. Involute of a circle: x = cos θ + θ sin θ y = sinθ - θ cos θ