Consider the following matrix, As[ cos θ sin θ -L-sin θ cos θ J, for some...
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)...
Consider the area bounded by θ 0 and θ = π/2 and cos θ + sin θ. Calculate this area
2.13 Probiems 73 216 Prove de Moivre's formula (cos θ + j sin θ)" = cos(n θ) + j sin(ne). where n is an integer 217 Use de Moivre's formula, given by Eq. (2.80), to develop the rectangular and polar form representations of the (2.80) following complex numbers: 2.18 Show that 219 Determine the roots of the following second-degree polynomials (a) (G)-2s2 -4s + 10,
2.13 Probiems 73 216 Prove de Moivre's formula (cos θ + j sin θ)" =...
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...
An ellipsoid, S, is parameterised by r = a sin θ cos
φi + a sin θ sin
φj + b cos θk 0
≤ θ ≤ π 0 ≤ φ ≤ 2π
i. Find the surface element dS, such that
dS points OUT of the ellipsoid.
ii. Hence determine the following surface integral over the
ellipsoid:
//rds JJs
Matrix operations 22. Suppose you are given a matrix of the form cos(() - sin(0) R(0) = sin() cos(0) Consider now the unit vector v = [1,0)" in a two dimensional plane. Compute R(O)v. Repeat your computations this time using w = [0, 1]". What do you observe? Try thinking in terms of pictures, look at the pair of vectors before and after the action of R(O). 23. You may have recognised the two vectors in the previous question to...
LINEAR ALGEBRA: PLEASE FOLLOW THE COMMENT and please
tell me what is the rotate matrix and why there is cos@ and -sin@ i
think it should be cos@ and sin@ on the first row
For each of the following linear operators on R2,
find the matrix representation of the transformation
with respect to the homogeneous coordinate
system:
(a) The transformation L that rotates each vector
by 120◦ in the counterclockwise direction
(b) The transformation L that translates each point
3...
3. If T2 = r3 cos(0) sin(d) and v2 = sin(0) cos(O)f + r sin(0)θ + r2 sin(d)φ compute the following (a) ▽T, (b) ▽.
Problem 2. (15 points in total) Polarization rotator. The Jones vector for an arbitrary linearly polarized state at an angle θ with respect to the horizontal is cos a sin e Starting from the above Jones vector, please prove that an optical filter described by a Jones matrix cos asin a -sin α cos α makes linearly polarized light rotate about an angle α.
Problem 2. (15 points in total) Polarization rotator. The Jones vector for an arbitrary linearly polarized...
e-30 sin() -1 backward substitution method 4. Given A = sin(t) cos(t) tanto find the following 0 a. Matrix of minors (2pts) b. Matrix of cofactors (2pts) c. Adjoint matrix (2pts) d. Determinant of A (2pts) Inverse of A using the Adjoint matrix. (2pts) e. 1 v T.