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...
6. The ground state of the hydrogen atom has the form (r)= Ae/a0 where ao is the Bohr radius, A is a constant and r is the radial distance of the electron from the nucleus. Find the constant A.
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...
Problem 7 If A cos 0,+B sin oft = rsin(0 -0) = R cos(t-8), (a) determiner and in terms of A and B , and (b) the relationship among R,r, 8 and 2 Ans. (a) r = VA’ + Bº, tan 0 4 b)r=R, 0 = 8-5, tan tan 8 +1 = 0 Problem 8 (a) Define the steady state solution of a given SLDE. (b) Consider the motion of a 2-kg mass in a (m,c,k)-system under a cyclic load....
Al. Practice with complex numbers: Every complex number z can be written in the form z r + iy where r and y are real; we call r the real part of z, written Re z, and likewise y is the imaginary part of z, y - Im z We further define the compler conjugate of z aszT-iy a) Prove the following relations that hold for any complex numbers z, 21 and 22: 2i Re (2122)(Re z) (Re z2) -...
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) ▽.
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)" =...
The only ideas that can be used include: area ABCI-RA2(A+B+C-lpi), the Pythagorean theorem: Cos c cos a cos b. Vectors-dot product cross product, sin A-sin a/sin c; cos A-cos a sin b/sin c; spherical law of sines, spherical law of cosines for sides and spherical law of cosines for angles Let r be the radius of the incircle of triangle ABC on the unit sphere S. If all the angles in triangle ABC are right angles, what is the exact...
r(t) = 10 sin(300t + 60°) cos(x - 90) sin(x) v(t) 10 cos(300t- 30°) ω = 300, Mag = 10, θ =-30° v 102-30 MLP Mcos(P) +jMsin(P) v = 10 cos(-30) +j10 sin(-30) Step 1: Convert to Cosine Step 2: Identify Frequency, Magnitude, and Phase Step 3: Convert to real/imag form = 8.66-J5 Step 4: Solve Step 5: Convert Back