Let g: R3 R3 be the cylindrical coordinate transformation g(r, 0, 2) (r cos e,r sin...
1 Cylindrical coordinate system Given the relation of the cylindrical coordinate system r=r cos pi+r sin øj + zk (1) Lets define vectors er, eg, and ez, that indicate the direction of the vectors in the cylindrical coordinate system. Using the definition ar e = pt=r, p2 = 0, p3 (2) (a) Find a matrix for calculating er, er and e, in terms of i, j, and k. Invert the relation for expressing i, j, and k in terms of...
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) ▽.
o 1 0 -1 Exercise 2. Let A= in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R3, set g(W) = WT AT ER. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]C,B in the - 1 bases C = {1} and B { 8.00 } ? (ii) Let f : R3 → R be the function defined by f() = 7T Aw E R. Show that...
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...
Find the Jacobian of the transformation. r = 3er sin(20), y=e-3r cos(20) a (x, y) a(r, o) - Need Help? Read it Master It Talk to a Tutor
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...
Establish the identity 1 - cos 0 sin 0 + sin 0 1 - cos 0 = 2 csc 0. Which of the following shows the key steps in establishing the identity? 1 - cos e sin 0 ОА. + sin e 1 cos e 1 - cos e B + sin e 1 - cos 0 sin e (1 - cos 0)2 + sine 2 = 2 csc 6 sin 0(1 - cos ) cOS (1 - cos 02...
Can i have help here please?, Thanks New image For a three-link cylindrical manipulator, derive the Jacobian with respect to base Coordinate frame (Paul's method) and with respect to the reference frame (veetor cross produet method Link 0 -90 0 Question 2 Given a coordinate frame 0 -10 2 T=1-1 0 0 10 different What is the differential transformation dA correspon +11+2k and rotation δ made with respect 0.11+0j+0k Given: sin α, sin Qa, cosQI -cosa, sint, cost) sin o...
cos(()dr - (r sin(O) - e)de = 0, r(0) = 1 (make r the subject of the formula)
1 1 0 -1 Exercise 2. Let A = 0 1 0 in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R”, set g(ū) = WT AV E R. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]c,b in the bases C = {1} and B = { 9 8 B |}? (ii) Let f: R3 + R be the function defined by f(w) = vſ Aw...