3. [4 marks] Compute the Jacobi matrix of the cornposite mapping z with a - ucosv and y u sin v. ...
3. (a) (3 marks) If multiplication by matrix A rotates a vector v in the x-y plane through an angle 0, what is the effect of multiplying v by A?. (b) (3 marks) Describe the geometric effect of multiplying a vector x by the following matrix (cos0 – sin? 0 -2 sin 0 cos 0 ) | 2 sin 6 cos sin0 – cosa e ) (c) (4 marks) In three dimensional space, find a matrix that rotates a vector...
)Given a 4-node element in x-y plane as shown here: Node X 3 3 1 8 a) Using the shape functions in u-v plane, determine an expression for mapped points from u-v to x-y, i.e. x- x(u, v) and y -y(u, v), for points within the 4-node element in u-v plane. Then, determine value of x and y for a point with (u, v)-(0.3,0.3). (10 points) b) Determine the value of Jacobian matrix, [J], and its determinant for such mapping...
dz Find when u = 0, v = 2, if z = sin (xy)+xsin (y), x=u2 +2V2, and y= uv. du az = du 1 = 0, V=2 (Simplify your answer.)
y 1) (4 marks) Algebraically invert the Lorentz transformation, I' (x- ut) (1) V (2) (3) 7(t - ux/c) where y = (1 - u2/)-1/2. In other words, solve for x, y, z and t in terms of X',V, :' and t'. Note: I expect you to simplify your expressions appropriately. 2
(1 point) 5x2 — 5у, v %3D 4х + Зу, f(u, U) sin u cos v,u = Let z = = and put g(x, y) = (u(x, y), v(x, y). The derivative matrix D(f ° g)(x, y) (Leaving your answer in terms of u, v, x, y ) (1 point) Evaluate d r(g(t)) using the Chain Rule: r() %3D (ё. e*, -9), g(0) 3t 6 = rg() = dt g(u, v, w) and u(r, s), v(r, s), w(r, s). How...
V X2 + y2 and θ u(r(z, y), θ(x, y))--sech2 r tanh r sin θ 6. [Sec. I 1.5] Letr tan l (y/z) be the usual polar rectangular coordinates relationships. Furthermore, define and u(r(z, y),θ(z, y)) sech2 r tanh r cos θ Show that tanh r
Give a parametric description of the form r(u, v) = (x(u,v),y(u,v),z(u,v)) for the following surface. The cap of the sphere x² + y2 + z = 36, for 3/3 sz56 Select the correct choice below and fill in the answer boxes to complete your choice. (Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than 21. Type exact answers in terms of ..) A. Fu.v) = (6 sin u cos V,6 sin...
Find the best approximation to z by vectors of the form C7 V + c2V2. 3 1 3 -1 -6 1 z = V2 4 0 -3 3 1 The best approximation to z is . (Simplify your answer.) - 15 - 8 8 - 1 Let y = , and v2 Find the distance from y to the subspace W of R* spanned by V, and vą, given 1 0 1 - 15 3 3 - 13 09 that...
4. Let f(x, y, z) = rytan'() + z sin(xy), < = wy=v²v, z = ". Find fu and , using the chain rule.
10. Consider the surface S parameterized by w r= (cos y, sin v, u + sin v), -3 <u <3, 050 < 27 *** (a) Write a linear equation for the tangent plane to the surface at (0,1,1) (b) Compute the surface area of S.