1. Let r sin cos F(r, 0, 0) = r sin 0 sin r cos for r≥ 0, 0 € [0,π) and Є [0,2π). Compute det JF (r, 0, 0). For which values of the different variables is this Jacobian matrix singular?

If u = x2 + 2x -y and v = -2xy-2y, then the vorticity vector is

Q2. If ) is a covariant vector in Cartesian coordinates x',x². Find its components in polar coordinates x', x where the transformation law from the Cartesian coordinate (x',x²) to polar coordinates (x', x²) are x' = x'cos(x²) and x² = x'sin(x²) %3D

Q1. Let u (₁,1), v= (2, 2) V=R² and ke R. Then we define addition and scalar multiplication on Vas; u+v= (₁+y₁-1, 2+92-1) and a.u = (ay₁, 2₁). (i). Compute, u+v and k.u, if u = (-1,3), v= (2,0) and k = 5. (ii). Show that V is not a vector space with given addition and scalar mul- tiplication.

1.2 If R₁ = 2i-j+k, R₂ = i +3j - 2k, R3 = -2i+j3k and R4 = 3i+2j+5k. Determine scalars x, y, and z such that R4 = xR₁ +yR₂ + zR3.