Question

1. Are £i and C2 skew lines? Explain your answer and find the distance between them if they are skew lines. 3 marks 2. Let S be the region given by S-((z, y) E R: z2 + y2 4,z? + y2-4y2 0,#2 0, y 20} 1 mark (a) Sketch the region S; (b) Consider the change of variables given by u2 , a2 +y-4y. Describe the region S as set in terms of the variables u and v. Call this region in the u and v 1 mark variables T; (c) Define f:S Tby f(x, y) (u(x, y), v(x, y)). Compute the Jacobian matrix and the Jacobian (the determinant of the Jacobian matrix) for the coordinate transformation given by f. 3 mark 3. Let f : R3 → R2 and g : R2 → R2 be given by T1 "( Ult42 Let S be the region given by S ((x,y) E R: z2+4,z2+ 4y 0, 20,v 20 (a) Sketch the region S 1 mark Consider the change of variables given by uy, -4y. Describe egion S as set in terms of the variables u and v. Call this region in the u and v 1 mark (c) Define f: S- T by f(x, y) - (u(z, y), v(z, y). Compute the Jacobian matrix and the Jacobian (the determinant of the Jacobian matrix) for the coordinate transformation variables T given by f.

Answer 1

1

6 -0 F3

:2. 2 26 2321