7.1. (Sec. 7.2) A transfonnation from rectangular to polar coordinates is

(a) Prove w 2 = r.y~. [Hint: Compute in turn Y; + Y;_I'(Y;+Y;_I) +Y;-2' and so forth.]

(b) Show that the Jacobian is w" -I cos"-2 /II cos" -3 /12 .. , cos /I" -2' [Hint: Prove

where x denotes elements whose explicit values are not needed.]

y = w sin 01, w cos 0 sin 02, - y3 w cos 0, cos 0 sin 03, Yn-1=wcos cos 02 cos 8-2 sin 8,,-1 = y, w cos 0, cos ... cos -2 cos 0-1- - where - < 0 , i = 1,...,n2,