Question

Let 0< a<b<e<d for a, b, c, d E R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation x=au + bu, y=cu + dv. (a) Sketch D in the r-y plane for the case ad -bc > 0. (a) Sketch D in the r-y plane for the case ad bc < 0. (c) Calculate the area of D. Show all working.

Answer 1

S-s(u, U 0s us1, 0 susi} so, s is shown beiow Now, given the linear transformation whex, "D'is the xegion in the xy-plane that is the image of S under the Linear transformation The line u=1 in the ỵy-plane is given as using transFormat eliminote 'o' rom the equatioms to obtain Likewise, For line u 0

NDw, Fo line v-1 Eliminate u From both the equations, to obtan, oy-cx - ad- bc and, For Line v-o au and y cu , eliminate 'u', to obtain, (o) Cven, ad- bc >0, so the region D in the xy-plane is given by the lines (1) to (4), bounding the since a, b, c, d >0 such that d>b and c> a so, the sbpes gven by the lines are positive ie and >. Also, ad-bc 0, he efore d-So divida both sides by ab >o) , so, lines given by (1) and (2) are a. parallel and have dope grea ter than ins (3) and (H), ard lines (3) and (4) are also parale)

Thus, he sketch is given as ()(ad-bc) ad-bc〈0, there foe (b) since, L 〈0 ( divide both sides by ab Ca Cl S0, lines given by 1) and (2) have slope less than lings (3) and (4) . Tids the stetch ỉ0,-

Consider the egon 'I OF part (a), Г.e. . C) (0,0 To calculete the area 0 Determine the Jacobian associaled with the transfor Ou U LA ou Olv -ad-bc o( u, v) Using the linearg transformation to map D' to rgim 'S'

dudu so. the area unde YHengion is given as lad-bcl, i.e. For egion in part (a) A-ad-bc and,ad-bc 0) for gien in part (b)