Question

Let 0 < a <b<e<d for a, b, c, d E R. Consider the set S={(u, ujo < u < 1, 0<u<1) and let D be the region in the r-y plance that is the image of S under the variable transformation (a) Sketch D in the r-y plane for the case ad - be>0. (a) Sketch D in the r-y plane for the case ad - be < 0. (c) Calculate the area of D. Show all working.

Answer 1

Now given the linear tronsformahion Whee,' D, is the region in the χy-plane that is the image of S under the Linear transpormation lhe Line us in the xy-plane is given as using tronsformahion eliminate 'u' crom the equatioms to obtain dz-by = ad-bc ,or す Likewse, For line u0 ỵEbu and yedU

Now, for line v-1 Eliminate 'u' Fyom both the equations, to obtain, al and, For Line v-o ỵ au and y: cu , eliminate 'Lúto obtain. (a) vn, ad- bc >0, so the yegion D in the xy-plone is given by the leso 4), bounding the Since a, b, c) d > 0 such ihat d > b and c > a so, the shpes gien by the lines are positive ie and C. Also, ad-bc >0, therefore, d d - S >o (divida both sides by ab o so, lines given by (i) and (2) a , re parallel and have slope greater than ines (3) and (4!), and lines (3) and (4) are also paralel

Thus, the sketh is given as ()(ad-bc) На (b nce, ad-bc 0, therefoe d-LCO ( divide both sides by ab>0) b a so, lines given by ) and (2) have slope less than lines (3) and (4) . Ћиs ½he sketch isayA

(C) Consider the region D 0F poyt (o), l'e- .> (0,0 To calculate the area Determine the Jacobian associaled wih the transformoti m,i.e, Lu au 0u olu OU (x,4) o( u, v) Using the linearg transformathion to map D' to regim 'S'

A ad-bo dd u 0 so, the area under the vegion is given os lad-bc), i.e For region in part (a) A - ad-bc and, (', ad-bc >o) for region in port (b) A- bc-ad ad-bc < o)