UCLA: Math 32B Week 4 Worksheet Spring 20

1. (a) Find a change of coordinates map G that takes the unit square [0, 1] ⇥ [0, 1] to the parallelogramwith vertices (0, 0), (2, 1), (1, 2)(3, 3).

(b) Find the Jacobian of G.

(c) Find a change of coordinates map G0 that takes the unit square [0, 1] ⇥ [0, 1] to the parallelogramwith vertices (2, 1), (4, 2), (3, 3), (5, 4).

(d) Find the Jacobian of G0 and give a geometric explanation for the similarity between the Jacobianof G and that of G0.

2. Consider the region D defined by 1 x2 � y2 4 and 0 y 3x

5. In this problem you’ll set up an

integral to compute

ZZ

Dex

2�y2

dA.

Consider the change of coordinates G(u, v) =⇣v2+

u

2v,v

2� u

2v

⌘. Recall from class that the inverse of

this coordinate change is given by G�1(x, y) = (x2 � y2, x+ y)

(a) Find a region R of the uv-plane so that G : R ! D is a change of coordinates map (so G is ontoand one-to-one on the interior of R). Hint: Start by finding 4 curves in the uv-plane that map tothe 4 curves forming the boundary of D.

(b) Give an iterated integral in uv-coordinatees to compute

ZZ

Dex

2�y2

dA (No need to compute the

actual integral, but it is an integral you can compute).

3. Consider the region of the part of the first quadrant D defined by 1 x2 + y2 4 and 1/10 xy 1/2and y � x. There is a change of coordinates G that takes the rectangle [1, 4]⇥ [1/10, 1/2] in the uv-planeto D, and the inverse of this change of coordinates is given by G�1(x, y) = (x2 + y2, xy).

Solutions

II

II

UCLA: Math 32B Week 4 Worksheet Spring 20

(a) What is the absolute value of the Jacobian of G�1? (It should be a function of x and y). Payattention to signs!

(b) Compute

ZZ

Dy2 � x2 dA

(c) Bonus problem: Note that the system of inequalites 1 x2 + y2 4 and 1/10 xy 1/2 definesfour di↵erent regions of the plane. Each of these regions can be described by a change of coordinatesG that takes the rectangle [1, 4] ⇥ [1/10, 1/2] in the uv-plane to the region where again inverse ofthis change of coordinates is given by G�1(x, y) = (x2 + y2, xy), but for each of these regions Gitself has a di↵erent formula. Find all 4 formula for G and say which of these four regions goes withwhich formula.

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b ie's L H e i 3

c This is thesameparallelogram but shiftedby 12,11 Sowe useGu v 2 l t Glu v 2 2ut v I t ut wandwe alsohave

d Je 11 3

Nb I don't recommendcallingthismap6 sincethematrix

Yi fro is usuallycalledG

Za I ET IE ET If 4 Ey If If 21u

so comesp to xl y LLikewise comesp A H y dy O corresponds to Vz In O v2 u O u v2 u fu

y 2 comesp to IT f Eu a f itr Kun I a use revert LILY I fado

Henceweusethepositivesquareroot

b 2612 IT 264g Iz Zu t7642 Yu 264g Iz t Izu2sotheJacobianis

det6 I Ef Lutz EH jZFf e TDA J e ldetdlu.nldudu en Iv dudu

r r

igeld 1 log2

Za det 12 41 262 y4

Since y x in this region IdelLG I't 212 y4 215 x4b Thus bythe changeofvariables formula

y a duly I t ldxdy Itzhak f H H th YeoD GDlt n242 H 11,4 10,41

c u x2ty2xy

solve furx y

u x2 y2 YIHu 6212 v2Xi x2u 2 02 u M

2

x ft

yV intermsofx

choosingthesigns f 2pmfor I t lu u

Il t t t

T1 t

IV