Question
Asked Nov 18, 2019
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can u help me with pullbacks and integration plz

2) Use a pullback to evaluate
xy dxdy
R
where R is the parallelogram with vertices at (-1,-2), (3, 0), (2,2), (-2, 0)
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2) Use a pullback to evaluate xy dxdy R where R is the parallelogram with vertices at (-1,-2), (3, 0), (2,2), (-2, 0)

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Step 1

Given that the integral,

ydrdy
JR
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ydrdy JR

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Step 2

here, R is the parallelogram with vertices at (-1, -2), (3, 0), (2, 2), and (-2, 0).

Equation of line through the point (-1, -2), (3, 0) is

x - 2y = 3

Equation of line through the point (3, 0), (2, 2) is

2x + y = 6

Equation of line through the point (2, 2), (-2, 0) is

x - 2y = - 2

Equation of line through the point (-2, 0), (-1, -2) is

2x + y = - 4

Assume that,

x - 2y = u

and

2x + y = v

Then, u = 3 and u = - 2

           v = 6 and v = - 4

Also,  x - 2y = u

2x + y = v

This implies that,

1
(u+2v)
and
1
y v2u)
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1 (u+2v) and 1 y v2u)

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Step 3

Use the pullback (method of change of variable), for this find the value of Jacobian ...

J =
1
-2
2
1
= 5
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J = 1 -2 2 1 = 5

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