Substitution for double integrals

[WhataRecch]

Homework Statement

Find a transform T that maps the unit square in the u-v plane to a quadrilateral with corners (1,2), (3,3), (4,2) and (2,1) to the x-y plane.

Homework Equations

The Attempt at a Solution

I've been able to create the proper region in the x-y plane when I have the transform T, but I have no clue how to come up with the transform in this case. The furthest I've gotten was coming up with the equations for the four sides in the quadrilateral, but I still hit a dead end.

 

[lanedance]

so let's say
T(0,0) = (1,2)
T(1,0) = (4,2)
T(0,1) = (2,1)
T(1,1) = (3,3)

try writing out the transformation, T as a linear matrix M with offset u. take an intial point p.

then let's express it as
<br /> T(\vec{p}) = M \vec{p} + \vec{u}<br />

writing it out components explicitly
<br /> T(p,q) = \begin{pmatrix} a &amp; b \\ c &amp; d\end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix} + \begin{pmatrix} u\\ v \end{pmatrix}<br />
(p,q) is the initial point

then use the known transformations to find a,b,c,d,u,v

hopefully this is sufficient to represent your transformation