Question

@ganeshie8

Answer 1

I have to do some transformations so, R is the parallelogram with vertices (0,0), (4,3), (2,4), (-2,1).

Answer 2

So I have the equations \[y+x/2=5\]
\[y+x/2=0\]
\[y-3x/4=0\]
\[y-3x/4=5/2\] but these equations don't exactly give me the transformation that maps a rectangular region S in uv - plane onto R

Answer 3

\[0 < y-3x/4<5/2 \implies 0 <u<5/2\]
\[0<y+x/2<5 \implies 0 <v<5\]

Answer 4

:\

Answer 5

you want to transform that parallelogram into a rectangle ?

Answer 6

Yup

Answer 7

try this
```
u = 2x+4y
v = -3x+4y
```

Answer 8

How'd you get that

Answer 9

is it working ?

Answer 10

I see what you did, sec let me see

Answer 11

Wait, what are the intervals

Answer 12

plugin the four vertices and sketch the rectangle

Answer 13

Alright

Answer 14

(x, y) ----> (u, v)
```
u = 2x+4y
v = -3x+4y
```

(0,0) ---> (0, 0)
(-2, 1) ---> (0, 10)
(2, 4) ---> (20, 10)
(4, 3) ---> (20, 0)

Answer 15

Yeah that works

Answer 16

there are several ways to get that linear transformation equations
what method did they teach you in class ?

Answer 17

the most obvious one is solving below coefficients by plugging in points :
\[u = ax+by\\v = cx+dy\]

Answer 18

We were not shown a method for transformation, just were taught about jacobians

Answer 19

Wowwwww, that method could've saved me so much time lol

Answer 20

okay next step is to find the jacobian

Answer 21

No, that's all the problem asked :P

Answer 22

What did I do wrong anyways?

Answer 23

good, usually the main purpose of doing this kindof transformation is to simplify the bounds for evaluating a volume integral

Answer 24

i didnt follow what u did.. leme go back and read again

Answer 25

I found the equation for each line

Answer 26

I see that but what exactly are you trying to do with equations of sides ?

Answer 27

define the parallelogram

Answer 28

Then I could find u and v, idk I thought it would've worked

Answer 29

okay thats not so obvious yet.. above method of finding coefficients looks pretty to me :)

Answer 30

Btw, how did you know exact points to plug in for coefficients? @rational

Answer 31

`u=2x+4y`
`v=-3x+4y`

Answer 32

one vertex of given parallelogram is (-2, 1)
decide where you want to move this point to : say (0, 1) ?

then the equations become :
\[0 = -2a+1b\\1 = -2c+1d\]

similarly get two more equations by pluggin in another vertex and sovle

Answer 33

Ah, ok! Thanks :3

Answer 34

in our earlier transformation "I wanted" (-2, 1) to move to (0, 10). so i got below two equations :

\[0 = -2a+1b\\10 = -2c+1d \]

Answer 35

So, the amount you want it to move is arbitrary right

Answer 36

You're the one who want to make that parallelogram a rectangle

so you get to choose the dimensions of rectangle too

Answer 37

Cool cool, thanks :P

Answer 38

you can map that parallelogram to a rectangle of ANY dimension and jacobian reflects this... so nothing to wry