Question

find the image of the set S under the given transformation
S={(u.v)|0<=u<=3, 0<=v<=2}
x=2u+3v, y=u-v
step by step, please

Answer 1

Isn't this still linear algebra? ^.^

Answer 2

no, it's cal3

Answer 3

part find jacobian

Answer 4

you are very good at cal, right?

Answer 5

Well,
\[\huge (u,v) \rightarrow (x,y)\]
\[\huge \left[\begin{matrix}x \\ y\end{matrix}\right]=\left[\begin{matrix}2 & 3 \\ 1 & -1\end{matrix}\right]\left[\begin{matrix}u \\ v\end{matrix}\right]\]

Answer 6

so?

Answer 7

Partial derivatives... should be simple, no? ;)

Answer 8

\[\huge \left[\begin{matrix}\frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v}\end{matrix}\right]\]

Answer 9

i know how to find jacobian number, just don't understand how to use it in this part

Answer 10

typo...

Answer 11

ok, ignore that typo, i understand

Answer 12

But the best way to do this is visualise the corners :)

Answer 13

Or, you could just substitute...

Answer 14

Solve for u and v in terms of x and y.

Answer 15

It's interesting to note that the jacobian of a linear transformation is itself :D

Answer 16

so, to find the image of (u,v), first find the jacobian matrix , second, ?

Answer 17

I don't know this method you are using. :/

Answer 18

neither I, that's why I am here to make question:(

Answer 19

I just substitute, and express u in terms of x and y.

Answer 20

and v, too.

Answer 21

ok, show me steps, please

Answer 22

So...
\[\huge u = \frac{x+3y}{5}\]
\[\huge v = \frac{x-2y}{5}\]

Answer 23

You could arrive at these^ if you solve
x = 2u + 3v
y = u - v

as a system of equations of u and v, considering x and y as constants.

Answer 24

So just replace the u and v with these expressions of x and y
\[\large \bar{S}=\left\{(x,y) \ | \ 0\le x+3y \le 15 \ \ , \ \ 0 \le x-2y \le10\right \}\]

Answer 25

that's it? ha!! thanks a lot, talent

Answer 26

It's Terence :P

Answer 27

hey, I saw you use :P twice, what does it mean?

Answer 28

A playful tease :)
Like a tongue-out gesture.

Answer 29

ok, young boy, be helpful, especially to older like me. thanks a lot. good night

Answer 30

It's not even noontime yet o.O

Answer 31

night here!!!11:13pm