Question

How do I find the double integral by using transformation equations.

Given x-y =u

x+y =v

evaluate ſſR (x-y)^4*e^(x+y)dxdy

where R is the Square with vertices (1,2), (0,1), (2,1) and (1,0),.... anybody there to help???

ſ ſ is the double integral

Answer 1

looks like we can start with a simple replacement:

(x-y)^4 e^(x+y) --> u^4 e^v

swapping the dxdy part tho; we need to define dxdy in terms of dudv, or dvdu

Answer 2

u = x-y

du = 1dx


v = x+y
dv = 1dy

im sure that off

Answer 3

You need to multiply by the jacobian of the transformation \[dxdy=\frac{\partial(x,y)}{\partial(u,v)}dudv\]

Answer 4

yes, the jacobian seems to apply here :)

Answer 5

but what is he limit R need to apply here
??

Answer 6

R is actually a square vertices (1,2), (0,1), (2,1) and (1,0) ?

Answer 7

Given x-y =u, and x+y =v

and a lot of points (x,y), you can determine the transformation from xy to uv

Answer 8

(1,2), (0,1), (2,1) and (1,0)

x-y =u, and x+y =v
1 2 1 2

(1,2) -> (-1,3)

======================

x-y =u, and x+y =v
0 1 0 1

(0,1) -> (-1,1)


etc