Question

this is bit vague... but anyway...
if the boundaries of some region in xy plane were defined by some functions (for example y=4x etc etc...)

and we had the co-ordinate transformation u=g(x,y) and v=h(x,y)

now if we had to find the corresponding region in u-v plane, is it always the case that the boundary y=4x (which might become something like u=x^2) is also a boundary of the region in uv plane?

my brains a bit fuzzy... let me know if you atleast understood what i'm trying to say

Answer 1

I can imagine a transformation where everything gets mapped to one point, and we lose the "boundary". I imagine there are conditions on the transformation that guarantee that you preserve the boundary, and "inside points" remain "inside points" in the transformed space.

Answer 2

thanks!! so you do get what i'm trying to say...

the general procedure for evaluating double integrals by changing coordinates involves transforming a function y=f(x) defining the boundary to u=g(v),... and then taking it for granted that u=g(v) will turn out to be a boundary in the transformed region...
i just want to check if i'm missing something obvious about this whole business... or do i have the procedure wrong?

Answer 3

@phi

" there are conditions on the transformation that guarantee that you preserve the boundary, and "inside points" remain "inside points" in the transformed space. "

are there such conditions? can i put that into my notes as a remark?

Answer 4

@ganeshie8 any thoughts?

Answer 5

I don't know the conditions, but I guess most reasonable transformations will work. In the real world, if you ever try to solve a problem using a transformation, you will know or be able to check that it works.

Answer 6

thanks!! :)