Hello all,
As per the title, I'm a stuck on a question, that involves a change of coordinate system. I am keen to do through the question myself afterhopefully resolving my queries. Here is the question, and below it the queries:
x and y are related to u and v by: x = u+v, y = uv
A region, S, of the xy plane corresponds to the trianglein the uv plane with vertices (u,v) = (0,0) (0,1) (1,1)
Find the area of S in the xy plane.
Queries:
1. Is it explicit that the Jacobian determinant method is onlyvalid for going from or to the orthogonal/Cartesian to new system (say(a,b)). Or does the Jacobian cope when both sets of variables arenon-orthogonal ie not involving Cartesian?
2. From using the Jacobian method (sorry I am clueless about inserting equations on website fields so will type in long hand):
dxdy = mod(del(x,y)/del(u,v))*dudv
I get a new double integral, where f(x,y) must be set =1whenever one requires the 2D area of the planar domain/not the usual volumebeneath the surface. However it seems unsolvable:
double integral, limits 0 to 1, 2y to y respectively of(dxdy)/(u-v)
Basically I clearly don't really understand the double integralchange of variable method and would be very grateful for some assistance!
Also, if someone could provide instruction on how to insertproperly formatted equations into a website field like a number of the regular contributors on this forum, I would be doubly grateful.