The 2D case of Jocabian determinants is easy:
If you want to express the area of dxdy by dudv, where (x,y) and (u,v) are different coordinate systems and d stands for differentiate.
dxdy= (dx/du dy/dv - dx/dv dy/du) dudv
which makes sense in geometry. The Jocabian Determinant in the parenthesis is in deed the area of dxdy in coordinate (u,v)
But how about n-dimensional case?
Is there a general proof for transformation of coordinates?
I figured it out
Determinant shares the same property with "volume"
x -> ax V->aV
x -> x+y V->V
x,y -> y,x V->-V
So it is reasonable to define a volumn in Rn by its determinant