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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) dudvwhich 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 outDeterminant shares the same property with "volume"x -> ax V->aVx -> x+y V->Vx,y -> y,x V->-VSo it is reasonable to define a volumn in Rn by its determinant
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