The Jacobian is a magnification (or reduction) factor that relates the area of a

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The Jacobian is a magnification (or reduction) factor that relates the area of a small region, or neighborhood, near a point (u, v) in R2 (uv-plane), to the area of that region near the point () in R (ry-plane), where of the preimage T:x=g(u, e) and y=h(u,v) is a one-to-one transformation Suppose Sis a little rectangle in the uu-plane with vertices (0,0) (Au, 0), (1 v), (0,At) The preimage of S under the transformation given above is a small region R in the ry-plane The arrows () below, and the picture, indicate the respective preimages of each of the following points (0,0) = O O, x = g(u, v) y = h(u, v) P' o' (a) (3 pts) Write down the coordinates for each of the points O', P.Q The text instead uses the term image and writes T(S) R. The reason for the discrepancy is delicat T : { unknowns in the rs and ys } { unknowns in the us and vs) (Algebraic Paradigm) T : { known values in the us and vs) { corresponding z-and y-values) (Geometric Paradigm) and relevant to the field of Algebraic Geometry. The transformation T can be described in two ways: or

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