2. First, read this definition. The Jacobian of a transformation T given by x = g(u, v) and y = h(u, v) is (x,y) (u, v) =xuyvxvYu - Assuming the transformation T is C¹ (meaning g and h have continuous partial derivatives) and T is a one to one transformation from a region R in the xy-plane to a region S in the uv-plane, then √ f(x, y)dA = f(z(u, v), y(u, v)) | [] (x,y) |(u, v) dudv Note the absolute value around the Jacobian) This is how we can do a substitution in a double integral! Now answer the following questions. (a) If R is the square with vertices (0,2), (1, 1), (2, 2), (1,3) and u = xy, v = x + y, sketch the new region in the uv-plane. Be sure to label your graph and provide coordinates of all vertices. 1 (b) Now find the Jacobian of the transformation (c) Now use the information above to integrate the following double integral. R - Y dA x + y

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