Consider the change of variables T(x, y) = (u(x, y), v(x, y)), where u(x, y) =...

 

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Consider the change of variables T(x, y) = (u(x, y), v(x, y)), where u(x, y) = x + 2y and v(x, y) = x-2y. If z and w are related by the equation z(x, y) = w(u, v) = w(r+2y, x-2y): a) Compute the first order partial derivatives of z, that is(x, y) and (z,y), in terms of w, u and u only, using the Chain rule. This means, for some real constants a, b, c, d, Əz (r, y) =a- дг dw du b) Verify that dw (u, v) + b (u, v) and Əv 20 (x,y) + dz 2 Əz Əy Әх (x, y) = c дz ду c) Write z(x, y) = sin(x-2y) + (x - 2y)³ in terms of u and u and verify that dw du Əz (x,y) + (x, y) =0 Əy (x, y) = 4 (u, v) du əw (u, v) + d(u, v) ὃν Consider the change of variables T(x, y) = (u(x, y), v(x, y)), where u(x, y) = x + 2y and v(x, y) = x-2y. If z and w are related by the equation z(x, y) = w(u, v) = w(r+2y, x-2y): a) Compute the first order partial derivatives of z, that is(x, y) and (z,y), in terms of w, u and u only, using the Chain rule. This means, for some real constants a, b, c, d, Əz (r, y) =a- дг dw du b) Verify that dw (u, v) + b (u, v) and Əv 20 (x,y) + dz 2 Əz Əy Әх (x, y) = c дz ду c) Write z(x, y) = sin(x-2y) + (x - 2y)³ in terms of u and u and verify that dw du Əz (x,y) + (x, y) =0 Əy (x, y) = 4 (u, v) du əw (u, v) + d(u, v) ὃν