SEhahon from R to R determined by (x1, x2) + (yı, y2), where %3D (*) Compute the Jacobian determinant of this transformation, and find the inverse (where it exists) by solving the system of equations in (+) for x1 and x2. Examine what the transformation does to the rectangle determined by 1 SIS2, 1/2 s x252/3. Draw a figure! 7. Consider the linear transformation T: (x, y) ► (u, v) from R² to R determined by u =. v = cx + dy, where a, b, c, and d are constants, not all equal to 0. Suppose the Jacobian determinant ofT is 0. Then, show that T maps the whole of R onto a straight line through the origin of the uv-plane. ax+by, O 8. Consider the transformation T:R? → R² defined by T(r, 8) = (r cos e,r sin 0). (a) Compute the Jacobian determinant J of T. (b) Let A be the domain in the (r, 0) plane determined by 1 2n. Show that J 0 in the whole of A, yet T is not one-to-one in A. 9. Give sufficient conditions on f and g to ensure that the equations (K x)8 = a '(K 'x)f = n can be solved for x and y locally. Show that if the solutions are x = F (4, v), y = G(u, v), and if f, g, F, and G are C', then se 1 xe r aF aG J ay ne %3! ng where J denotes the Jacobian determinant of f and g w.r.t u and v. M 10. (a) Consider the system of equations 1+ (x +y)u - (2+ u)!+" = 0 0 = (1-n(Kx + 11) - nz Use Theorem 2.7.2 to show that the system defines u and v as functions of x and y in an open ball around (x, y, u, v) = (1, 1, 1, 0). Find the values of the partial derivatives of the two functions w.r.t. x when x = 1, y = 1, u = 1, v = 0. %3D (b) Let a and b be arbitrary numbers in the interval [0, 1]. Use the intermediate value theorem (see e.g. EMEA, Section 7.10) to show that the equation 0 = (1-9»³D – n has a solution in the interval (0, 1]. Is the solution unique? (c) Show by using (b) that for any point (x, y), x E [0, 1], y e (0, 1, there exist solutions u and u of the SYstem. Are u and y uniquely determined?
Unitary Method
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Can you help with question 10? I attached theorem 2.7.2!!
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