Can you help with question 10? I attached theorem 2.7.2!! SM 6. Consider the transformation from R? to R? determined by (x1, x2) → (yi, y2), whereZylr = K(*)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 doesto the rectangle determined by 1 sxI S2, 1/2 s x2 2/3. Draw a figure!7. Consider the linear transformation T: (x, y) → (u, v) from R² to R² determined by u = ax+by,v = cx + dy, where a, b, c, and d are constants, not all equal to 0. Suppose the Jacobiandeterminant of T is 0. Then, show that T maps the whole of R? onto a straight line through theorigin of the uv-plane.SM 8. Consider the transformation T: R → R defined by T(r, 8) = (r cos 0 , 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 27. 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 equationsu = f(x, y), v = g(x, y)can be solved for x and y locally. Show that if the solutions are x = F (u, v), y = G(u, v), andif f, g, F, and G are C', thende%3DaG8e Inenewhere J denotes the Jacobian determinant of f and g w.r.t u and v.M 10. (a) Consider the system of equations1+ (x + y)u – (2 + u)'+v = 00 = (1-1)»³(Kx +1) – nzUse Theorem 2.7.2 to show that the system defines u and v as functions of x and y in anopen ball around (x, y, u, v) = (1, 1, 1, 0). Find the values of the partial derivatives of thetwo functions w.r.t. x when x = 1, y = 1, u = 1, v = 0.(b) Let a and b be arbitrary numbers in the interval [0, 1]. Use the intermediate value theorem(see e.EMEA, Section 7.10) to show that the equationhas 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 uand v of the system. Are u and v uniquely determined?PassengerQueue.javaer.javaRemoved f(x, y) are the m x m and m xn Jacobian matrices of f(x, y) with respect to y and x,respectively. From (4) we can obviously find the partials of y1,... Ym with respect to x1,..., Xn provided f(x, y) is invertible.The matrix f,(x, y) is square. Its determinant is called the Jacobian determinant of("Sa(yı..... Ym)f(x, y) w.r.t. y and is commonly denoted byIn the argument leading to (4) we assumed that (2) defines y1, ..., ym as differenti-able functions of x1, ., xn. The following theorem, one of the most important results inmathematical analysis, gives sufficient conditions for this to be the case:THEOREM 27.2 (THE IMPLICIT FUNCTION THEOREM GENERAL VERSIONSuppose f = (fı,.., fm) is a C' function of (x, y) in an open set A in R" × R",and consider the m-dimensional vector equation f(x, y) = 0. Let (xº, yº) be aninterior point of A satisfying f(x, y) = 0. Suppose that the Jacobian determinantof f w.r.t. y is different from 0 at (x°, yº), i.e.#0 at (x, y) = (x°, yº)(5)a(yı, ...., Ym)%3DThen there exist open balls B1 in R" and B2 in R" around xº and yº, respectively,with B1 x B2 C A, such that If (x, y)| # 0 in B1 × B2, and such that for eachx in B1 there is a unique y in B2 with f(x, y) = 0. In this way y is defined"implicitly" on Bị as a Cl function g(x) of x. The Jacobian matrix y = g'(x) =(ag: (x)/ax;) is%3D(9)NOTE 1 Suppose fi,..., fm are C' functions of (x, y). Then the elements of the matricesin (6) are all C'-l functions, and it follows that g = (g1, . .., gm) is C"Transformations and Their InversesMany economic applications involve functions that map points (vectors) in R" to points(vectors) in R™. Such functions are often called transformations or mappings. For ex-ample, we are often interested in how an m-vector y of endogenous variables depends onan n-vector x of exogenous variables, as in the implicit function theorem.Consider a transformation f : A → B where A C R and BC R". Suppose the rangePassengerQueue.javaRemoved|甲V9

Name: Can you help with question 10? I attached theorem 2.7.2!! SM 6. Consid
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Description: Can you help with question 10? I attached theorem 2.7.2!! SM 6. Consider the transformation from R? to R? determined by (x1, x2) → (yi, y2), whereZylr = K(*)Compute the Jacobian determinant of this transformation, and find the inverse (where it exist

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Can you help with question 10? I attached theorem 2.7.2!! SM 6. Consider the transformation from R? to R? determined by (x1, x2) → (yi, y2), whereZylr = K(*)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 doesto the rectangle determined by 1 sxI S2, 1/2 s x2 2/3. Draw a figure!7. Consider the linear transformation T: (x, y) → (u, v) from R² to R² determined by u = ax+by,v = cx + dy, where a, b, c, and d are constants, not all equal to 0. Suppose the Jacobiandeterminant of T is 0. Then, show that T maps the whole of R? onto a straight line through theorigin of the uv-plane.SM 8. Consider the transformation T: R → R defined by T(r, 8) = (r cos 0 , 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  27. 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 equationsu = f(x, y), v = g(x, y)can be solved for x and y locally. Show that if the solutions are x = F (u, v), y = G(u, v), andif f, g, F, and G are C', thende%3DaG8e Inenewhere J denotes the Jacobian determinant of f and g w.r.t u and v.M 10. (a) Consider the system of equations1+ (x + y)u – (2 + u)'+v = 00 = (1-1)»³(Kx +1) – nzUse Theorem 2.7.2 to show that the system defines u and v as functions of x and y in anopen ball around (x, y, u, v) = (1, 1, 1, 0). Find the values of the partial derivatives of thetwo functions w.r.t. x when x = 1, y = 1, u = 1, v = 0.(b) Let a and b be arbitrary numbers in the interval [0, 1]. Use the intermediate value theorem(see e.EMEA, Section 7.10) to show that the equationhas 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 uand v of the system. Are u and v uniquely determined?PassengerQueue.javaer.javaRemoved f(x, y) are the m x m and m xn Jacobian matrices of f(x, y) with respect to y and x,respectively. From (4) we can obviously find the partials of y1,... Ym with respect to x1,..., Xn provided f(x, y) is invertible.The matrix f,(x, y) is square. Its determinant is called the Jacobian determinant of(&#34;Sa(yı..... Ym)f(x, y) w.r.t. y and is commonly denoted byIn the argument leading to (4) we assumed that (2) defines y1, ..., ym as differenti-able functions of x1, ., xn. The following theorem, one of the most important results inmathematical analysis, gives sufficient conditions for this to be the case:THEOREM 27.2 (THE IMPLICIT FUNCTION THEOREM GENERAL VERSIONSuppose f = (fı,.., fm) is a C' function of (x, y) in an open set A in R&#34; × R&#34;,and consider the m-dimensional vector equation f(x, y) = 0. Let (xº, yº) be aninterior point of A satisfying f(x, y) = 0. Suppose that the Jacobian determinantof f w.r.t. y is different from 0 at (x°, yº), i.e.#0 at (x, y) = (x°, yº)(5)a(yı, ...., Ym)%3DThen there exist open balls B1 in R&#34; and B2 in R&#34; around xº and yº, respectively,with B1 x B2 C A, such that If (x, y)| # 0 in B1 × B2, and such that for eachx in B1 there is a unique y in B2 with f(x, y) = 0. In this way y is defined&#34;implicitly&#34; on Bị as a Cl function g(x) of x. The Jacobian matrix y = g'(x) =(ag: (x)/ax;) is%3D(9)NOTE 1 Suppose fi,..., fm are C' functions of (x, y). Then the elements of the matricesin (6) are all C'-l functions, and it follows that g = (g1, . .., gm) is C&#34;Transformations and Their InversesMany economic applications involve functions that map points (vectors) in R&#34; to points(vectors) in R™. Such functions are often called transformations or mappings. For ex-ample, we are often interested in how an m-vector y of endogenous variables depends onan n-vector x of exogenous variables, as in the implicit function theorem.Consider a transformation f : A → B where A C R and BC R&#34;. Suppose the rangePassengerQueue.javaRemoved|甲V9

Accepted Answer

Given the system of equations is
1+x+yu-2+u1+v=02u-1+xyeux-1=0
Let us consider two functions…