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9 Nov 2019
please i need Help in Real analysis!!
Consider the polar coordinate map (x, y) = pi (r, theta) = (r cos theta, r.sin theta) defined on R2, and its behavior on the set A = [0, 1] x [0, 2 pie]. Obtain the reassuring information that the image D = pi (A), which is the unit disk D = {(x, y) : x2 +y2 1}, has content. Investigate the manner in which pi maps the boundary of A. Show that the boundary of D is the image under pi of only one side of A, and that the other three sides of A gets mapped into the interior of D. Given that the area of the circular disk {(x, y) : x2 + y2 1} is equal to pi, find the areas of the elliptical disks given by: Let B = {(u, v) : 0 u + v 2, 0 v - u 2}. By using the transformation (x, y) (u, v) = (x - y, x + y), evaluate the integral Let Psi : R2 rightarrow R2 be defined by (u, v) = Psi (x, y) = (x2 - y2, x2 + y2). Note that the inverse image under Psi of the line u = a > 0 is a hyperbola, and the inverse image under Psi of the line v = c > 0 is a circle. Show that Psi is not injective on R2, but its restriction to Q = {(x, y) : x > 0. y > 0} is an injective map onto {(u, v) : v > |u|}. Let Psi be the inverse of the restriction Psi |Q and show that if 0
please i need Help in Real analysis!!
Consider the polar coordinate map (x, y) = pi (r, theta) = (r cos theta, r.sin theta) defined on R2, and its behavior on the set A = [0, 1] x [0, 2 pie]. Obtain the reassuring information that the image D = pi (A), which is the unit disk D = {(x, y) : x2 +y2 1}, has content. Investigate the manner in which pi maps the boundary of A. Show that the boundary of D is the image under pi of only one side of A, and that the other three sides of A gets mapped into the interior of D. Given that the area of the circular disk {(x, y) : x2 + y2 1} is equal to pi, find the areas of the elliptical disks given by: Let B = {(u, v) : 0 u + v 2, 0 v - u 2}. By using the transformation (x, y) (u, v) = (x - y, x + y), evaluate the integral Let Psi : R2 rightarrow R2 be defined by (u, v) = Psi (x, y) = (x2 - y2, x2 + y2). Note that the inverse image under Psi of the line u = a > 0 is a hyperbola, and the inverse image under Psi of the line v = c > 0 is a circle. Show that Psi is not injective on R2, but its restriction to Q = {(x, y) : x > 0. y > 0} is an injective map onto {(u, v) : v > |u|}. Let Psi be the inverse of the restriction Psi |Q and show that if 0