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10 Nov 2019
1. Let F : R2-R2 be the transformation defined by F(z,y)- (2+ ??.??-z?) for all (z.yje R2 a. Show that F is smooth and find its differential matrix [dF(x,y)]. Use the Inverse Function Theorem to prove b. Let V ((ry) E R2:r,y >0. Show that the restriction Fly is one-to-one and that the image W FV) c. Find a formula for the function G : W ? R2 such that (Go F)(z,y) = (z,y) for all (z,y) E V and show that d. Verify that [dG(u, v) dFG(u, for (u, ) E W using parts (a) and (c) above. that F has a smooth local inverse at a-(a,a2) if a 0 and a2 0 is open (explicitly describe w). G is smooth directly. Find the differential matrix [dG(u, v)] for (u v) E W using this formula.
1. Let F : R2-R2 be the transformation defined by F(z,y)- (2+ ??.??-z?) for all (z.yje R2 a. Show that F is smooth and find its differential matrix [dF(x,y)]. Use the Inverse Function Theorem to prove b. Let V ((ry) E R2:r,y >0. Show that the restriction Fly is one-to-one and that the image W FV) c. Find a formula for the function G : W ? R2 such that (Go F)(z,y) = (z,y) for all (z,y) E V and show that d. Verify that [dG(u, v) dFG(u, for (u, ) E W using parts (a) and (c) above. that F has a smooth local inverse at a-(a,a2) if a 0 and a2 0 is open (explicitly describe w). G is smooth directly. Find the differential matrix [dG(u, v)] for (u v) E W using this formula.
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