M3220-1 Tenth Homework Assignment

MATH 3220 § 1 TENTH HOMEWORK ASSIGNMENT Due Friday,

A. Treibergs November 9, 2007.

MATH 3220 § 1	TENTH HOMEWORK ASSIGNMENT	Due Friday,
A. Treibergs		November 9, 2007.

You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.

Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor.

9.6[1, 2*, 3, 7, 8*, 9]

Please do the following additional exercises.

A*. Let F : R² → R² be given by

x = u² - v², y = 2 u v;
Find an open set U ⊆ R² such that (3,4) ∈ U and V = F(U) is an open set, and find a C¹ function G : V → U such that
G o F (u,v) = (u,v) for all (u,v)∈U and
F o G (x,y) = (x,y) for all (x,y)∈V.
Find the differential dG(F(3,4)).
(G is a local inverse. Solve for G and check its properties. Do not use the Inverse Function Theorem, which guarantees the existence of local inverse near (3,4) assuming F is continuously differentiable near (3,4) and dF(3,4) is invertible.)

You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.
Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor. 9.6[1, 2, 3, 7, 8, 9] Please do the following additional exercises. A. Let F : R² → R²* be given by x = u² - v², y = 2 u v; Find an open set U ⊆ R² such that (3,4) ∈ U and V = F(U) is an open set, and find a C¹ function G : V → U such that G o F (u,v) = (u,v) for all (u,v)∈U and F o G (x,y) = (x,y) for all (x,y)∈V. Find the differential dG(F(3,4)). (G is a local inverse. Solve for G and check its properties. Do not use the Inverse Function Theorem, which guarantees the existence of local inverse near (3,4) assuming F is continuously differentiable near (3,4) and dF(3,4) is invertible.)