Stanford University Polar Coordinates & ?Orientations Diffeomorphism Proof Real Analysis proof. Need some complete proof about these two questions. Thank y

Stanford University Polar Coordinates & ?Orientations Diffeomorphism Proof Real Analysis proof. Need some complete proof about these two questions. Thank you. 9.1. Define f: R2 → R by
x3 – 3xy2
f(x,y)=r(cos 30)
x2 + y2
and f(0,0) = 0. Prove that f is Lipschitz. (Note and hint: This is the same function as
problem 5.1(a) and in lecture for a different purpose. f is not a C function, but you can
follow the proof from lecture that C functions on bricks are Lipschitz. Namely, if the
derivative of f is bounded, you can integrate the derivative you can show that its derivative
is bounded and integrate it. The origin is a special case because f is not differentiable there;
although it does have directional derivatives.)
9.2. Define a region UPC R2 that consists of two particular open squares:
π π 2 1 2 2
U
6’6 N 3
Also define a map ğ: U2 + R2 as follows:
(s, t) = g(x,y) = (x+2(siny),y+2(sinx)).
The goal of this problem is to understand g as a change of variables on U2.
(a) Draw (s,t) coordinates in the region U2 the (x,y) plane as you did in problem 8.1. Draw
a pair of coordinate directional vectors a/as and a /at at least one point (x,y) in each
of the open squares of U2. (Note: You can use graphing software such as the excellent
Desmos to help you. However, if it does not label everything that you want, you should
print it out or copy it to add labels somehow, or add explanation to make a complete
answer. Also, if using color, please make the curves with constant s one color and the
curves with constant t a second color.)
(b) Prove that ğ is a Cl diffeomorphism from U2 to its image V2 = g(U2). (Note: It is a bit
tricky to prove that g is one-to-one onto its image. You can either give a computational
proof; or for this part of the question, you can give an informal argument referring to
your answer in part (a).)
(e) Find orientations of U2 and V2 to make g an oriented diffeomorphism from Ű2 to V2.

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