5. Let ф: S1 S2 be a diffeomorphism. a. Show that S is orientable if and...
*1. Let S2((x, y, z) e R3:xy+2 be the unit sphere and let A: S2 S, be the (antipodal) map A(x, y, z)-(-x,-y,-z). Prove that A is a diffeomorphism.
(a) A diffeomorphism : S1 S2 is area-preserving if the area of any region Rc S is equal to the area of 4(R). Show that if is area-preserving and conformal, then is a local isometry (b) Show that the Mercator's projection (defined in do Carmo, chapter 4-2, exercise 16) is not area ea-preserving (c) Lambert's cylindrical projection projects L = S2 \ {N,S}, the unit sphere minus the north pole N and the south pole S, into a the unit...
please i need the question 15 for the detailed proof and explaination ! thanks ! 233 42 Isometries, Conformal Maps 14, we say that a differentiable map ф: S,--S2 preserves angles when for every p e Si and every pair vi, v2 E T (S,) we have cos(u, 2) cos(dp, (vi). do,()). Prove that pis locally conformal if and only if it preserves angles. 15. Letp: R2 R2 be given by ф(x, y)-(u (x, y), u(x, y), where u and...
2. 9 marks] Strings. Consider the following definitions on strings Let U be the set of all strings. Let s be a string. The length or size of a string, denoted Is, is the number of characters in s Let s be a string, and ie N such that 0 Si< Is. We write si] to represent the character of s at index i, where indexing starts at 0 (so s(0 is the first character, and s|s -1 is the...
I need help trying to understand what (S1) and (S2) are saying. Maybe in other words or pictures because the book is more confusing 3.1.1. Let M CR" be a nonempty set and 1 s k n. Then k . Then M is a -dimensional regular surface (briefly, regul each point xo there ar kf class CP (p)i nd amapping of class C e M there exist an open set AC such that (SI) there exists an open set U...
please explain steps. I know U(f,P)-L(f,P)= something that *16. Let S = {S1, S2, ..., Sk} be a finite subset of [a,b]. Suppose that f is a bounded function on [a, b] such that f(x) = 0 if x € S. Show that f is integrable and that sa f = 0.
2. 9 marks] Strings. Consider the following definitions on strings Let U be the set of all strings Let s be a string. The length or size of a string, denoted Is, is the number of characters in s Let s be a string, and i e N such that 0 < ί < sl. We write s[i] to represent the character of s at index i, where indexing starts at 0 (so s 0] is the first character, and...
Let us start with the usual conditional probability exercise Let Sn be the random walk S, -So + 61 +...+ En such that Ei €{+1} are iid with P(Ei=1) So = x (0,N) Z. p. Let 1. Show that P(S In S0, S1, ..., Sn 1) P(Sn In Sn 1) Hint Start with P(S, Ir. S DO, S I1...,S-I I -1) / 0 ill I; - I;+11 1. Thal is, if the sequence of steps is not possible for the...