Term Singular Point Diffeomorphism and Regular Value Theorem Worksheet

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Please solve the Question and show all the steps. you may need the note(pdf) but you cannot copy from it. Thank you so muchI prefer it to be typed. I tip for quality work

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In the questions below you need to justify your answers rigorously.
(1) Let D : R” + RM be a smooth map. Define the term singular point of a map
9. Give an example of a map : RR that has a unique singular point.
(i) Can a diffeomorphism : R2 + R2 have singular points?
(ii) Give an example of a smooth map : R2 + R2 whose only singular point is
(0,0).
(iv) Give an example of a non-constant smooth map Ý : R2 + R2 such that every
point (x,y) € R2 is singular.
(v) Give an example of a smooth map : R2 + R2 such that the set of its
singular points is the parabola {(x,y) ER?: y = 2*}.
(1) Let : R + Rº be a smooth map of the form,
(11, 12, 13, 14) = (01:02 +03:14, 11, 12, 13, 14)),
where f is an unspecified smooth function. Find a function f : R4 + R
such that the map and the pre-image 0-1(1, 2) satisfy the conditions of the
Regular Value Theorem. Verify that your answer satisfies these conditions.
(i) Choose any point p < 0-1(1, 2) in your answer for (b)0). Write down its coordinates and construct a basis for the tangent space T,Σ, where Σ = 0-(1,2). Justify your answer. Purchase answer to see full attachment Tags: Math Problems Diffeomorphism Term Singular Point Regular Value Theorem User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.