6. Let S be an explicit surface in R*** defined by a C¹ map F: R" →R" so S={(x.F(x)): xЄR*} = img G. where G: R" → R+ is defined by G(x) = (x,F(x)) for all x € R". The theorems on tangent spaces for surfaces are quite difficult to prove except for explicit surfaces. Explicit surfaces are easier to study because they have a simple description for all of the differentiable curves lying on the surface: Lemma. Let y: R→ R be a curve. Then y is differentiable with y(R) CS if and only if there exists a differentiable curve g: R-R" such that y(t)=(g(t), Fog(t)) for all te R. You will prove D7 Theorem 2 using this lemma but first you will analyze the lemma's proof. (6a) Here is an essentially correct but incomplete proof of one direction of this lemma. 1. Assume g: R→ R" is differentiable such that y(t) = (g(t), Fog(t)) for all tЄR. 2. By definition of S, for all tЄR, y(t)=(g(t), F(g(t))) = S so y(R) CS. 3. Moreover, since g is differentiable, y is differentiable. Identify 1 line which is not properly justified and suggest how to fix it. Choose the line with the most important missing justifications. Your answer should be brief. (6b) Here is an essentially correct but incomplete proof of the other direction of this lemma. 1. Assume y: R→R"+" is differentiable and y(R) CS. 2. Let : R+ →R" be the standard projection onto the first n coordinates so (x1,xn+m)=(x1,...,x). 3. By definition of it and the explicit surface S, every pЄS satisfies p =((p), F(л(p))). 4. As y(R)CS, this implies for teR that y(t)=(xy(t), F(xy(t))) 5. Define g: R→R" by g=oy so it follows that y(t)=(g(t), F(g(t))) for all t = R. 6. Since x and y are differentiable, g is differentiable so we are done. Identify 1 line which is not properly justified and suggest how to fix it. Choose the line with the most important missing justifications. Your answer should be brief. (6c) Now, fix p€ S so p = (a, F(a)) for some a € R. Prove that T,SC img dGa. (Revised 2020-11-19) Hint: Use the chain rule and the lemma. (6d) Prove that img dG, CT,S. This concludes the proof of D7 Theorem 2. (Revised 2020-11-19) Hint: Briefly justify why an element v of img dG, can be written as (w,dF,(w)) for some wЄ R". Then construct a simple curve on S with velocity v at p.