Question: Problem 3. Suppose fe Co (R) and let S = f-({0}). If the gradient Vf(x) = (01f(x), .... Onf(I) ) does not equal the zero

Problem 3. Suppose fe Co (R") and let S = f-({0}). If the gradient Vf(x) = (01f(x), .... Onf(I) ) does not equal the zero vector for any x ( S, then show that S is a submanifold of R" of dimension n - 1. Use this result to prove that the n-sphere S" C R"+ is a n-dimensional submanifold of Ra+1

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