Please see the attachment.
Only question 1 is answered in this solution.
Given a subset of R^n, how can we determine if it forms a submanifold? One way is to exhibit the set as the preimage of a continuous map from R^n to R whose derivative map is nonzero on the level set. (This is a special case of a more general theorem.) The solution comprises 1 page written in Word with equations in mathtype illustrating the use of this theorem in 3 separate cases.