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Identifying submanifolds of R^n

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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.

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