Answered: can we have a polynomial Q(x,y,z) such…

can we have a polynomial Q(x,y,z) such that the subspace {(x,y,z):Q(x,y,z)=0} is a smooth submanifold diffeomorphic to T^2(torus)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 47CR: Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}

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Question

can we have a polynomial Q(x,y,z) such that the subspace {(x,y,z):Q(x,y,z)=0} is a smooth submanifold diffeomorphic to T^2(torus) in step by step with out using chatgpt.

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Step 1: We use Implicit mapping theorem to solve this

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Step 2: Constructing the required polynomial.

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Follow-up Question

Actually, it is more theoretical to me could you show me mathematically? And how do you define Q(x,y,z) can you see it again? finally, do we have another type of Torus other than the product you have used? Just I want to be sure and it will be good to send it me in more mathematical way. Thanks!

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