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(x 2 ,y2 ,z2 ,v'2yz,hzx,hxy) is a submanifold of JR. 6 . P 2 ? 8. Show that the composition of two embeddings is an embedding and that the product of two embeddings is an embedding. m+ 1 . CHAPTER 2 THE DERIVATIVES OF DIFFERENTIABLE MAPS In the first chapter we defined differential manifolds and differentiable maps between them.

6. m+l). m+l is the inclusion and "I : ffi. then [lo "f]x =(lo 1)'(0) is perpendicular to x. 7. For v E ffi. 3. be v rotated through an angle 1r /2. 4. 4 Whitney's Embedding Theorem Revisited. 2 at times. As with all of Chapter 3, this section may be omitted at a first reading. 1. ) A compact manifold Mm of dimension m may be immersed in IR 2 m and embedded in JR 2 m+ 1 . 4. Proof. Assume that the compact manifold Mm is embedded in ]Rn, which we denote by M ~ IRn, and let JRn-l = {x E IRnlxn = O}.

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