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  1. (1) Let $f : \mathbb R^3\to \mathbb R$ becomes a smooth function. Prove that $f^{-1}(m_0)$ is 2-dimensional manifolds if $m_0$ is the regular value of $f$. (2) Define $F : \mathbb R^2 \to \mathbb R$ as below, then answer the following questions. $$F(x,y) = x^2 - y^2$$ (a) Scratch the surface of $Gr(F)$. (b) Define $f : Gr(F) \to \mathbb R$ as $f(x,y,z) = z$. Decide whether $f$ is a Morse function. If $f$ is a Morse function, find all the critical points and the Morse index at that points. (3) and (4) to be continued ~