Smoothness of a map to a circle I am reading "An introduction to Manifolds" by Loring Tu. And trying to solve one exercise. Prove that the map $F:\mathbb R\to S^1$,$F(t)=(cost,sint)$ is $C^\infty$.I am attaching the screenshot of the exercise.Also is there a book (or a solution manualof Loring Tu) where i can find examples of the things that are given in this book so that i can use that book as a reference.
 A: Tu has solutions at the back of the book and in particular has one for this because it is starred.



Btw, I think you can use Section 11 to answer this. Of course this is inadmissible, but

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*otherwise I don't really have anything else to say for this answer.


*I think what follows is the more natural way to answer this exercise. Like we're just changing the range, so why would this change affect this smoothness, right?

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*2.1. But I guess this exercise at this point in the book is good in trying to think about the use of charts and stuff in order to understand the concepts in Subection 6



Anyway here goes:
After showing that $\iota \circ F: \mathbb R \to S^1 \to \mathbb R^2$ is smooth, for $\iota$ is inclusion from circle to plane (i.e. $\iota: S^1 \to \mathbb R^2$, $\iota(a,b)=(a,b)$), we can can say $F$ itself is smooth, by Theorem 11.15, because $S^1$ is a regular/an embedded $k$-submanifold of $\mathbb R^2$ (with $k=1$). Here, the '$f$' is $\iota \circ F$ while the '$\tilde f$' is $F$.
