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\large{2014 Fall Math 5041 \quad Test 2 \qquad Nov 11, 2014}
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\large{Name: }
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1. Let $X \subset \mathbb{R}^N$ be an embedded submanifold. Show that almost every vector space $V$ of a fixed dimension $l$ in $\mathbb{R}^N$ intersects $X$ transversely. 

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2. Let $N$ be a closed embedded submanifold of $M$. Show that every smooth vector field $X \in \Gamma(N, TN)$ can be extended to a smooth vector field on $M$.

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3. Let $X$ be a compact manifold, and let $Y$ be a connected manifold. We assume $\dim X =\dim Y$. For a smooth map $f: X \to Y$ such that $\mathsf{deg}_2(f) \neq 0$, show that $f$ is surjective.

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4. Let $X$ be a vector field on a m-dimensional manifold $M$. Suppose $X_p \neq 0$. Show that there is a coordinate chart $(U, \varphi)$ with $\varphi(p) = 0 \in \mathbb{R}^m$ such that
$$
	X_q = \frac{\partial}{\partial x_1}
$$
for all $q \in U$. Here $x_1$ is the first coordinate of $(x_1, x_2, \ldots, x_m) \in \mathbb{R}^m$.

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