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I need help with the following question. I am not sure how to begin. Any help will be appreciated. Thank you!

For any smooth manifold $M,$ the tangent bundle $TM$ is an oriented manifold.

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Are you comfortable with the fact that it's a manifold? – Sammy Black Apr 8 '13 at 5:05
I know what a manifold is. I'm not confortable with oriented things I guess. – Susan Apr 8 '13 at 5:07
I think what Sammy means is: Are you familiar with how $TM$ is a manifold. That is, do you know what the usual charts are on $TM$? You'll need this in order to work through Alex Youcis' hint. – Jesse Madnick Apr 8 '13 at 5:12
Also, could you perhaps change the title to make it more specific to the question, perhaps involving the words "tangent bundle" and "oriented"? Because the differential-topology tag already tells us the subject matter, there's no need to repeat it in the title. Thank you! – Jesse Madnick Apr 8 '13 at 5:14
This sound more like a topic in differential manifolds, not differential topology. The later subject is primarily surgery theory on high dimensional manifolds. – Bombyx mori Apr 8 '13 at 5:18

Hint: Take an atlas on $M$, say $\{(U,\varphi)\}$ and look at the corresponding atlas on $TM$, $\displaystyle \left(U,\frac{\partial}{\partial \varphi}\right)$. What is the determinant of these overlap maps' Jacobians?

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