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 Feb11 accepted Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book) Feb10 comment Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book) I've actually been looking at this proof at a more careful level. And I realized Tubular Neighborhood and Retraction are exactly the tools that Lee used to solve the question I have. I guess I will choose @Potato as the best answer if nobody gives a better answer in the following couple of days! Thanks man, @Potato! Feb9 comment Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book) I saw this proof before. But since I am not aware of Whitney approximation theorem or the smooth retraction mentioned in the proof, I skipped it. Bott, Tu's book's proof requires less background knowledge. But as I explained, I found I can't get around that point. I still wish that someone might be able to tell me how it is possible. Feb9 asked Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book) Nov20 awarded Critic Nov20 awarded Supporter Nov15 accepted Confused at the definition for integration of differential form over manifold Nov14 asked Confused at the definition for integration of differential form over manifold Nov13 comment How to prove this formula for Lie derivative for differential forms @FlybyNight I believe $\phi_t$ in my notation is the $f$ in your notation. And $\phi_t^* w$, viewed as a whole thing, is exactly the $w^*$ in your notation. $\phi_t^*$ is the mapping that takes $w$ to $w^*$. But never mind, I know how to prove this thing now. It's only a simple computation once you know what all those mappings really are. Thanks anyway. Nov13 revised How to prove this formula for Lie derivative for differential forms edited body Nov13 comment How to prove this formula for Lie derivative for differential forms @FlybyNight huh? $w$ here is a differential form. Nov12 asked How to prove this formula for Lie derivative for differential forms Sep26 accepted How do I see that the tangent bundle of torus is trivial Sep26 comment How do I see that the tangent bundle of torus is trivial Thanks, @Neal, for providing so many ways of seeing it Sep26 comment How do I see that the tangent bundle of torus is trivial @RyanBudney Thanks! I guess that's one way of doing it. Sep26 comment How do I see that the tangent bundle of torus is trivial what do you mean by taking one vector field in the direction of each "factor"? Sep26 asked How do I see that the tangent bundle of torus is trivial Sep10 accepted How to show the height function of a torus has 4 critical points Sep10 comment How to show the height function of a torus has 4 critical points Thank you. I wasn't aware of this form before. Sep10 comment How to show the height function of a torus has 4 critical points Thanks, @t.b.! I will take a look.