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I need a help in how to prove that the n-dimensional sphere ia an orientable: How we can prove this?

Sila
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    There’s nothing in your question to indicate that it is related to Mathematica, it looks purely like a math question to me. Are you sure you’re on the right site? –  Jul 17 '19 at 07:25
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    Show that there exists a volume form. This might help: https://math.stackexchange.com/questions/1284234/volume-form-on-n-1-sphere-sn-1 – studiosus Jul 17 '19 at 10:26
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    What is your definition of oriantable? – Elad Jul 17 '19 at 10:28

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Take a unit sphere $S_n$. Construct a vector field $\mathbf n=\mathbf x$ defined on $S_n$. Then you can easily show that this vector field is continuous, has unit length, and normal to $S_n$. So by definition you have defined orientation to your surface, and this the surface is orientable.

Vasily Mitch
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