The euclidean space $\Bbb R^n$ is orientable as a manifold.

Question

I know that

The euclidean space $\Bbb R^n$ is orientable as a manifold.

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I think that it is orientable because it has a nowhere vanishing $n$-form.

But I am not sure.

Please can you explain me more formally and mathematically?

Thank you:)

Answer 1

One very simple way of doing this is showing that

If a manifold has an atlas with exactly one chart, then it is orientable.

Of course, this applies to euclidean spaces.

Answer 2

You know that $\mathbb{R}^n$ has a natural set of global coordinates. Use these to write down a pointwise basis for $\Lambda^1_x(\mathbb{R}^n)$. Now use those to write down a basis for $\Lambda^n_x(\mathbb{R}^n)$. The coordinates are global. What does that tell you about your basis of $\Lambda^n_x(\mathbb{R}^n)$?