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I have several subtle and vague questions about differential forms and the term "euclidean space" that perhaps you can help me to work out.

Firts of all, if in cartesian coordinates [tex]dx_i[/tex] is that covector that when acting on a tangent vector [tex]X=X_i\frac{\partial}{\partial x_i}[/tex] of [tex]\mathbb{R}^n[/tex] gives its i-th component [tex]dx_i(X)=X_i[/tex], what is [tex]dr[/tex] in spherical coordinates? If I'm right it should return the component [tex]X_r[/tex] for a vector [tex]X=X_r\frac{\partial}{\partial r}+X_{\theta}\frac{\partial}{\partial \theta}+...[/tex]. That implies that it should be possible to talk about a vector in the direction of [tex]r[/tex] in the [tex](r,\theta,...)[/tex] space (for example the unit vector [tex]\frac{\partial}{\partial r}[/tex]). So the space [tex](r,\theta,...)[/tex] is in complete foot of equality with the space [tex](x_1,x_2,...)[/tex] and we can consider vectors in both spaces. So, a final question arises: if both are totally equivalent and only the physical interpretation can make a difference in their meaning, are both euclidean spaces? If the answer is affirmative: should we endow the spherical space [tex](r,\theta,...)[/tex] with the euclidean distance even though it has no physical meaning?

I'm not sure if I had made my points and doubts clear, so please let me know.

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# Differential forms and Euclidean space

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