Differential forms and Euclidean space

1. Aug 25, 2010

Fernsanz

Hi,

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 $$dx_i$$ is that covector that when acting on a tangent vector $$X=X_i\frac{\partial}{\partial x_i}$$ of $$\mathbb{R}^n$$ gives its i-th component $$dx_i(X)=X_i$$, what is $$dr$$ in spherical coordinates? If I'm right it should return the component $$X_r$$ for a vector $$X=X_r\frac{\partial}{\partial r}+X_{\theta}\frac{\partial}{\partial \theta}+...$$. That implies that it should be possible to talk about a vector in the direction of $$r$$ in the $$(r,\theta,...)$$ space (for example the unit vector $$\frac{\partial}{\partial r}$$). So the space $$(r,\theta,...)$$ is in complete foot of equality with the space $$(x_1,x_2,...)$$ 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 $$(r,\theta,...)$$ 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.

Last edited: Aug 25, 2010