Hello,

i have read in wikipedia, that a orientation form (volume form) of the sphere S^m is given by: $\displaystyle w=\sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \wedge...\wedge dx_{i-1} \wedge dx_{i+1} \wedge...\wedge dx_{n+1}$

I don't understand the notation. Ok w is a map from the sphere into the set of alternating tensors:

$\displaystyle w:S^m->\Lambda^m (S^m)$.

but what is $\displaystyle w(x_1,...,x_m+1)=$?

and what does dx_i mean?

Can you please explain it to me?

Regards