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then[itex]\alpha \wedge \beta = \sum_{i,j} \alpha_i(x) \beta_j(x) dx_i \wdge dx_j \in \Omega^2[/itex]

NOW THE STEP I DON'T FOLLOW - he jumps to this in the lecture notes:

[itex]\alpha \wedge \beta = \sum_{i<j} (\alpha_i \beta_j - \beta_j \alpha_i) dx_i \wedge dx_j[/itex]

the subscript on the sum was either i<j or i,j - could someone tell me which as well as explaing where on earth this step comes from.

(ii) could someone explain the Leibniz rule for exterior derivative [itex]d: \Omega^k \rightarrow \Omega^{k+1}[/itex]

i.e. why [itex]d(\alpha^k \wedge \beta^l)=d \alpha^k \wedge \beta^l + (-1)^k \alpha^k \wedge d \beta^l[/itex]

note that the superscript on the differential form indicates that it's a k form or and l form

my main problem here is where the (-1)^k comes from

cheers for your help