 Solution

A Grassmann variable is an odd generator of a Grassmann algebra: ${\eta }_i{\eta }_j=-{\eta }_j{\eta }_i$, so ${\eta }_i^2=0$. For one generator, any function is $f(\eta )=a+b\eta$. The Berezin integral is the linear operationThus integration extracts a coefficient, rather than assigning a length or volume. For many generators it extracts the coefficient of the highest-degree monomial with the sign fixed by the order of the measure. Odd coefficients and Grassmann derivatives must retain their order; exchanging two odd objects changes the sign.

This operation is invariant under odd translations, because a translation only changes terms of lower degree. For an invertible ordinary matrix $S$ and $\eta =S\xi$, the Grassmann change-of-variables formula isThe inverse Jacobian determinant, rather than the ordinary commuting-variable Jacobian, compensates for the factor $\det S$ multiplying the top monomial. Integration agrees with the appropriate ordered Grassmann derivatives, but the orientation must be specified when combining barred and unbarred variables.