Abstract
We now proceed to define integration with respect to odd variables.

The fundamental theorem of calculus for odd variables
Let us consider just one odd variable. This will be sufficient for our purposes for now. Following the direct analogy with integration of functions over a circle the fundamental theorem of calculus states

\(\int D\theta \frac{\partial f(\theta)}{\partial \theta} =0\).

We use the notation \(D\theta \) for the measure rather than \(d\theta \) as the measure cannot be associated with a one-form. We will discuss this in more detail another time.

Definition of integration
Recall that the general form of a function in one odd variable is

\(f(\theta) = a + \theta b\),

with a and b being real numbers. Thus from the fundamental theorem we have

\(\int D\theta b =0\).

In particular this implies

\(\int D\theta =0\).

Then we have

\(\int D\theta f(\theta) = a \int D\theta + b \int D\theta \:\: \theta = b \int D\theta\:\: \theta \).

Thus to define integration all we have to do is define the normalisation

\(\int D\theta\:\: \theta\).

The choice made by Berezin was to set this to unity. Other choices are also just as valid. Thus,

\(\int D\theta f(\theta) = b\).

Integration for several odd variables
For the case of more than one odd variable one simply uses

\(\int D(\theta_{1}, \theta_{2} , \cdots \theta_{n})f(\theta) = \int D\theta_{1} \int D \theta_{2} \cdots \int D\theta_{n} f(\theta)\).

example Consider two odd variables.

\(\int D(\overline{\theta}, \theta) \left( f_{0} + \theta \:f + \overline{\theta}\: \overline{f} + \theta \overline{\theta}F \right) = F \).

The general rule is that (taking care with signs) the integration with respect to the measure \(D(\theta_{1}, \theta_{2} , \cdots \theta_{n})\) of a function picks out the coefficient of the \(\theta_{1}, \theta_{2} , \cdots \theta_{n}\) term.

Integration and differentiation are the same!
From the above we see that differentiation with respect to an odd variable is the same as integration with respect to the odd variable. This explains why we cannot associate a “top-form” with the measure. This will become more apparent when we discuss changes of variables.

What next?
Next we will examine how changing variables in the integration effects the measure. We will see that things look “upside down” as compared with the integration of real variables. This is anticipated by the equivalence of integration and differentiation.