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Chapter 2
Rules of calculus.
2.1 Superalgebras.
$\mathrm{A}$ (commutative associative) {\it superalgebra} is a vector space
$$
A=A_{even}\oplus A_{odd}
$$
with a given direct sum decomposition into even and odd pieces, and a map
$$
A\times A\rightarrow A
$$
which is bilinear, satisfies the associative law for multiplication, and
$$
A_{even}\times A_{even}\ \rightarrow\ A_{even}
$$
$$
A_{even}\times A_{odd}\ \rightarrow\ A_{odd}
$$
$$
A_{odd}\times A_{even}\ \rightarrow\ A_{odd}
$$
$$
A_{odd}\times A_{odd}\ \rightarrow\ A_{even}
$$
$\omega\cdot\sigma = \sigma\cdot\omega$ if either $\omega$ or $\sigma$ are even,
$\omega\cdot\sigma = -\sigma\cdot\omega$ if both $\omega$ and $\sigma$ are odd.
We write these last two conditions as
$$
\omega\cdot\sigma=(-1)^{\deg_{\sigma}\deg_{\omega}}\sigma\cdot\omega.
$$
Here $\deg\tau=0$ if $\tau$ is even, and $\deg\tau=1(\mathrm{m}\mathrm{o}\mathrm{d}\ 2)$ if $\tau$ is odd.
2.2 Differential forms.
A {\it linear} differential form on a manifold, $M$, is a rule which assigns to each $p\in M$ a linear function on $TM_{p}$. So a linear differential form, $\omega$, assigns to each $p$ an element of $TM_{p}^{*}$. We will, as usual, only consider linear differential forms which are smooth.
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{\it CHAPTER 2. RULES OF CALCULUS}.
The superalgebra, $\Omega(M)$ is the superalgebra generated by smooth functions on $M$ (taken as even) and by the linear differential forms, taken as odd.
Multiplication of differential forms is usually denoted by $\wedge$. The number of differential factors is called the {\it degree} of the form. So functions have degree zero, linear differential forms have degree one.
In terms of local coordinates, the most general {\it linear} differential form has an expression as $a_{1}dx_{1}+\cdots+a_{n}dx_{n}$ (where the $a_{i}$ are functions). Expressions of the form
$$
a_{12}dx_{1}\wedge dx_{2}+a_{13}dx_{1}\wedge dx_{3}+\cdots+a_{n-1,n}dx_{n-1}\wedge dx_{n}
$$
have degree two (and are even). Notice that the multiplication rules require
$$
dx_{i}\wedge dx_{j}=-dx_{j}\wedge dx_{i}
$$
and, in particular, $dx_{i}\wedge dx_{i}=0$. So the most general sum of products of two linear differential forms is a differential form of degree two, and can be brought to the above form, locally, after collections of coefficients. Similarly, the most general differential form of degree $k\leq n$ in $n$ dimensional manifold is a sum, locally, with function coefficients, of expressions of the form
$$
dx_{i_{1}}\wedge\cdots\wedge dx_{i_{k}},\ i_{1}<\cdots