- Define the cotangent space $T_a^*\mathbb{R}^n$. Define the differential of a dunction $f$ at the point $a, df \in T_a^*\mathbb{R}^n$. Write down the explicit formula for the deffertial $df$ in terms of partial derivatives.
- Define a differential form $\omega$ on an open set $U \subset \mathbb{R}^n$.
- Define the coboundary operator $D: \Omega^k(U) \rightarrow \Omega^{k+1}(U)$.
My definitions are:
- Let $M$ be a smooth manifold & $a$ a point in $M$. Then $T_a(M)$ is defined as the tangent space at $a$. The cotangent space is then the dual space of $T_a(M)$, at $a$, i.e $T_a^*(M)$. The elemems in $T_a^*(M)$ are linear functions on $T_a(M)$.
The differential, $df$, of a function $f$ at $a$ is an element of the duals space $T_a^*(\mathbb{R}^n)$ defined by the following notation
$$df({\underline{v}}) = {\underline{v}}(f).$$
The formula is
$$df = \frac{\partial f}{\partial x_1}dx_1, \cdots , \frac{\partial f}{\partial x_n}dx_n.$$
- Let $U \subset \mathbb{R}^n$ be an open subset. A $k$ form $\omega \in \Omega^k(U)$ is an expression
$$\omega = \sum_{1 \leq i_1 < \cdots < i_k \leq n} \omega_{i_1, \cdots i_k}(x_1, \cdots x_n)dx_{i_1} \wedge \cdots \wedge dx_{i_k}.$$
- The definition of the coboundary operator is
$$d\left( \sum_{1 \leq i_1 < \cdots < i_k \leq n} \omega_{i_1, \cdots i_k}(x_1, \cdots x_n)dx_{i_1} \wedge \cdots \wedge dx_{i_k} \right)$$
$$ = \sum_{1 \leq i_1 < \cdots < i_k \leq n} \{d\omega_{i_1, \cdots i_k}(x_1, \cdots x_n)\} \wedge dx_{i_1} \wedge \cdots \wedge dx_{i_k}.$$
Are these correct?