Definition 6.7.1 (Exterior derivative). Let $U \subset \R^n$ be an open subset. The exterior derivative $d: A^k(U) \to A^{k+1}(U)$ is defined by the formula

$$ \underbrace{d\varphi}{(k+1)-form} (P_x(\vec v_1, \dots , \vec v{k+1})) = \lim_{h\to 0}\frac{1}{h^{k+1}}\int_{\partial P_x(h\vec v_1, \dots, h \vec v_{k+1})}\varphi. $$

This makes sense because we are integrating the $k$-form $\varphi$ over the boundary of a $k+1$ dimensional parallelogram.

The limit exists because of how we defined boundary. The terms from different faces cancel out because they have opposite orientation.

Example 6.7.3 (Exterior derivative of a function). When $\varphi$ is a 0-form field (a function), the exterior derivative is just a function. So $df$ is the work form of $\nabla f$.

$$ df(P_x(\vec v)) = \lim_{h\to 0}\frac{1}{h}\int_{\partial P_x(h\vec v)}f = \lim_{h\to 0}\frac{f(x + h\vec v) - f(x)}{h} = [Df(x)] \vec v. $$

Theorem 6.7.4 (Computing the exterior derivative). Let

$$ \varphi = \sum_{1 \leq i_1 < \dots < i_k \leq n}a_{i_1, \dots, i_k}dx_{i_1} \wedge \dots \wedge dx_{i_k} $$

be a $k$-form of class $C^2$ on an open subset $U\subset \R^n$.

1. The limit in Definition 6.7.1 exits and defines a $(k+1)$ form.
2. The exterior derivative is linear over $\R$: if $\varphi$ and $\psi$ are $k$-forms on $U\subset \R^n$, and $a$ and $b$a re numbers (not function), then

$$ d(a\varphi + b\psi) = ad\varphi + b\psi. $$

3. The exterior derivative of a constant form is 0.

4. The exterior derivative of the 0-form $f$ is given by the formula $df = [Df] = \sum_{i=1}^n (D_if) dx_i$.

5. If $f: U \to \R$ is a $C^2$ function, then

$$ d(fdx_{i_1}\wedge \dots \wedge dx_{i_k}) = df \wedge dx_{i_1} \wedge \dots \wedge dx_{i_k}. $$

These rules let us compute the exterior derivative of any $k$-form.