definition of exterior derivative of \(C^\infty\) \(q\)-form over \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of exterior derivative of \(C^\infty\) \(q\)-form over \(C^\infty\) manifold with boundary.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\( t\): \(: M \to \Lambda_q (T M)\), \(\in \Omega_q (T M)\)
\(*d t\): \(: M \to \Lambda_{q + 1} (T M)\), \(\in \Omega_{q + 1} (T M)\)
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Conditions:
\(\forall (U \subseteq M, \phi) \in \{\text{ the charts for } M\} \text{ such that } t = \sum_{j_1 \lt ... \lt j_q} t_{j_1, ..., j_q} d x^{j_1} \wedge ... \wedge d x^{j_q} (d t = \sum_{j_1 \lt ... \lt j_q} d t_{j_1, ..., j_q} \wedge d x^{j_1} \wedge ... \wedge d x^{j_q})\)
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2: Note
Let us see that it is well-defined.
Let us see that the formula does not depend on the choice of chart.
Let \((U' \subseteq M, \phi')\) be any other chart such that \(U \cap U' \neq \emptyset\).
On \(U'\), \(t = \sum_{j_1 \lt ... \lt j_q} t'^{j_1, ..., j_q} d x'^{j_1} \wedge ... \wedge d x'^{j_q}\).
On \(U \cap U'\), \(t = \sum_{j_1 \lt ... \lt j_q} t_{j_1, ..., j_q} d x^{j_1} \wedge ... \wedge d x^{j_q} = \sum_{j_1 \lt ... \lt j_q} t'^{j_1, ..., j_q} d x'^{j_1} \wedge ... \wedge d x'^{j_q}\), but \(\sum_{j_1 \lt ... \lt j_q} d t_{j_1, ..., j_q} \wedge d x^{j_1} \wedge ... \wedge d x^{j_q} = \sum_{j_1 \lt ... \lt j_q} d t'_{j_1, ..., j_q} \wedge d x'^{j_1} \wedge ... \wedge d x'^{j_q}\), by the proposition that for any \(C^\infty\) \(q\)-form over any \(C^\infty\) manifold with boundary that is expressed as multiplication of function by wedge product of \(q\) exterior derivatives of functions in any 2 ways, the expressions with the heading functions replaced by its exterior derivatives (and the multiplications replaced by wedge products) represent the same object.
So, \(d t\) does not depend on the choice of chart.
\(d\) is \(\mathbb{R}\)-linear, which means that \(d (r_1 t_1 + r_2 t_2) = r_1 d t_1 + r_2 d t_2\) for each \(r_1, r_2 \in \mathbb{R}\), because \(t_l = \sum_{j_1 \lt ... \lt j_q} t_{l, j_1, ..., j_q} d x^{j_1} \wedge ... \wedge d x^{j_q}\), \(d (r_1 t_1 + r_2 t_2) = d (r_1 \sum_{j_1 \lt ... \lt j_q} t_{1, j_1, ..., j_q} d x^{j_1} \wedge ... \wedge d x^{j_q} + r_2 \sum_{j_1 \lt ... \lt j_q} t_{2, j_1, ..., j_q} d x^{j_1} \wedge ... \wedge d x^{j_q}) = d (\sum_{j_1 \lt ... \lt j_q} (r_1 t_{1, j_1, ..., j_q} + r_2 t_{2, j_1, ..., j_q}) d x^{j_1} \wedge ... \wedge d x^{j_q}) = \sum_{j_1 \lt ... \lt j_q} d (r_1 t_{1, j_1, ..., j_q} + r_2 t_{2, j_1, ..., j_q}) \wedge d x^{j_1} \wedge ... \wedge d x^{j_q} = \sum_{j_1 \lt ... \lt j_q} (r_1 d t_{1, j_1, ..., j_q} + r_2 d t_{2, j_1, ..., j_q}) \wedge d x^{j_1} \wedge ... \wedge d x^{j_q} = r_1 \sum_{j_1 \lt ... \lt j_q} d t_{1, j_1, ..., j_q} \wedge d x^{j_1} \wedge ... \wedge d x^{j_q} + r_2 \sum_{j_1 \lt ... \lt j_q} d t_{2, j_1, ..., j_q} \wedge d x^{j_1} \wedge ... \wedge d x^{j_q} = r_1 d t_1 + r_2 d t_2\).
In fact, although the definition requires \(t\) to be expressed with the increasing orders terms with \(j_1 \lt ... \lt j_q\), even when \(t = \sum_{(l_1, ... l_q)} t_{l_1, ..., l_q} d x^{l_1} \wedge ... \wedge d x^{l_q}\) with any orders terms, \(d t = \sum_{(l_1, ... l_q)} d t_{l_1, ..., l_q} \wedge d x^{l_1} \wedge ... \wedge d x^{l_q}\), because while as \(d\) is \(\mathbb{R}\)-linear, \(d t = d (\sum_{(l_1, ... l_q)} t_{l_1, ..., l_q} d x^{l_1} \wedge ... \wedge d x^{l_q}) = \sum_{(l_1, ... l_q)} d (t_{l_1, ..., l_q} d x^{l_1} \wedge ... \wedge d x^{l_q})\), as \(t_{l_1, ..., l_q} d x^{l_1} \wedge ... \wedge d x^{l_q} = sgn \sigma t_{l_1, ..., l_q} d x^{j_1} \wedge ... \wedge d x^{j_q}\) where \(j_1 \lt ... \lt j_q\) and \(\sigma\) is the permutation that permutates \((l_1, ..., l_q)\) to \((j_1, ..., j_q)\), \(d (t_{l_1, ..., l_q} d x^{l_1} \wedge ... \wedge d x^{l_q}) = d (sgn \sigma t_{l_1, ..., l_q} d x^{j_1} \wedge ... \wedge d x^{j_q}) = d (sgn \sigma t_{l_1, ..., l_q}) \wedge d x^{j_1} \wedge ... \wedge d x^{j_q} = sgn \sigma d t_{l_1, ..., l_q} \wedge d x^{j_1} \wedge ... \wedge d x^{j_q} = d t_{l_1, ..., l_q} \wedge d x^{l_1} \wedge ... \wedge d x^{l_q}\).
In fact, when \(t = \sum_{j \in J} f_{j, 0} d f_{j, 1} \wedge ... \wedge d f_{j, q}\) for any index set, \(J\), and any \(C^\infty\) functions, \(\{f_{j, l} \vert j \in J, l \in \{0, ... q\}\}\), \(d t = \sum_{j \in J} d f_{j, 0} \wedge d f_{j, 1} \wedge ... \wedge d f_{j, q}\), because as \(d\) is \(\mathbb{R}\)-linear, \(d t = \sum_{j \in J} d (f_{j, 0} d f_{j, 1} \wedge ... \wedge d f_{j, q})\), but applying the proposition that the exterior derivative of the wedge product of any \(C^\infty\) \(q_1\)-form and any \(C^\infty\) \(q_2\)-form over any \(C^\infty\) manifold with boundary is the wedge product of the exterior derivative of the \(q_1\)-form and the \(q_2\)-form plus \(-1\) to power \(q_1\) the wedge product of the \(q_1\)-form and the exterior derivative of the \(q_2\)-form, \(= \sum_{j \in J} d f_{j, 0} \wedge d f_{j, 1} \wedge ... \wedge d f_{j, q} + (-1)^0 f_{j, 0} d (d f_{j, 1} \wedge ... \wedge d f_{j, q})\), but \(d (d f_{j, 1} \wedge ... \wedge d f_{j, q}) = 0\), by iteratively applying that proposition and the proposition that the double exterior derivative of any \(C^\infty\) \(q\)-form over any \(C^\infty\) manifold with boundary is \(0\).
