Operations on Differential Forms: Sum and Wedge Product
📂Geometry Operations on Differential Forms: Sum and Wedge Product Definition Sum + + + Suppose ω = ∑ I a I d x I , φ = ∑ I b I d x I \omega = \sum\limits_{I} a_{I} dx_{I}, \varphi = \sum\limits_{I} b_{I} dx_{I} ω = I ∑ a I d x I , φ = I ∑ b I d x I k k k
ω + φ : = ∑ I ( a I + b I ) d x I
\omega + \varphi := \sum\limits_{I}\left( a_{I} + b_{I} \right)dx_{I}
ω + φ := I ∑ ( a I + b I ) d x I
Wedge Product ∧ \wedge ∧ Let ω = ∑ I a I d x I \omega = \sum\limits_{I} a_{I}dx_{I} ω = I ∑ a I d x I φ = ∑ J b J d x J \varphi = \sum\limits_{J} b_{J}dx_{J} φ = J ∑ b J d x J k k k s s s wedge product of these two is defined as:
ω ∧ φ = ∑ I , J a I b J d x I ∧ d x J
\omega \wedge \varphi = \sum \limits_{I,J} a_{I}b_{J} dx_{I} \wedge dx_{J}
ω ∧ φ = I , J ∑ a I b J d x I ∧ d x J
Explanation Let M M M n n n differential manifold . (If the concept of differential manifold is challenging, think of it as M = R n M=\mathbb{R}^{n} M = R n ∧ \wedge ∧ φ 1 , φ 2 \varphi_{1}, \varphi_{2} φ 1 , φ 2 T p ∗ M T_{p}^{\ast}M T p ∗ M alternating function φ 1 ∧ φ 2 \varphi_{1} \wedge \varphi_{2} φ 1 ∧ φ 2 Moreover, the wedge product itself is an alternating function . That is, for,
φ i ∧ φ j = − φ j ∧ φ i and φ i ∧ φ i = 0
\varphi_{i} \wedge \varphi_{j} = - \varphi_{j} \wedge \varphi_{i} \quad \text{and} \quad \varphi_{i} \wedge \varphi_{i} = 0
φ i ∧ φ j = − φ j ∧ φ i and φ i ∧ φ i = 0
it is easy to demonstrate according to the definition.
( φ 1 ∧ φ 2 ) ( v 1 , v 2 ) = det [ φ i ( v j ) ] = ∣ φ 1 ( v 1 ) φ 1 ( v 2 ) φ 2 ( v 1 ) φ 2 ( v 2 ) ∣ = − ∣ φ 2 ( v 1 ) φ 2 ( v L ) φ 1 ( v 1 ) φ 1 ( v 2 ) ∣ = − ( φ 2 ∧ φ 1 ) ( v 1 , v 1 )
\begin{align*}
\left(\varphi_{1} \wedge \varphi_{2}\right)\left(v_{1}, v_{2}\right)
=&\ \det \left[\varphi_{i}\left(v_{j}\right)\right] \\
=&\ \begin{vmatrix}
\varphi_{1}\left(v_{1}\right) & \varphi_{1}\left(v_{2}\right) \\
\varphi_{2}\left(v_{1}\right) & \varphi_{2}\left(v_{2}\right)
\end{vmatrix} \\
=&\ -\begin{vmatrix}
\varphi_{2}\left(v_{1}\right) & \varphi_{2}\left(v_{L}\right) \\
\varphi_{1}\left(v_{1}\right) & \varphi_{1}\left(v_{2}\right)
\end{vmatrix}
\\ =&\ -\left(\varphi_{2} \wedge \varphi_{1}\right)\left(v_{1}, v_{1}\right) \\
\end{align*}
( φ 1 ∧ φ 2 ) ( v 1 , v 2 ) = = = = det [ φ i ( v j ) ] φ 1 ( v 1 ) φ 2 ( v 1 ) φ 1 ( v 2 ) φ 2 ( v 2 ) − φ 2 ( v 1 ) φ 1 ( v 1 ) φ 2 ( v L ) φ 1 ( v 2 ) − ( φ 2 ∧ φ 1 ) ( v 1 , v 1 )
( φ 1 , φ 1 ) ( v 1 , v 2 ) = det [ φ 1 ( v i ) ] = ∣ φ 1 ( v 1 ) φ 1 ( v 2 ) φ 1 ( v 1 ) φ 1 ( v 2 ) ∣ = 0
\begin{align*}
\left(\varphi_{1}, \varphi_{1}\right)\left(v_{1}, v_{2}\right)
=&\ \det \left[\varphi_{1}\left(v_{i}\right)\right] \\
=&\ \begin{vmatrix}
\varphi_{1}\left(v_{1}\right) & \varphi_{1}\left(v_{2}\right) \\
\varphi_{1}\left(v_{1}\right) & \varphi_{1}\left(v_{2}\right)
\end{vmatrix} \\
=&\ 0
\end{align*}
( φ 1 , φ 1 ) ( v 1 , v 2 ) = = = det [ φ 1 ( v i ) ] φ 1 ( v 1 ) φ 1 ( v 1 ) φ 1 ( v 2 ) φ 1 ( v 2 ) 0
By definition, the wedge product of a k k k s s s k + s k+s k + s
Moreover, while d x i ∧ d x i = 0 dx_{i} \wedge dx_{i} = 0 d x i ∧ d x i = 0 i i i ω ∧ ω = 0 \omega \wedge \omega = 0 ω ∧ ω = 0 ω = x 1 d x 1 ∧ d x 2 + x 2 2 d x 3 ∧ d x 4 \omega = x_{1}dx_{1}\wedge dx_{2} + x2_{2} dx_{3} \wedge dx_{4} ω = x 1 d x 1 ∧ d x 2 + x 2 2 d x 3 ∧ d x 4
ω ∧ ω = 2 x 1 x 2 d x 1 ∧ d x 2 ∧ d x 3 ∧ d x 4
\omega \wedge \omega = 2x_{1}x_{2} dx_{1} \wedge dx_{2} \wedge dx_{3} \wedge dx_{4}
ω ∧ ω = 2 x 1 x 2 d x 1 ∧ d x 2 ∧ d x 3 ∧ d x 4
Example Let ω = x 1 d x 1 + x 2 d x 2 \omega = x_{1}dx_{1} + x_{2}dx_{2} ω = x 1 d x 1 + x 2 d x 2 R 3 \mathbb{R}^{3} R 3 φ = x 1 d x 1 ∧ d x 2 + x 2 d x 1 ∧ d x 3 + d x 2 ∧ d x 3 \varphi = x_{1}dx_{1}\wedge dx_{2} + x_{2} dx_{1}\wedge dx_{3} + dx_{2} \wedge dx_{3} φ = x 1 d x 1 ∧ d x 2 + x 2 d x 1 ∧ d x 3 + d x 2 ∧ d x 3 R 3 \mathbb{R}^{3} R 3
ω ∧ φ = ( x 1 d x 1 + x 2 d x 2 ) ∧ ( x 1 d x 1 ∧ d x 2 + x 2 d x 1 ∧ d x 3 + d x 2 ∧ d x 3 ) = x 1 2 d x 1 ∧ d x 1 ∧ d x 2 + x 1 x 2 d x 1 ∧ d x 1 ∧ d x 3 + x 1 d x 1 ∧ d x 2 ∧ d x 3 + x 1 x 2 d x 2 ∧ d x 1 ∧ d x 2 + x 2 2 d x 2 ∧ d x 1 ∧ d x 3 + x 2 d x 2 ∧ d x 2 ∧ d x 3 = ( x 1 − x 2 2 ) d x 1 ∧ d x 2 ∧ d x 3
\begin{align*}
\omega \wedge \varphi
=&\ (x_{1}dx_{1} + x_{2}dx_{2}) \wedge (x_{1}dx_{1}\wedge dx_{2} + x_{2} dx_{1}\wedge dx_{3} + dx_{2} \wedge dx_{3}) \\
=&\ x_{1}^{2} \cancel{dx_{1} \wedge dx_{1}\wedge dx_{2}} + x_{1}x_{2}\cancel{dx_{1} \wedge dx_{1}\wedge dx_{3}} + x_{1}dx_{1} \wedge dx_{2} \wedge dx_{3} \\
& + \cancel{x_{1}x_{2}dx_{2} \wedge dx_{1}\wedge dx_{2}} + x_{2}^{2} dx_{2} \wedge dx_{1}\wedge dx_{3} + x_{2}\cancel{dx_{2} \wedge dx_{2} \wedge dx_{3}} \\
=&\ (x_{1}-x_{2}^{2}) dx_{1} \wedge dx_{2} \wedge dx_{3}
\end{align*}
ω ∧ φ = = = ( x 1 d x 1 + x 2 d x 2 ) ∧ ( x 1 d x 1 ∧ d x 2 + x 2 d x 1 ∧ d x 3 + d x 2 ∧ d x 3 ) x 1 2 d x 1 ∧ d x 1 ∧ d x 2 + x 1 x 2 d x 1 ∧ d x 1 ∧ d x 3 + x 1 d x 1 ∧ d x 2 ∧ d x 3 + x 1 x 2 d x 2 ∧ d x 1 ∧ d x 2 + x 2 2 d x 2 ∧ d x 1 ∧ d x 3 + x 2 d x 2 ∧ d x 2 ∧ d x 3 ( x 1 − x 2 2 ) d x 1 ∧ d x 2 ∧ d x 3
Properties Let ω \omega ω k k k φ \varphi φ s s s θ \theta θ r r r
( ω ∧ φ ) ∧ θ = ω ∧ ( φ ∧ θ ) ( ω ∧ φ ) = ( − 1 ) k s ( φ ∧ ω ) ω ∧ ( φ + θ ) = ω ∧ φ + ω ∧ θ , if r = s
\begin{align}
(\omega \wedge \varphi) \wedge \theta =&\ \omega \wedge (\varphi \wedge \theta)
\\ (\omega \wedge \varphi) =&\ (-1)^{ks} (\varphi \wedge \omega)
\\ \omega \wedge (\varphi + \theta) =&\ \omega \wedge \varphi + \omega \wedge \theta,\quad \text{if } r=s
\end{align}
( ω ∧ φ ) ∧ θ = ( ω ∧ φ ) = ω ∧ ( φ + θ ) = ω ∧ ( φ ∧ θ ) ( − 1 ) k s ( φ ∧ ω ) ω ∧ φ + ω ∧ θ , if r = s
Proof Proof(1) Let ω = ∑ I a I d x I \omega = \sum\limits_{I} a_{I} dx_{I} ω = I ∑ a I d x I φ = ∑ J b J d x J \varphi = \sum\limits_{J} b_{J} dx_{J} φ = J ∑ b J d x J ϑ = ∑ L c L d x L \vartheta = \sum\limits_{L} c_{L} dx_{L} ϑ = L ∑ c L d x L
( ω ∧ φ ) ∧ ϑ = ( ∑ I a I d x I ∧ ∑ J b J d x J ) ∧ ∑ L c L d x L = ( ∑ I , J a I b J d x I ∧ d x J ) ∧ ∑ L c L d x L by definition of ∧ = ∑ I , J , L ( a I b J ) c L ( d x I ∧ d x J ) ∧ d x L by definition of ∧ = ∑ I , J , L ( a I b J ) c L d x I ∧ ( d x J ∧ d x L ) by property of ∧
\begin{align*}
(\omega \wedge \varphi) \wedge \vartheta
=&\ \left( \sum\limits_{I}a_{I}dx_{I} \wedge \sum\limits_{J}b_{J}dx_{J} \right) \wedge \sum\limits_{L} c_{L}dx_{L} \\
=&\ \left( \sum\limits_{I,J}a_{I}b_{J} dx_{I} \wedge dx_{J} \right) \wedge \sum\limits_{L} c_{L}dx_{L} & \text{by definition of } \wedge\\
=&\ \sum\limits_{I,J,L} (a_{I}b_{J})c_{L} (dx_{I} \wedge dx_{J}) \wedge dx_{L} & \text{by definition of } \wedge\\
=&\ \sum\limits_{I,J,L} (a_{I}b_{J})c_{L} dx_{I} \wedge (dx_{J} \wedge dx_{L}) & \text{by property of } \wedge\\
\end{align*}
( ω ∧ φ ) ∧ ϑ = = = = ( I ∑ a I d x I ∧ J ∑ b J d x J ) ∧ L ∑ c L d x L I , J ∑ a I b J d x I ∧ d x J ∧ L ∑ c L d x L I , J , L ∑ ( a I b J ) c L ( d x I ∧ d x J ) ∧ d x L I , J , L ∑ ( a I b J ) c L d x I ∧ ( d x J ∧ d x L ) by definition of ∧ by definition of ∧ by property of ∧
Since the functions a , b , c a, b, c a , b , c
( ω ∧ φ ) ∧ ϑ = ∑ I , J , L a I ( b J c L ) d x I ∧ ( d x J ∧ d x L ) = ∑ I a I d x I ∧ ∑ J , L b J c L d x J ∧ d x L by property of ∧ = ω ∧ ( φ ∧ ϑ ) by property of ∧
\begin{align*}
(\omega \wedge \varphi) \wedge \vartheta
=&\ \sum\limits_{I,J,L} a_{I}(b_{J}c_{L}) dx_{I} \wedge (dx_{J} \wedge dx_{L}) \\
=&\ \sum\limits_{I} a_{I} dx_{I} \wedge \sum\limits_{J,L} b_{J}c_{L} dx_{J} \wedge dx_{L} & \text{by property of } \wedge\\
=&\ \omega \wedge ( \varphi \wedge \vartheta) & \text{by property of } \wedge\\
\end{align*}
( ω ∧ φ ) ∧ ϑ = = = I , J , L ∑ a I ( b J c L ) d x I ∧ ( d x J ∧ d x L ) I ∑ a I d x I ∧ J , L ∑ b J c L d x J ∧ d x L ω ∧ ( φ ∧ ϑ ) by property of ∧ by property of ∧
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Proof(2) Let ω = ∑ I a I d x I \omega = \sum\limits_{I} a_{I} dx_{I} ω = I ∑ a I d x I φ = ∑ J b J d x J \varphi = \sum\limits_{J} b_{J} dx_{J} φ = J ∑ b J d x J
ω ∧ φ = ∑ I , J a I b J d x I ∧ d x J = ∑ I , J a I b J d x i 1 ∧ ⋯ ∧ d x i k ∧ d x j 1 ∧ ⋯ ∧ d x j s = ∑ I , J a I b J ( − 1 ) k d x j 1 ∧ d x i 1 ∧ ⋯ ∧ d x i k ∧ d x j 2 ∧ ⋯ ∧ d x j s = ∑ I , J a I b J ( − 1 ) k ( − 1 ) k d x j 1 ∧ d x j 2 ∧ d x i 1 ∧ ⋯ ∧ d x i k ∧ d x j 3 ∧ ⋯ ∧ d x j s ⋮ = ∑ I , J a I b J ( − 1 ) k s d x j 1 ∧ d x j 2 ∧ ⋯ ∧ d x j s ∧ d x i 1 ∧ ⋯ ∧ d x i k = ( − 1 ) k s ∑ I , J a I b J d x I ∧ d x J = ( − 1 ) k s φ ∧ ω
\begin{align*}
\omega \wedge \varphi
=&\ \sum\limits_{I,J} a_{I}b_{J} dx_{I} \wedge dx_{J} \\
=&\ \sum\limits_{I,J} a_{I}b_{J} dx_{i_{1}} \wedge \dots \wedge dx_{i_{k}} \wedge dx_{j_{1}} \wedge \dots \wedge dx_{j_{s}} \\
=&\ \sum\limits_{I,J} a_{I}b_{J} (-1)^{k} dx_{j_{1}} \wedge dx_{i_{1}} \wedge \dots \wedge dx_{i_{k}} \wedge dx_{j_{2}} \wedge \dots \wedge dx_{j_{s}} \\
=&\ \sum\limits_{I,J} a_{I}b_{J} (-1)^{k}(-1)^{k} dx_{j_{1}} \wedge dx_{j_{2}} \wedge dx_{i_{1}} \wedge \dots \wedge dx_{i_{k}} \wedge dx_{j_{3}} \wedge \dots \wedge dx_{j_{s}} \\
&\ \vdots \\
=&\ \sum\limits_{I,J} a_{I}b_{J} (-1)^{ks} dx_{j_{1}} \wedge dx_{j_{2}} \wedge \cdots \wedge dx_{j_{s}} \wedge dx_{i_{1}} \wedge \dots \wedge dx_{i_{k}}\\
=&\ (-1)^{ks} \sum\limits_{I,J} a_{I}b_{J} dx_{I} \wedge dx_{J} \\
=&\ (-1)^{ks} \varphi \wedge \omega
\end{align*}
ω ∧ φ = = = = = = = I , J ∑ a I b J d x I ∧ d x J I , J ∑ a I b J d x i 1 ∧ ⋯ ∧ d x i k ∧ d x j 1 ∧ ⋯ ∧ d x j s I , J ∑ a I b J ( − 1 ) k d x j 1 ∧ d x i 1 ∧ ⋯ ∧ d x i k ∧ d x j 2 ∧ ⋯ ∧ d x j s I , J ∑ a I b J ( − 1 ) k ( − 1 ) k d x j 1 ∧ d x j 2 ∧ d x i 1 ∧ ⋯ ∧ d x i k ∧ d x j 3 ∧ ⋯ ∧ d x j s ⋮ I , J ∑ a I b J ( − 1 ) k s d x j 1 ∧ d x j 2 ∧ ⋯ ∧ d x j s ∧ d x i 1 ∧ ⋯ ∧ d x i k ( − 1 ) k s I , J ∑ a I b J d x I ∧ d x J ( − 1 ) k s φ ∧ ω
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Proof(3) Let ω = ∑ I a I d x I \omega = \sum\limits_{I} a_{I} dx_{I} ω = I ∑ a I d x I φ = ∑ J b J d x J \varphi = \sum\limits_{J} b_{J} dx_{J} φ = J ∑ b J d x J ϑ = ∑ J c J d x J \vartheta = \sum\limits_{J} c_{J} dx_{J} ϑ = J ∑ c J d x J
ω = ω ∧ ( φ + ϑ ) = ω ∧ ( ∑ J ( b J + c J ) d x J ) by definition of + = ( ∑ I a I d x I ) ∧ ( ∑ J ( b J + c J ) d x J ) = ∑ I , J a I ( b J + c J ) d x I ∧ d x J by definition of ∧
\begin{align*}
\omega
=&\ \omega \wedge (\varphi + \vartheta) \\
=&\ \omega \wedge \left( \sum\limits_{J} (b_{J} + c_{J}) dx_{J} \right) & \text{by definition of } + \\
=&\ \left( \sum\limits_{I} a_{I} dx_{I} \right)\wedge \left( \sum\limits_{J} (b_{J} + c_{J}) dx_{J} \right) \\
=&\ \sum\limits_{I,J} a_{I}(b_{J} + c_{J})dx_{I}\wedge dx_{J} & \text{by definition of } \wedge\\
\end{align*}
ω = = = = ω ∧ ( φ + ϑ ) ω ∧ ( J ∑ ( b J + c J ) d x J ) ( I ∑ a I d x I ) ∧ ( J ∑ ( b J + c J ) d x J ) I , J ∑ a I ( b J + c J ) d x I ∧ d x J by definition of + by definition of ∧
Since the functions a , b , c a, b, c a , b , c
ω = ∑ I , J a I b J d x I ∧ d x J + ∑ I , J a I c J d x I ∧ d x J = ω ∧ φ + ω ∧ ϑ by definition of ∧
\begin{align*}
\omega
=&\ \sum\limits_{I,J} a_{I}b_{J}dx_{I}\wedge dx_{J} + \sum\limits_{I,J} a_{I}c_{J}dx_{I}\wedge dx_{J} \\
=&\ \omega \wedge \varphi + \omega \wedge \vartheta & \text{by definition of } \wedge\\
\end{align*}
ω = = I , J ∑ a I b J d x I ∧ d x J + I , J ∑ a I c J d x I ∧ d x J ω ∧ φ + ω ∧ ϑ by definition of ∧
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