k-형식의 외미분

k-형식의 외미분

Exterior Differential of k-form

정의1

$\omega = \sum\limits_{I} a_{I} dx_{I}$를 $n$차원 미분다양체 $M$ 위의 $k$-형식이라고 하자. $\omega$의 외미분exterior differential $d\omega$를 다음과 같이 정의한다.

$$ d\omega := \sum\limits_{I} da_{I} \wedge dx_{I} $$

이때 $\wedge$는 쐐기곱이다.

설명

$da_{I}$는 $1$-형식이고 $dx_{I}$는 $k$-형식이므로, $d\omega$는 $(k+1)$-형식이다.

예제

$\omega$를 다음과 같이 주어진 $\mathbb{R}^{3}$에서의 $1$-형식이라고 하자.

$$ \omega = xyz dx + yz dy + (x + z)dz $$

그러면 $d\omega$는 다음과 같다.

$$ \begin{align*} d\omega &= d(xyz) \wedge dx + d(yz) \wedge dy + d(x+z)\wedge dz \\ &= (yzdx + xzdy + xydz) \wedge dx + (zdy + ydz) \wedge dy + (dx + dz) \wedge dz \\ &= xzdy \wedge dx + xydz \wedge dx + ydz \wedge dy + dx \wedge dz \\ &= -xzdx \wedge dy - xydx \wedge dz - ydy \wedge dz + dx \wedge dz \\ &= -xzdx \wedge dy + (1 - xy) dx \wedge dz - ydy \wedge dz \end{align*} $$

성질

(a) $\omega_{1}, \omega_{2}$가 $k$-형식일 때,

$$ d(\omega_{1} + \omega_{2}) = d\omega_{1} + d\omega_{2} $$

(b) $\omega$가 $k$-형식, $\varphi$가 $s$-형식일 때,

$$ d(\omega \wedge \varphi) = d\omega \wedge \varphi + (-1)^{k}\omega \wedge d\varphi $$

(c) $d(d\omega) = d^{2}\omega = 0$

(d) $N, M$이 각각 $n, m$차원 미분다양체, $\omega$가 $M$ 위의 $k$-형식, $f : N \to M$가 미분가능한 함수일 때,

$$ d(f^{\ast} \omega) = f^{\ast}(d\omega) $$

이때 $f^{\ast}\omega$는 $\omega$의 풀백이다.

증명

(a)

$\omega_{1} = \sum\limits_{I}a_{I}dx_{I}$, $\omega_{2} = \sum\limits_{I}b_{I}dx_{I}$라고 하자. 그러면 미분형식의 합과 쐐기곱의 성질에 의해,

$$ \begin{align*} d(\omega_{1} + \omega_{2}) &= d\left( \sum\limits_{I}(a_{I} + b_{I})dx_{I} \right) \\ &= \sum\limits_{I} d (a_{I} + b_{I}) \wedge dx_{I} \\ &= \sum\limits_{I} (d a_{I} + d b_{I}) \wedge dx_{I} \\ &= \sum\limits_{I} (d a_{I} \wedge dx_{I} + d b_{I} \wedge dx_{I}) \\ &= \sum\limits_{I} d a_{I} \wedge dx_{I} + \sum\limits_{I} d b_{I} \wedge dx_{I} \\ &= d\omega_{1} + d\omega_{2} \end{align*} $$

(b)

$\omega = \sum\limits_{I}a_{I}dx_{I}$, $\varphi = \sum\limits_{J}b_{J}dx_{J}$라고 하자. 그러면

$$ \begin{align*} d(\omega \wedge \varphi) &= d\left( \sum\limits_{I}a_{I}dx_{I} \wedge \sum\limits_{J}b_{J}dx_{J} \right) \\ &= d\left( \sum\limits_{I,J}a_{I}b_{J} dx_{I} \wedge dx_{J} \right) \\ &= \sum\limits_{I,J} d\left( a_{I}b_{J} \right) \wedge dx_{I} \wedge dx_{J} \\ &= \sum\limits_{I,J} \left( b_{J}da_{I} + a_{I}db_{J} \right) \wedge dx_{I} \wedge dx_{J} \\ &= \sum\limits_{I,J} b_{J}da_{I} \wedge dx_{I} \wedge dx_{J} + \sum\limits_{I,J} a_{I}db_{J}\wedge dx_{I} \wedge dx_{J} \\ &= \left( \sum\limits_{I} da_{I} \wedge dx_{I} \right) \wedge \sum\limits_{J}b_{J} dx_{J} + (-1)^{k}\sum\limits_{I,J} a_{I}dx_{I}\wedge db_{J} \wedge dx_{J} \\ &= d\omega \wedge \varphi + (-1)^{k} \left( \sum\limits_{I}a_{I}dx_{I} \right)\wedge \left( \sum\limits_{J} db_{J}\wedge dx_{J} \right) \\ &= d\omega \wedge \varphi + (-1)^{k} \omega \wedge d\varphi \\ \end{align*} $$

(c)

  • Case 1. $k=0$

$\omega$가 $0$-형식이라고 하자.

$$ \omega = f : M \to \mathbb{R} $$

그러면

$$ \begin{align*} d(d\omega) = d(df) &= d\left( \sum\limits_{i=1}^{n} \dfrac{\partial f}{\partial x_{i}}dx_{i} \right) \\ &= \sum_{i} d\left( \dfrac{\partial f}{\partial x_{i}} \right) \wedge dx_{i} \\ &= \sum_{i} \left( \sum_{j} \dfrac{\partial }{\partial x_{j}} \dfrac{\partial f}{\partial x_{i}} dx_{j}\right) \wedge dx_{i} \\ &= \sum_{i, j} \left( \dfrac{\partial^{2} f}{\partial x_{j} \partial x_{i}} dx_{j}\right) \wedge dx_{i} \\ &= \begin{cases} 0 & i = j \\ 0 & i \ne j \end{cases} \\ &= 0 \end{align*} $$

이때 $i=j$인 경우 $dx_{i} \wedge dx_{i} = 0$이므로 $0$이다. $i \ne j$인 경우에는

$$ \begin{align*} \left( \dfrac{\partial^{2} f}{\partial x_{j} \partial x_{i}} dx_{j}\right) \wedge dx_{i} + \left( \dfrac{\partial^{2} f}{\partial x_{i} \partial x_{j}} dx_{i}\right) \wedge dx_{j} \end{align*} $$


  1. Manfredo P. Do Carmo, Differential Forms and Applications, p8-9 ↩︎

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