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--- 수학:그라스만_대수 [2026/04/20 13:33] – [함께 보기] admin
+++ 수학:그라스만_대수 [2026/04/20 13:50] (current) – [적분] admin
@@ Line 24: / Line 24: @@
 즉 사실상 적분은 미분과 동일하다:
 $$\int d\theta = \frac{\partial}{\partial \theta}.$$
+여기에 따라오는 성질로서 다음과 같은 것들이 있다:
+$$\int d\theta = 0$$
+$$\int \theta d\theta = 1.$$
 ======가우스 적분======
 $n\times n$ 행렬 $M$에 대해
-$$\det M = \int d\theta_1 d\bar{\theta}_1 d\theta_1 d\bar{\theta}_1 \ldots d\theta_n d\bar{\theta}_n \exp\left( \sum_{i,j=1}^n \bar{\theta}_i M_{ij} \theta_j \right).$$
+$$\det M = \int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \ldots d\theta_n d\bar{\theta}_n \exp\left( \sum_{i,j=1}^n \bar{\theta}_i M_{ij} \theta_j \right).$$
+$n=2$인 경우를 예로 들면 이렇게 계산한다:
+\begin{eqnarray*}
+\int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \ldots d\theta_n d\bar{\theta}_n \exp\left( \sum_{i,j=1}^n \bar{\theta}_i M_{ij} \theta_j \right) &=&
+\int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \exp\left( \bar{\theta}_1 M_{11} \theta_1 + \bar{\theta}_1 M_{12} \theta_2 + \bar{\theta}_2 M_{21} \theta_1 + \bar{\theta}_2 M_{22} \theta_2 \right)\\
+&=& \int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \frac12 \left( \bar{\theta}_1 M_{11} \theta_1 + \bar{\theta}_1 M_{12} \theta_2 + \bar{\theta}_2 M_{21} \theta_1 + \bar{\theta}_2 M_{22} \theta_2 \right)^2\\
+&=& \int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \frac12 \left( \bar{\theta}_1 M_{11} \theta_1 \bar{\theta}_2 M_{22} \theta_2 + \bar{\theta}_1 M_{12} \theta_2 \bar{\theta}_2 M_{21} \theta_1 + \bar{\theta}_2 M_{21} \theta_1 \bar{\theta}_1 M_{12} \theta_2 + \bar{\theta}_2 M_{22} \theta_2 \bar{\theta}_1 M_{11} \theta_1 \right)\\
+&=& \frac{\partial}{\partial\theta_1} \frac{\partial}{\partial\bar{\theta}_1} \frac{\partial}{\partial \theta_2} \frac{\partial}{\partial\bar{\theta}_2} \left( \bar{\theta}_1 M_{11} \theta_1 \bar{\theta}_2 M_{22} \theta_2 + \bar{\theta}_1 M_{12} \theta_2 \bar{\theta}_2 M_{21} \theta_1 \right)\\
+&=& \frac{\partial}{\partial\theta_1} \frac{\partial}{\partial\bar{\theta}_1} \frac{\partial}{\partial \theta_2} \frac{\partial}{\partial\bar{\theta}_2} \left(  \bar{\theta}_2 \theta_2 \bar{\theta}_1 \theta_1 M_{11}M_{22} - \bar{\theta}_2 \theta_2 \bar{\theta}_1 \theta_1 M_{12} M_{21} \right)\\
+&=& M_{11} M_{22} - M_{12} M_{21}.
+\end{eqnarray*}
 반대칭 행렬 $A$에 대해 ($A_{ij}+A_{ji}=0$)