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--- 수학:범함수 [2020/01/09 17:23] – [감쇠 오일러 방정식의 분석] admin
+++ 수학:범함수 [2020/01/10 11:25] – [첫 번째 항] admin
@@ Line 60: / Line 60: @@
 - \int \rho(\vec{r}') \int^{\rho(\vec{r}')} \frac{P(\rho')}{\rho'^2} d\rho' d\vec{r}'
 \right\}\\
-&=&
+&\approx&
 \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}') + \epsilon \delta (\vec{r}'-\vec{r}) \right]
 \left[ \int^{\rho(\vec{r}')} \frac{P(\rho')}{\rho'^2} d\rho'
@@ Line 73: / Line 73: @@
 &=&
 \frac{P[\rho(\vec{r})]}{\rho(\vec{r})} + \Psi[\rho(\vec{r})]
+\end{eqnarray*}
+====두 번째 항====
+\begin{eqnarray*}
+\frac{\delta}{\delta \rho(\vec{r})} \int \rho(\vec{r}') \Phi(\vec{r}') d\vec{r}'
+&=& \frac{\delta}{\delta \rho(\vec{r})} \int \rho(\vec{r}') \int \rho(\vec{r}'') U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'\\
+&=& \lim_{\epsilon\to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}') + \epsilon \delta(\vec{r}-\vec{r}') \right]
+\left[ \rho(\vec{r}'') + \epsilon \delta(\vec{r}-\vec{r}'') \right] U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'
+- \int \rho(\vec{r}') \int \rho(\vec{r}'') U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'
+\right\}\\
+&\approx&
+\int \rho(\vec{r}') \delta(\vec{r}-\vec{r}'') U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'
++\int \rho(\vec{r}'') \delta(\vec{r}-\vec{r}') U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'\\
+&=& \int \rho(\vec{r}') U(|\vec{r}'-\vec{r}|) d\vec{r}' + \int \rho(\vec{r}'') U(|\vec{r}''-\vec{r}|) d\vec{r}''\\
+&=& 2\Phi(\vec{r})
 \end{eqnarray*}