# 普通和偏微分方程的数值方法 Numerical Methods for Ordinary and Partial Differential Equations  MATH336

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$d W=P d V$
and define the work done by the fluid along $\Gamma$ to be
$$\mathbb{W}(\Gamma)=\int_{\Gamma} d W=\int_{\Gamma} P d V$$
We likewise define the heating 1-form
$$d Q=C_{V} d T+\Lambda_{V} d V$$
and define the net heat gained by the fluid along $\Gamma$ to be
$$\mathbb{Q}(\Gamma)=\int_{\Gamma} d Q=\int_{\Gamma} C_{V} d T+\Lambda_{V} d V$$

## MATH336 COURSE NOTES ：

Therefore
\begin{aligned} \mathbb{Q}^{-}=-\int_{V_{3}}^{V_{4}} \Lambda_{V} d V &=-R T_{1} \log \left(\frac{V_{4}}{V_{3}}\right) \ &=-R T_{1} \log \left(\frac{V_{1}}{V_{2}}\right)>0 \end{aligned}
The work is
\begin{aligned} \mathbb{W} &=\mathbb{Q}^{+}-\mathbb{Q}^{-} \ &=R\left(T_{2}-T_{1}\right) \log \left(\frac{V_{2}}{V_{1}}\right) \ &>0 \end{aligned}
and for later reference we deduce fromthat
$$\mathbb{W}=\left(1-\frac{T_{1}}{T_{2}}\right) \mathbb{Q}^{+} \quad \text { for a Carnot cycle of an ideal gas. }$$