所以我们也能从另一途径得到这个结果，但一个更简单，更直接的方式是直接，从波耳兹曼的结果入手得到熵。

So we could get this result from another pathway, too, but a more simple and direct way is just to start directly from the Boltzmann result for entropy.

youdao

你们告诉他们关于波耳兹曼的墓碑等事情。

You told them about Boltzmann's tombstone and so forth.

youdao

以此为基础，再加上波耳兹曼发现的结果，为了让我们得到并理解他的成果。

Based on that plus the result that Boltzmann fought and paid so dearly for, so that we would have it and understand it.

youdao

The West of the Germanic tribes that had toppled the Roman empire was weak and it was divided.

弱小分裂的西方日耳曼部族,推翻了罗马帝国

耶鲁公开课 - 古希腊历史简介课程节选

These Germanic and other tribes, who were largely located north and northeast of Europe, came down--I should say not Europe, but the Roman Empire-- came down and ultimately destroyed it by invasion.

那些日耳曼部落和其他部落,主要位于欧洲北部和东北部,的入侵,我应该说不是位于欧洲,而是位于罗马帝国,入侵并且完全摧毁了罗马帝国

耶鲁公开课 - 古希腊历史简介课程节选