举个简单的例子，想想两个人跳华尔兹，或者弹钢琴二重唱。
As simple examples, think of two people waltzing, or playing a piano duet.
二重对数是对数的积分。
看两只猫的滑稽二重唱！
追求揭示了现代大学的二重性。
在和简表演二重唱时，玛丽唱得非常好。
那么，使用格林公式，我们去计算二重积分。
So, using Green's theorem, the way we'll do it is I will, instead, compute a double integral.
这是极坐标系下的二重积分。
关于在xy坐标系里建立二重积分有问题吗？
OK, any questions about how to set up double integrals in xy coordinates?
我们已经学过，二重积分和线积分。
We've learned about double integrals, and we've learned about line integrals.
就二重积分来讲,它是对区域里函数值求总和。
The way we actually think of the double integral is really as summing the values of a function all around this region.
也就是最终要摆脱曲面积分，回到常规的二重积分。
And this is finally where I have left the world of surface integrals to go back to a usual double integral.
然后观察二重积分,看看能不能使两式相等。
Next, I should try to look at my double integral and see if I can make it equal to that.
第二重要的，上帝将生命之气吹进亚当的鼻孔。
Second and significantly, God himself blows the breath of life into Adam's nostrils.
如果旋度在原点有定义，你就可以试试了，计算二重积分。
So, if a curl was well defined at the origin, you would try to, then, take the double integral.
我和迈克唱二重唱。
我们已经做过的一个例子是，计算四分之一单位圆上的二重积分。
One example that we did, in particular, was to compute the double integral of a quarter of a unit disk.
先看看简单些的曲线的情形，这样我们解决二重积分会简单许多。
So maybe we first want to look at curves that are simpler, that will actually allow us to set up the double integral easily.
这个古典作品的特点是花腔女高音和次女高音之间的一段二重奏。
This classic piece features a duet between a coloratura soprano and and a mezzo-soprano.
做三重积分，和二重一样，当然，我们会有更多的坐标系。
When we do triple integrals in space, well, it is the same kind of story, except now we have, of course, more coordinate systems.
邻近的大厅安装着汽轮机，这是核电厂里第二重要的单元设备。
In the neighboring hall there is the turbine - the second most important unit in a nuclear power plant.
研究者称，发现导致肾癌的第二重要致病基因是一个重大进步。
The researchers said the discovery of a second major gene linked to renal cancer was a major advance.
信息治理是指与企业第二重要的资源即信息资源相关的决策活动。
Information governance represents the decision making activities for the second-most important resource of the enterprise - its information resources.
如果你还不是完全清楚，建议你去复习一下，如何建立二重积分。
OK, if that's not completely clear to you, then I encourage you to go over how we set up double integrals again.
这种二重性从一开始便存在，就在谁发现了艾滋病致病病毒问题上。
Such dualism was there from the beginning, in the question of who discovered the AIDS-causing virus.
另一件关于二重积分的是，我们已经讲过了，如何做更复杂的变量变换。
OK, now another thing we've seen with double integrals is how to do more complicated changes of variables.
那么我就能名正言顺地,用R上的某个函数的二重积分来替代通量的线积分。
Then I can actually -- --replace the line integral for flux by a double integral over R of some function.
但是你知道，它给了你一个例证，其中你可以，把复杂的线积分化成简单些的二重积分。
But, you know, it gives you an example where you can turn are really hard line integral into an easier double integral.
主办此次峰会的加拿大在周二重申其支持采取进一步制裁，但是语气有所缓和。
Canada, which is hosting the meeting, has until now said it backs the idea of more sanctions, but softened its tone on Tuesday.
主办此次峰会的加拿大在周二重申其支持采取进一步制裁，但是语气有所缓和。
Canada, which is hosting the meeting, has until now said it backs the idea of more sanctions, but softened its tone on Tuesday.
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