由于表面是二维的，所以结果是二重积分。
Because a surface is a two-dimensional object, that will end up being a double integral.
这也就是二重积分的意义。
这是由于我们才刚开始做二重积分的缘故。
这边是平面上普通的二重积分。
要懂得如何计算一个函数的二重积分。
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.
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.
这段演唱听上去象是1983年迈克尔杰克逊跟保罗-麦卡特尼的二重唱，“说，说，说”。
The music sounds like the 1983 Michael Jackson-Paul McCartney duet, "Say, Say, Say."
区域R的面积是函数1在R上的二重积分。
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.
他的新技术完美地体现在他与当年的妻子玛丽·福特的二重奏《高高的月亮》之中。
His new techniques were showcased in hit songs such as "how High the Moon," a duet with his then-wife, Mary Ford.
就是做F·dS或是F·ndS的二重积分，为了能建立积分，需要用到曲面的几何性质，这与该曲面的类型有关。
Double integral of F.dS or F.ndS if you want, and to set this up, of course, I need to use the geometry of the surface depending on what the surface is.
二重对数是对数的积分。
求S上zdxdy的二重积分。
这份长达80页的手稿不仅是贝多芬的一部主要作品的唯一钢琴演奏版，也是他创作的为数不多的钢琴二重奏乐谱之一。
Not only is the 80-page document a working manuscript for the only piano version of a major work by Beethoven, it is one of his few compositions for a piano duet.
追求揭示了现代大学的二重性。
文章认为，解决实践理性二重性的关键不是要证明利己主义是完全错误的，而是要创建正义的社会制度、辩证地处理好个人利益与普遍利益的关系。
The key for us to resolve dualism of practical reason is to construct the just social system and harmonize individual interest and universal interest, but not to illustrate that egoism is fully wrong.
研究者埃玛·格雷格和史蒂芬·普鲁厄特·琼斯说，鹩莺的鸣叫是对伯劳鸟叫声的回应，所以它们几乎是在唱二重唱。
The researchers, Emma Greig and Stephen Pruett-Jones, say that the wren's song responds to the butcherbird's song, so that the two are almost singing a duet.
庄严的二重唱。
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.
不管是线积分或是二重积分，也不管它们表示的是功还是通量，计算它们的方法实际上是一样的。
And whether these line integrals or double integrals are representing work, flux, integral of a curve, whatever, the way that we actually compute them is the same.
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.
看两只猫的滑稽二重唱！
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.
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.
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.
唱吧，再和我唱一次，这奇怪的二重唱。
唱吧，再和我唱一次，这奇怪的二重唱。
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