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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.

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二重对数是对数的积分。

The dilogarithm is an integral of the logarithm.

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他可能只是患有二重性记忆错误。

He might just suffer from Reduplicative Paramnesia.

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看两只猫的滑稽二重唱！

Look at the Duetto buffo di due gatti!

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那么，怎么计算这个二重积分呢？

Now, how do we compute that double integral?

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如何用二重积分来表示面积呢？

How do you express the area as a double integral?

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这边是平面上普通的二重积分。

This side here is a usual double integral in the plane.

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追求揭示了现代大学的二重性。

The pursuit reveals the duality of the modern college.

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没有必要对二重积分重新命名了。

The double integral side does not even have any kind of renaming to do.

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要懂得如何计算一个函数的二重积分。

You know how to compute a double integral of a function.

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一个是建立并计算二重积分。

One is setting up and evaluating double integrals.

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但是让我们用二重积分来做。

But let's do it as a double integral.

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这也就是二重积分的意义。

That is what a double integral means.

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这是由于我们才刚开始做二重积分的缘故。

That is because we are only doing double integrals so far.

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这也就说明了，可以用极坐标做二重积分。

The claim is we are able, to do double integrals in polar coordinates.

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求S上zdxdy的二重积分。

We will end up with double integral on S of z dxdy.

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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.

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在和简表演二重唱时，玛丽唱得非常好。

Mary sang very well when duetting with Jane.

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区域R的面积是函数1在R上的二重积分。

The area R is the double integral over R of a function one.

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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.

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这是极坐标系下的二重积分。

Anyway, that is double integrals in polar coordinates.

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来想想二重积分的其他用途吧。

Let's think about various USES of double integrals.

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