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.
丹尼斯赞扬了富勒在 Valse à laLoie 里的二重唱。
二重对数是对数的积分。
追求揭示了现代大学的二重性。
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.
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.
这是极坐标系下的二重积分。
如果是一条闭曲线，也可以用二重积分来代替的。
If it is a closed curve, we should be able to replace it by a double integral.
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.
要计算二重积分，要做的就是要利用切面。
So, to compute this integral, what we do is actually we take slices.
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.
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.
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.
请说，你想知道极坐标系下的积分边界，这是一个二重积分。
Yes? In case you want the bounds for this region in polar coordinates, indeed it would be double integral.
当计算二重积分时，要多了解这些符号的具体含义。
OK, we'll come up with more concrete notations when we see how to actually compute these things.
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.
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.
举个例子，区域R的面积是dA的二重积分，便于理解，在这里写成。
So, for example, the area of region is the double integral of just dA, 1dA or if it helps you, one dA if you want.
称之为区域R上fdA的二重积分,会向大家解释这些符号的含义的。
So, we'll call that the double integral of our region, R, of f of xy dA and I will have to explain what the notation means.
也就是二重和三重积分的内容，以及平面和空间中的向量积分。
And so that was stuff about double and triple integrals and vector calculus in the plane and in space.
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.
第一点就是，有些你以为是用一重积分来做的，但却通常是用二重积分来完成的。
The first one that I will mention is actually something you thought maybe you could do with a single integral, but it is useful very often to do it as a double integral.
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.
计算质量可看成二重或三重积分，这取决于空间维数，取决于密度函数，dA还是。
And mass will be double or triple integral, depending on how many dimensions you have, dV of whatever density function you have, dA or dV.
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.
那么就是在这个区域的对xdA的二重积分，当然可能和密度有关系，但在这认为密度均为。
So, it's one over the area times the double integral of xdA, well, possibly with the density, 1 but here I'm thinking uniform density one.
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