That is what a double integral means.
这也就是二重积分的意义。
But let's do it as a double integral.
但是让我们用二重积分来做。
Now, how do we compute that double integral?
那么，怎么计算这个二重积分呢？
How do you express the area as a double integral?
如何用二重积分来表示面积呢？
We will end up with double integral on S of z dxdy.
求S上zdxdy的二重积分。
This side here is a usual double integral in the plane.
这边是平面上普通的二重积分。
You know how to compute a double integral of a function.
要懂得如何计算一个函数的二重积分。
Now I have everything I need to compute my double integral.
至此我已经得到了，用来计算二重积分的所有量。
The area R is the double integral over R of a function one.
区域R的面积是函数1在R上的二重积分。
We've seen various formulas for how to set up the double integral.
我们已经学过，如何建立这种二重积分的公式。
Use geometry or you need to set up for double integral of a surface.
总之，就是用几何方法或是在曲面上建立二重积分。
The double integral side does not even have any kind of renaming to do.
没有必要对二重积分重新命名了。
If it is a closed curve, we should be able to replace it by a double integral.
如果是一条闭曲线，也可以用二重积分来代替的。
Because a surface is a two-dimensional object, that will end up being a double integral.
由于表面是二维的，所以结果是二重积分。
Next, I should try to look at my double integral and see if I can make it equal to that.
然后观察二重积分,看看能不能使两式相等。
So, using Green's theorem, the way we'll do it is I will, instead, compute a double integral.
那么，使用格林公式，我们去计算二重积分。
So, if a curl was well defined at the origin, you would try to, then, take the double integral.
如果旋度在原点有定义，你就可以试试了，计算二重积分。
One example that we did, in particular, was to compute the double integral of a quarter of a unit disk.
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。
The introduction of one way to solve the problem of definite integral by means of double integral.
介绍利用二重积分解决有关定积分问题的一种方法。
Then I can actually -- --replace the line integral for flux by a double integral over R of some function.
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。
Yes? In case you want the bounds for this region in polar coordinates, indeed it would be 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。
And this is finally where I have left the world of surface integrals to go back to a usual double integral.
也就是最终要摆脱曲面积分，回到常规的二重积分。
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.
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。
In this paper, the symmetry of double integral in symmetric domain and its application are briefly introduced.
文章简单介绍了在对称区域上重积分的对称性及其应用。
The way we actually think of the double integral is really as summing the values of a function all around this region.
就二重积分来讲,它是对区域里函数值求总和。
But, you know, it gives you an example where you can turn are really hard line integral into an easier 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。
Double integral is a kind of common conversion technology, with high precision, strong anti-jamming capability, etc.
双积分是一种常用的转换技术，具有精度高，抗干扰能力强等优点。
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.
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 maybe we first want to look at curves that are simpler, that will actually allow us to set up the double integral easily.
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, now, if I compare my double integral and, sorry, my triple integral and my flux integral, I get that they are, indeed, the same.
比较这个二重积分的话，抱歉。。。，比较这个三重积分和通量积分，就可以看到，它们是一样的。
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