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。
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。
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。
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。
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。
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, that means that the double integral for flux through the top of R vector field dot ndS becomes double integral of the top of R dxdy.
这就是说通量的二重积分,顶部R•ndS的二重积分,变成了Rdxdy的二重积分。
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.
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。
To give better advice on teaching, futher discussion, is made on how to simplify the double integral operation with symmetry, precisely and effectively.
为更好地指导教学，文章还对如何准确、有效地利用对称性简化二重积分的计算作了进一步的探讨。
Then, yes, we can apply Green's theorem and it will tell us that it's equal to the double integral in here of curl F dA, 0 which will be zero because this is zero.
那就可以使用格林公式了，并且我们知道,它就等于的二重积分，结果为0，因为旋度F等于。
From the theory and examples, the article points out why in the symmetric block the calculation of the double integral is easy to go wrong and offers some methods to avoid errors.
从理论和实例两个方面，指出了对称区域上二重积分计算中易出现问题的原因，并给出了避免错误的方法。
So, one of them says the line integral for the work done by a vector field along a closed curve counterclockwise is equal to the double integral of a curl of a field over the enclosed region.
其中一种说明了，在向量场上，沿逆时针方向，向量做的功等于，平面区域上旋度F的二重积分。
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。
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。
This side here is a usual double integral in the plane.
这边是平面上普通的二重积分。
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。
No matter which form it is, it relates a line integral to a double integral Let's just try to see if we can reduce it to the one we had yesterday.
不管哪种形式，都把线积分和二重积分联系在一起，来看看，能不能通过化简得到昨天的公式。
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.
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。
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, 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。
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.
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。
One way to think about it, if you're really still attached to the idea of double integral as a volume what this measures is the volume below the graph of a function one.
一种考虑这个问题的办法是，如果你还觉得，二重积分是求体积的话，那这个度量的，就是函数1的图形下的体积。
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.
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。
How do you express the area as a double integral?
如何用二重积分来表示面积呢？
That is just going to be, if you look at this paraboloid from above, all you will see is the unit disk so it will be a double integral of the unit disk.
这就会变成…，如果俯视这个抛物面，所看到的就是单位圆盘，这就应该是单位圆上的二重积分了。
To compute things, Green's theorem, let's just compute, well, let us forget, sorry, find the value of a line integral along the closed curve by reducing it to double integral.
用格林公式计算…，只是计算…，让我们忘记…，应该是，算沿闭曲线的线积分值，可以通过二重积分来算。
So, switching the area, moving the area to the other side, I'll get double integral of xdA is the area of origin times the x coordinate of the center of mass.
那么，改变一下区域，把这块移到另一侧，我们得到对xdA的双重积分,是原点那的圆面积乘质心的x的坐标。
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