The paper introduces a process of 3 1/2 bite digital voltage using double-integral style A/D transform 5G14433chip as the core.
介绍了选用双积分型A/D转换器5G14433芯片为核心设计组装一个3（1/2）位数字电压表的实现过程。
Now, how do we compute that double integral?
那么，怎么计算这个二重积分呢？
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。
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。
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。
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。
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。
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。
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。
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。
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。
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。
This side here is a usual double integral in the plane.
这边是平面上普通的二重积分。
And this is finally where I have left the world of surface integrals to go back to a usual double integral.
也就是最终要摆脱曲面积分，回到常规的二重积分。
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。
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。
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。
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。
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, 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。
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。
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。
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。
There is a similar thing with the divergence theorem, of course, with flux and double integral of div f, you can apply exactly the same argument.
有一个和散度定理很像的东西，当然，对于通量和div,f的二重积分，都可以使用类似的理论。
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。
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。
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 will end up with double integral on S of z dxdy.
求S上zdxdy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。
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