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。
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。
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。
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。
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。
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。
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。
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.
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。
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。
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。
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。
On a 64-bit system, integral types are converted to 64-bit integral types, and single precision floating point types are promoted to double precision.
在 64位系统上，整型被转换成 64 位的整型值，单精度的浮点类型被转换成双精度的浮点类型。
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。
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。
As a new installation method for big structures, double girder integral lift is mainly composed of fixed girder, moveable girder, work anchor, moveable anchor, jacks and guide rail.
双梁式整体提升法是一种新型的大型结构安装方法，主要由固定梁、活动梁、钢绞线、工作锚、工具锚、导向装置及千斤顶组成，与现在常用的安装方法相比，具有价廉、安全、高效和易推广等特点。
And it also points out the mistakes in some Higher Mathematics Reference Books concerning indefinite integral, double integral and curve integral calculations.
同时还指出了几本高等数学参考书中关于不定积分、二重积分、曲线积分计算中出现的错误。
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