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。
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。
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。
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。
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。
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。
In this paper, a theorem which turns double integral into linear integral is given, and its application is discussed.
给出把一类二重积分化为曲线积分的一个定理，讨论定理的一些应用。
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。
And both the continuous double integral method and the virtual work method used in this paper can solve this problem well.
文中利用连续二次积分法和虚功法很好地解决了这一问题。
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 yellow lead making process is essentially a subject with multivariable coupling, double integral action, and nonminimun phase characteristics.
铅粉机是一个多变量耦合的、具有双重积分和本质非最小相位特性的对象。
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。
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。
This paper solves some special estimating value problems of double integral with the help of the optimality principle and graphical solution of the linear programming.
利用线性规划论的最优性原理和图解法解决一些特殊二重积分的估值问题。
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。
This article explores the four ways for solving integral inequality with the nature of definite integral, mean value theorem of differentials, Schwarz inequality and double integral.
本文利用定积分的性质、微分中值定理、施瓦兹不等式、二重积分等内容，研究了积分不等式的四种证法。
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