That we can turn the sums into integrals.
以至于我们可以把求和变成积分。
So then we have integrals instead of sums.
那么我们用积分代替求和。
So, line integrals we know how to evaluate.
我们知道如何计算线积分。
OK, anyway, let's move on to line integrals.
再转到线积分。
Some of them compute different line integrals.
它们中的一些在做不同的线积分。
What if I add all of the small line integrals?
那如果将所有线积分都加起来呢？
I would just calculate three easy line integrals.
我只需计算三个简单的线积分。
OK. Let's see more ways of taking flux integrals.
来看看更多的取通量积分的方法。
One is setting up and evaluating double integrals.
一个是建立并计算二重积分。
You don't have to set up all these line integrals.
不用建立这些线积分。
Let's think about various USES of double integrals.
来想想二重积分的其他用途吧。
Let's start right away with line integrals in space.
现在开始学习空间线积分了。
So conceptually it is very similar to line integrals.
从概念上来看，这与线积分相似。
Anyway, that is double integrals in polar coordinates.
这是极坐标系下的二重积分。
So, the first topic will be setting up double integrals.
先来说说关于建立二重积分的事。
That is because we are only doing double integrals so far.
这是由于我们才刚开始做二重积分的缘故。
The other one is setting up and evaluating line integrals.
另一个是建立并计算线积分。
Other kinds of integrals we have seen are triple integrals.
我们学过的另一种积分是三重积分。
So, in fact, I have to break this into two different integrals.
实际上，要拆成两个不同的积分。
Unfortunately, it is not quite as simple as with line integrals.
很遗憾，它的过程不像线积分那样简单。
So now we're going to triple integrals in spherical coordinates.
现在，在球坐标中进行三重积分。
OK, so I proved that my two line integrals along C1 and C2 are equal.
我证出了分别沿着C1和C2的两个线积分是相等的。
But most of the time we need to learn how to set up double integrals.
但在大多数情况下，我们需要了解怎样建立二重积分。
The advantage of this method is you don't have to write any integrals.
这种方法的优点是，不用写出任何积分。
The claim is we are able, to do double integrals in polar coordinates.
这也就说明了，可以用极坐标做二重积分。
OK, any questions about how to set up double integrals in xy coordinates?
关于在xy坐标系里建立二重积分有问题吗？
We've learned about double integrals, and we've learned about line integrals.
我们已经学过，二重积分和线积分。
It is not a surprise that you will get the same answer for both line integrals.
对它们两个做线积分，得到相同结果，就不令人感到吃惊了。
See, this is actually good practice to remember how we set up triple integrals.
这是建立三重积分的绝佳练习。
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