所以，我们不能直接使用格林公式。
这是格林公式的一种表示。
对于它们中的每一个，使用格林公式。
也就是通量的格林公式——散度公式。
一种是直接计算，另一种则是格林公式。
One is direct calculation, and the other one is Green's theorem.
请记得做功的例子，可以用格林公式的。
格林公式是另一种可以,避免计算线积分的方法。
So, Green's theorem is another way to avoid calculating line integrals if we don't want to.
它说了什么呢？它是三维空间中通量的格林公式。
能看到什么？，可以发现做功的格林公式。
What have we seen? Well, we have seen Green's theorem for work.
这是封闭曲线，所以我们可以用格林公式。
That's a closed curve. So, I would like to use Green's theorem.
平面上的就是格林公式。
我们已经了解了，格林公式的两种表达形式。
So, we've seen the statement of Green's theorem in two forms.
那么，使用格林公式，我们去计算二重积分。
So, using Green's theorem, the way we'll do it is I will, instead, compute a double integral.
把式子都加起来了，就得到了完整的格林公式。
When we add things together, we get Green's theorem in its full generality.
昨天讲了格林公式。
现实生活中有一个方面，格林公式曾经非常有用。
So, there's one place in real life where Green's theorem used to be extremely useful.
于是，就有格林公式的推广，它描述了如下内容。
So, there is an extended version of Green's theorem that tells you the following thing.
下面证明格林公式，这么怪的公式，怎么得到的呢？
So, I want to tell you how to prove Green's theorem because it's such a strange formula that where can it come from possibly?
但是，如果曲线不是封闭的，不能直接使用格林公式。
But, you can't use Green's theorem directly if the curve is not closed.
利用富比尼定理建立了非光滑函数的格林公式、高斯公式和斯托克斯公式。
In this paper, we establish Green's formula, Gauss's formula and stokes's formula of nonsmooth functions with the help of the Fubini Theorem.
负责任地告诉你们，当一个区域有个洞的时候,就可以这样巧妙地使用格林公式。
OK, so basically that tells you, you can still play tricks with Green's theorem when the region has holes in it.
对数学分析中的格林公式、高斯公式、斯托克斯公式的条件做了进一步的探讨。
This paper further probes into the conditions of Green formula, Gauss formula, and Stoces formula in mathematical analysis.
这就是为什么，这个线积分，有着完美的定义，但却不能对它使用格林公式的原因。
And so that's why you have this line integral that makes perfect sense, but you can't apply Green's theorem to it.
如果运用一下格林公式，你就发现当沿着一个逆时针的曲线时，结果就是区域的面积。
And, now, if you apply Green's theorem, you see that when you have a counterclockwise curve, this will be just the area of the region inside.
这里的“格林”和格林公式的“格林”是同一个人，因为这是格林公式在空间中的表述。
The Green here is the same Green as in Green's theorem, because somehow that is a space version of Green's theorem.
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。
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.
如果不喜欢计算线积分，可以通过增加一条线积分让曲线封闭起来，然后就可以用格林公式来计算了。
Or, if you really don't like that line integral, you could close the path by adding some other line integral to it, and then compute using Green's theorem.
通过挖掘格林公式的内在涵义，将其和微积分基本公式牛顿——莱布尼兹联系了起来，给出两点注记。
For a better understanding of Green Formula, this paper has analyzed the internal connotation and connected it with calculus basic formula and provided two notes.
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。
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。
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.
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