也就是通量的格林公式——散度公式。
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。
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.
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.
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.
如果知道了，向量空间在单连通区域内处处有定义，那么就可以毫无顾忌地，在这个区域里使用格林公式了？
Well, if you know that your vector field is defined everywhere in a simply connected region, then you don't have to worry about this question of, can I apply Green's theorem to the inside?
我想让你们看到，格林公式是怎么在这个平面中运用的，但是也牵涉到功和旋度等等，这也是比较特别的地方。
Stokes' versus Green. I want to show you how Green's theorem for work that we saw in the plane, but also involved work and curl and so on, is actually a special case of this.
牛顿-莱布尼茨公式、高斯公式、格林公式和斯托克斯公式是积分学中非常重要的公式，相互间的联系非常紧密。
Newton-Leibniz's, Green's, Gauss's and Stokes's formula are important for integral theory and there is close relation each other.
为此先推导离散格林函数的权模估计和有限元解的渐近不等式展开，然后给出公式的证明。
For this, we derive the weighted estimates for discreet Green function and the asymptotic error expansion inequalities, and then the proofs of the formulas are given.
第三章介绍了非平衡格林函数方法及其在介观输运中的应用，推导了广泛应用的运动方程以及电流公式。
Chapter three introduces the nonequilibrium Green function method and its applications in mesoscopic transport, and the widely-employed Keldysh equation of motion with a current formula are induced.
本文给出了双粒子格林函数的能谱方程及其简单的推导过程，而且还推导出了在TDA近似下的顶角因子的二级近似公式。
We give the spectra equation of double particle Green Function and the process eliting it. As well as the second approximate formula of the TDA approximate diagram is also given.
本文采用并矢格林函数和场量变换方法，给出了圆形波导中偶极天线辐射的普遍公式。
The general formulas of radiation from dipole antenna in a circular waveguide are given by means of dyadic Green's function and field transformation.
本文从微扰计算的普遍公式出发，给出了节块格林函数法(NGFM)下微扰计算的具体公式。
Based on the generalized perturbation formula, the detailed formula for Nodal Green's Function Method (NGFM) is derived.
本文采用并矢格林函数和场量变换给出了椭圆波导中探针天线辐射的普遍公式。
In this paper, the general formulas of probe antenna radiation in an elliptic waveguide are given by means of dyadic Green's function and field transformation methods.
本文采用并矢格林函数，场量变换和反应概念讨论了矩形波导中线天线的辐射和互耦合，给出了场强和互阻抗的普遍公式。
The radiation and coupling of linear antennas in a rectangular waveguide are discussed by means of dyadic Green's function, field transformation and reaction concept.
本文采用并矢格林函数，场量变换和反应概念讨论了矩形波导中线天线的辐射和互耦合，给出了场强和互阻抗的普遍公式。
The radiation and coupling of linear antennas in a rectangular waveguide are discussed by means of dyadic Green's function, field transformation and reaction concept.
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