OK, so basically that tells you, you can still play tricks with Green's theorem when the region has holes in it.
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。
The Green here is the same Green as in Green's theorem, because somehow that is a space version of Green's theorem.
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。
But, you can't use Green's theorem directly if the curve is not closed.
但是，如果曲线不是封闭的，不能直接使用格林公式。
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.
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。
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.
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。
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.
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。
What it says is if s is a closed surface — Remember, it is the same as with Green's theorem, we need to have something that is completely enclosed.
它说的是，如果s是一个封闭曲面，这和格林定理是一样的，我们需要完全封闭的。
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?
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？
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。
So, there is an extended version of Green's theorem that tells you the following thing.
于是，就有格林公式的推广，它描述了如下内容。
So, that's the end of the proof. OK, so you see, the idea is really the same as for Green's theorem.
这就是最终的证明了，证明的思路和证明格林定理的是一样的。
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。
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。
In this paper, we study the existence of positive solutions to second - order nonlinear ordinary differential equations by using fixed point theorem in cones and Green's function.
本文主要运用锥不动点定理和格林函数研究二阶非线性常微分方程组正解的存在性。
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.
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.
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。
应用推荐