So if you had a closed surface you wouldn't know where to put your curve.
因此如果你有一个封闭的曲面，你就不知道怎么把曲线放进去了。
If we have a closed curve then the line integral for work is just zero.
如果给定一条封闭曲线，那么求所做功的线积分为零。
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。
And, I still want to compute the line integral along a closed curve.
但仍然想要沿着封闭曲线的线积分计算。
You know it's automatically OK because if you have a closed curve, then the vector field is, I mean, if a vector field is defined on the curve it will also be defined inside.
它必然成立，因为如果给出一条闭合曲线，然后向量空间是…，我是指，向量空间在曲线上有定义，当然在区域内部也有定义。
I need to be on a closed curve to do it.
我需要在一条封闭曲线上来做。
Well, you just take a closed curve in the plane.
只需在区域中取一条封闭曲线。
That is all pretty good.Let me tell you now what if I have to compute flux along a closed curve and I don't want to compute it.
很好，如果我们想做这样的一件事，我需要计算沿着一条闭曲线的通量，但又不想去计算。
If it is a closed curve, we should be able to replace it by a double integral.
如果是一条闭曲线，也可以用二重积分来代替的。
OK, so if I give you a curve that's not closed, and I tell you, well, compute the line integral, then you have to do it by hand.
如果给你们一条非封闭曲线，然后让你们计算线积分，你们必须动手一点点来计算。
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 remind ourselves that we are doing it along a closed curve, very often we put just a circle for the integral to tell us this is a curve that closes on itself.
为了提醒我们是在封闭曲线上做积分，经常在积分符号上加个圆圈，告诉我们，这条曲线自我封闭。
It just reminds you that you are doing it on a closed curve.
只是提醒你，是在封闭曲线上做积分。
It is true actually for any closed curve that the flux out of it is going to be twice the area of the region inside.
实际上这对任何闭曲线都适用，向外的通量还是面积的两倍。
Stokes says if I have a closed curve in space, now I have to decide what kind of thing it bounds.
空间上存在一条闭合曲线，我需要看出它是什么的边界。
If I move along this closed curve, I start at the origin.
沿这条闭曲线，从原点出发。
So now if I take, see, I can form a single closed curve that will enclose all of this region with kind of an infinitely thin slit here counterclockwise.
如果这么做的话，就能得到区域上的一条闭合曲线,它包围了这个区域。
If we apply Stokes' theorem to compute the work done by the electric field around a closed curve, that means you have a wire in there and you want to find the voltage along the wire.
如果用Stokes定理计算的话，电场对一条封闭曲线做的功，比如说，电厂中有一个金属线圈，想要知道金属线圈的电压是多少。
So, to say that a vector field with conservative means 0 that the line integral is zero along any closed curve.
一个保守的向量场就是说,沿任意闭曲线的线积分的结果是。
That means when you go along a closed curve, 0 well, the change of value of a potential should be zero.
这意味着，沿一条闭合曲线，势函数的变化量是。
The warping associated with the closed timelike curve could cause our slice to twist back on itself, making it impossible to divide all of space-time into distinct moments.
与封闭的时间型曲线相关的这种扭曲会导致我们的切分转回原处，不可能把所有的时空分成各自不同的时刻。
The space filling Z-order curve might not be the best choice for geodetic data, given the closed nature of the earth's surface.
对于大地数据，由于地区表面具有封闭的特性，因此空间填充z顺序曲线可能不是最好的选择。
A closed timelike curve seems to imply predestination: we know what is going to happen to us in the future because we witnessed it in our past.
一个封闭的时间型曲线可能暗示着命运注定：我们知道未来自己身上会发生什么状况，因为我们已经在自己的过去目睹了这些情景。
In general, events along a closed timelike curve cannot be compatible with an uninterrupted increase of entropy along the curve.
通常，沿着一个封闭的时间型曲线发生的事件不可能与沿着这个曲线不间断地增加的一致性相符合。
If we use a closed timelike curve to observe something about our future actions, those actions become predestined.
如果我们用一个封闭的时间型曲线来观察未来的行为，那么这些行为就成了预先注定的。
This ability vanishes as soon as someone builds a time machine and creates a closed timelike curve.
在有人制造出时间机器，又创造出一个封闭的时间型曲线后，这种预测能力很快就消失了。
Life on a closed timelike curve seems pretty drab.
在一个封闭的时间型曲线中，生活看来是很单调的。
That's a closed curve. So, I would like to use Green's theorem.
这是封闭曲线，所以我们可以用格林公式。
A closed plane curve resulting from the intersection of a circular cone and a plane cutting completely through it.
由完全相交的一个圆锥体和一个平面的交集所形成的闭合的平面曲线。
So, one of them says the line integral for the work done by a vector field along a closed curve counterclockwise is equal to the double integral of a curl of a field over the enclosed region.
其中一种说明了，在向量场上，沿逆时针方向，向量做的功等于，平面区域上旋度F的二重积分。
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