At the last minute, I threw them a curve ball by saying, "We're going to bring spouses."
在最后一刻，我给了他们一个惊喜，说：“我们会带配偶。”
Then I go clockwise along the inside curve, then back along the slit.
再顺时针沿着里面的曲线走，又回到这个裂口。
And so, if I go counterclockwise around this region, basically I go counterclockwise along the outer curve.
如果沿着区域边界逆时针走，基本上，是逆时针沿着外面的曲线走。
And when I am on the curve I am on the boundary of the surface, so there is a direction along the surface that is the curve and the other one is pointing into the surface.
当我在曲线上时，我是在曲面的边上的，所以一个是沿着曲面的边指，那个就是曲线的定向，另外一个就要指向曲面里面的那个方向。
I want my thumb to go along the curve so that is kind of towardS the right.
要让大拇指沿着曲线指，所以看上去差不多就是向右边。
How do I compute the line integral along the curve that goes all around here?
应该怎样沿着围绕这个区域的曲线，做线积分呢？
So, I will have some parametric curve that lives on this surface so, the question is, what's going to happen at any given time?
也就是我在此平面上有一个参数曲线，所以问题就是，在任意给定时刻会发生什么呢？
I don't see it that way at all. I think it's a sine curve, like most of our lives.
我并不这样认为，我认为那是一个正弦曲线，就像大部分人的人生一样，有起有伏。
Well, I claim this curve actually bounds another surface that is orientable.
我断言这条曲线，界定了另一个可定向的曲面。
I touch the padded curve of the ceiling for added balance as I walk, and I smile when I reach the cockpit.
我一边走，一边摸着机舱顶部的弯曲软垫，以增加平衡。到达驾驶舱时，我笑了。
Lines and Curves - a painting professor I once had said that every curve was made up of tiny straight lines, this stayed with me.
线与曲线-我的绘画教授曾经说过，每条曲线都是由很多短小的直线构成的。
And so I want to compute for the line integral along that curve.
那我想计算那条曲线上的线积分。
Flux, on the other hand, measures, when I go along the curve, roughly how much the field is going to across the curve.
从另一方面来看，“通量”度量的是，沿着曲线前进时，大致会有多少向量场通过曲线。
Let's say that I gave you this curve bounding this surface.
给你这个曲面和这个曲线作为它的边。
Then, I can actually find a curve that goes in that direction, and stays on the level.
那么可以找到一条这个方向的曲线，而且这曲线位于等值面上。
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。
See, I want a surface that stops on this curve, and doesn't go beyond it.
我想要一个落在这条曲线上的曲面，但是不超过这条曲线。
What if I give you a really complicated curve and then you have trouble finding the normal vector?
如果给你的是，一条很复杂的曲线，而又找不到法向量？
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。
The surgeon had followed with religious fervor the curve of her flesh, I promise you that.
我向你保证，她的外科医生以他们对忠实信仰的名义，已经尽力帮她修补面部的曲线了。
I need to be on a closed curve to do it.
我需要在一条封闭曲线上来做。
What I do at any point is project F to the tangent direction, I figure out how much F is going along my curve and then I sum these things together.
我所做的就是把F投影到切向量方向上，得出F沿着曲线的值,然后再把这些加起来。
Roughly-speaking the work measures, you know when I move along my curve, how much I am going with or against F.
粗略来讲，“功”度量的是，沿着曲线前进，F做多少功或克服F做多少功？
Let's say that I have a plane curve and a vector field in the plane.
有一条平面曲线和这个平面上的向量场。
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。
Now, as long as I am in the linear portion of the curve, I can generate a simple harmonic motion.
既然现在处在，曲线的线性区域内，可以弄出个简谐运动。
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。
If I move along this closed curve, I start at the origin.
沿这条闭曲线，从原点出发。
C So, let's say that I have a curve, c, in space.
假设有一条在空间中的曲线。
I get a curve, and I looked at the slope of that curve.
我得到了曲线，然后观察曲线的斜率。
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