双曲曲面，即通常所称的负曲率，因为它的表面同时向上和向下弯曲，就像一个马鞍。
The hyperbolic surface, which has so-called negative curvature because its surface curves up and down at the same time, like a saddle.
大多数天线被制成碟形，用以收集和反射无线电波至亚反射器，就像曲面镜将可见光聚焦于一点一样。
Most antennas are made in the shape of a dish to collect and reflect the radio waves to the sub reflector, in the same way as a curved mirror focuses visible light to one point.
弯月面容器中之静止液体的弯曲上表面，如果液体浸润了容器壁，那么弯曲面呈凹状。
The curved upper surface of a nonturbulent liquid in a container that is concave if the liquid wets the container walls.
注意了，正在处理的东西都在曲面上。
And here, see, we are using the fact that we are only looking at things that are on the surface.
当我在曲面上移动一点，会发生什么呢？
Let's think a little bit about what happens when I move a little bit on my surface.
每当做通量的曲面积分时，你要做两件事。
Whenever you do a surface integral for flux you have two parts of the story.
事实上，这就是所谓的确定曲面定向的过程。
So, in fact, this is called choosing an orientation of a surface.
这就是切平面的，或者说这个曲面的法向量。
OK, and that's going to be the normal vector to the surface or to the tangent plane.
这个曲面是一个双曲抛物面。
这条曲线界定的曲面叫,叫做Mobius带。
OK, so one way to find a surface that's bounded by this curve is to take what's called the Mobius strip.
通量之所以为零，是因为根本没有东西通过曲面。
Flux is going to be zero because nothing passes through the surface.
它看上去并不是十分地相切，因为它穿过了曲面。
So, it doesn't look very tangent because it crosses the surface.
下面我们要搞清楚，这个向量场是如何穿过曲面的。
Well, I want to figure out how much my vector field is going across that surface.
也就是，向量场是连续的，且在曲面s上处处可微。
That means you need your vector field to be continuous and differentiable everywhere on the surface s.
能找到一个以该环路为界，且不经过原点的曲面吗？
Can I find a surface that's bounded by this loop and that does not pass through the origin?
也就是最终要摆脱曲面积分，回到常规的二重积分。
And this is finally where I have left the world of surface integrals to go back to a usual double integral.
我在这儿画一个圈来提醒你们，这是一个封闭曲面。
I am going to put a circle just to remind you it is has got to be a closed surface.
注意，我们一直在描述，通量就是流过曲面的流体的量。
Remember, we've been describing, flux means how much fluid is passing through this surface.
上图是她彩色柜的一个面，褶皱逐步过渡到极小的曲面。
Pictured above is her multi-colored cabinet with a faceted, pleated design that gradually fades into a flat minimal surface.
如果是在空间中，考虑的就是曲面，而非曲线。
Now, if we are in space then you take flux for a surface, not for a curve.
我想要一个落在这条曲线上的曲面，但是不超过这条曲线。
See, I want a surface that stops on this curve, and doesn't go beyond it.
在散度定理中的约定是,将曲面的定向取为外法线的方向。
The convention in the divergence theorem is that we orient the surface with a normal vector pointing always outwards.
在这儿，只是在一个二维曲面上做积分，这里是一维曲线。
Here, you integrate only over a two-dimensional surface, and here, only a one-dimensional curve.
你也可以这样记，如果曲面在你的右边,那么法向量就向下。
You can remember, if it helps you, that if a surface is to your right then the normal vector will go down.
我们想要找出，电场对于一个封闭曲面向外的通量。
Let's say that we want to find the flux of the electric field out of a closed surface.
她大胆进入第三维空间，绘制出曲面表示植物的轮廓。
She ventured into the third dimension to produce graphs that map the contours of plants.
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。
So if you had a closed surface you wouldn't know where to put your curve.
有两种选择，当你要建立通量的积分时,就必须取定曲面的正向。
There are two choices. Basically, whenever you want to set up a flux integral you have to choose one side of the surface.
如果你在圆锥的内侧走的话，那么这样这个曲面就在你的左边了。
Let's say instead that you are walking on the inside of a cone following the boundary. Well, then the surface is to your left.
在另一端，我们有优雅和圆滑的，具有C2曲面的物品。
On the other end we have C2 objects of elegance and sophistication.
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