在这里单连通的假设是有用的。
And, that's where the assumption of simply connected is useful.
全空间去掉原点，它是单连通的吗？
OK, so space with the origin removed, OK, you think it's simply connected?
这就是球面上，对单连通的定义。
OK, so that's the definition of the surface of a sphere being simply connected.
总之，这是关于单连通的习惯性提醒。
OK, so anyway, that's the customary warning about simply connected things.
谁认为它不是单连通的？
这是检验,该区域不是单连通的一种方法。
So, that's one way to check just for sure that this thing is not simply connected.
谁认为它是单连通的？
稍后我们会知道，在一个单连通区域上也可以。
任何具有单连通表面的物体都能简化成一个球体。
Any object with a simply connected surface can be smoothed out to look like a sphere.
证明了皱褶立方体的单连通性与一致局部单连通性。
We prove that the crumpled cube is simply-connected and uniformly locally simply-connected.
那么我们是在哪里利用了，“定义在单连通区域”的假设呢？
OK, so where do we use the assumption of being defined in a simply connected region?
提出一种基于边界标注的单连通区域扫描线填充快速算法。
A novel edge - labeled algorithm for simple connected area scan filling is presented.
借助保形变换将单连通区域及其边界化为上半平面与实轴。
With the help of the conformal transformation, we transform a simply connected region and its boundary into the upper half plane and real axis.
该算法能处理多个区域的情况和区域为单连通或复连通区域的情况；
The improved algorithm can deal with multiple regions, single connected domain and multiple connected domain.
并由此讨论了紧致的非单连通黎曼流形上无穷多的闭测地线存在性问题。
Also, the paper discuss the existence of the infinite closed geodesics of a compact no-simply connected Riemannian manifold.
对于它，不用太担心，为了精确起见，需要向量场定义在一个单连通区域中。
Let's not worry too much about it. For accuracy we need our vector field to be defined in a simply connected region.
本文把在单连通区域上成立的曲线积分与路线无关性定理推广到复连通区域。
In this paper the theorem in which a curve integral is independent of the integral path on a single connected region is generalized.
证明了拟常曲率流形中二维极小子流形上一个单连通区域为稳定的充分条件。
Sufficient conditions for a simply-connected domain of 2-dimensional minimal submanifold immerged in 2 + p-dimensional manifold of quasi-constant curvature to be stable were proved.
好滴，目前这里，在有定义的区域里，减掉原点，即移去原点，就不是单连通的。
OK, so for this guy, domain of definition, which is plane minus the origin with the origin removed is not simply connected.
给出了单连通区域和多连通区域中的一般级数表达式及确定级数项系数的方法。
A series expansion and a method to determine the coefficients of the series expansion are given.
结合算例管网，分析了单连通管和双连通管管网结构在限额供热工况下的水力工况；
The hydraulic results of spare structure with single connecting pipe and double connecting pipes of calculating pipe network under limited-heating supply state are analyzed.
在这个例子里，圈饼表面就是非单连通的，并且这类物体的简化就会像是至少有一个洞的曲面。
In this example, the surface is not simply connected and any smoothed-out object looks like a torus with at least one hole.
他在1904年就提出猜想：在四维空间，所有单连通的封闭三维面都能转化为一个三维球体。
His conjecture, made in 1904, was that in this four-dimensional world, all closed three-dimensional surfaces that are simply connected could be transformed to look like a three-dimensional sphere.
本文提出一种算法实现单连通区域的线性四元树表示转换成区域边界的4 -方向链码描述。
An algorithm is presented for converting the linear quadtree representation of a simply connected region into a 4-direction chain code description of the region's boundary.
单连通和非单连通的概念，研究哪些环路能界定曲面，可以用来对空间内部物体的形状进行分类。
This concept of being simply connected or not, and studying which loops bound surfaces or not can be used to classify shapes of things inside space.
对比传统单连通区域扫描线填充法，新方法算法效率高，实现简单，对复杂区域的填充同样适用。
Compared with the normal algorithms, the new one is efficient in algorithm, simple in realization for scan filling of simple connected area and also applicable to the filling of the complex area.
该算法能统一处理互感和无互感线路、单连通树支集的特定网络以及树枝集中有分支的通用电网络。
The algorithm can deal with mutual inductance or non mutual inductance network and especial 'a stroke' tree set network or general offset-trees network.
文中还定义了平面单连通多边形区域的所谓分层三角剖分，并确定了此剖分下二次样条空间的维数。
Then, a new kind of so-called stratified triangulation of a simply connected planar polygonal region is introduced.
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?
一个向量场,如果定义在单连通区域并且旋度为零，那么它就是一个梯度场，并且其上的线积分与路径无关。
OK, so, we've seen that if we have a vector field defined in a simply connected region, and its curl is zero, then it's a gradient field, and the line integral is path independent.
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