对于曲面积分，也已经知道如何计算。
这把一个向量场的线积分，和另外一个向量场的曲面积分联系起来。
This relates a line integral for one field to a surface integral from another field.
平面中的通量和空间中的通量有很大区别，在平面中，通量仅仅是线积分的另一种形式，而在空间中，它表现为曲面积分。
Flux looks quite different in the plane and in space because, in the plane, it is just another kind of line integral, while in space it is a surface integral.
下面应该如何，在这样的曲面上建立通量的积分。
Now, how would we actually set up a flux integral on such a 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.
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.
它可以对任一个小平面使用-，比如说对于这条曲线的线积分，等于通过这个曲面的旋度通量。
What it says on each small flat piece — it says that the line integral along say, for example, this curve is equal to the flux of a curl through this tiny piece of surface.
这个曲面积分，不是同一个向量场。
OK, and that surface integral, well, it's not for the same vector field.
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.
计算这个曲面积分的方法，和其他任何曲面积分的一样。
And, the way in which you would compute the surface integral is just as with any surface integral.
就是做F·dS或是F·ndS的二重积分，为了能建立积分，需要用到曲面的几何性质，这与该曲面的类型有关。
Double integral of F.dS or F.ndS if you want, and to set this up, of course, I need to use the geometry of the surface depending on what the surface is.
的法向量似乎是指向外部。，And, the,normal, vector, to,s, seems, to, be, pointing, out wards, everywhere。，如果我们有一个法向量指向外部的封闭曲面，并且我们要求出它的一个通量积分，那么我们可以用一个三重积分来代替这个通量积分。
S if we have a closed surface with a normal vector pointing outwards, and we want to find a flux integral for it, well, we can replace that with a triple integral.
在我花了这么多时间来告诉你们，如何计算曲面积分之后，我打算告诉你们，如何避免计算它们。
After spending a lot of time telling you how to compute surface integrals, now I am going to try to tell you how to avoid computing them.
总之，就是用几何方法或是在曲面上建立二重积分。
Use geometry or you need to set up for double integral of a surface.
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.
如果在这个曲面积分中是以dx,dy,dz结尾的，则表明出现了很大的问题。
If you end up with a dx, dy, dz in the surface integral, something is seriously wrong.
给出曲面积分在空间坐标的正交变换下的一个计算公式。
A calculating formula for surface integrals under orthogonal transformation of space coordinates is given.
为了理解(5.9.22)中其余那些曲面积分的意义，我们首先考察对应于偶源的解。
To understand the meaning of the remaining surface integrals in (5. 9. 22) we first investigate the solution corresponding to a doublet source.
给出第二型曲面积分计算的几种方法，并举例说明了这几种方法的应用。
In this paper, we give a few methods for calculating second-kind surface integral and their applications.
对曲面积分中值定理，给出了一个新的证明，并举出相关例子加以应用。
In this paper, a new proving of the mean value theorem of integral on surface is given, with some application in related cases presented.
入口截面不消耗剪切功率，然后用上界定理与曲面积分方法首次得到了用余弦模拉拔方棒时变形力的解析解。
Then with the upper-bound theorem and the surface integral an analytical solution of the drawing stress was first obtained.
掌握常见的曲面方程的识记规律，不仅能轻松建立空间图形，而且为多元函数积分学的学习打下坚实的基础。
To master the law of common surface equation, not only can easily establish a space graphics, but can lay a solid basis for learning multi-function.
本文建立了一种特殊的第一型曲面积分与第一型曲线积分的转化公式，并通过实例表明该方法在解决问题时所带来的方便。
This paper gives the conversion formula from the first type surface integral to the first type curvilineal, and sets a example of using the method to solve exercises.
引入积分绝对平均曲率来描述可定向闭曲面的平均弯曲程度。
Integral absolute mean curvature is introduced to describe average curving of an orientable closed surface.
它建立了有向曲面上的曲面积分与它的边界曲线积分的关系。
It gives a relationship between a surface integral over an oriented surface and a line integral along a simple closed curve.
文章把这些方法推广到曲线积分和曲面积分中，并给出了证明。
This article popularizes these methods in the calculation of curvilinear integral and surface integral and gives proof of them as well.
水利行业经常要进行流量计算，这样就会遇到曲面拟合和曲面积分的问题。
The calculation of flow is often needed in water conservancy, and surface-fitting and double integration are usually met in order to solve this problem.
告诫学生使用高斯公式计算曲面积分时一定不能忽视条件，否则可能导致错误。
It reminds the learners to notice the formula premise when using Gauss Formula, to avoid obtain a wrong result.
本文探讨了对称性在第二类曲线积分和第二类曲面积分中的应用，给出了一些有用的结论，并举例说明。
On the base of these notions, the second mean valued theorems for the second type curve integral are proved.
本文探讨了对称性在第二类曲线积分和第二类曲面积分中的应用，给出了一些有用的结论，并举例说明。
On the base of these notions, the second mean valued theorems for the second type curve integral are proved.
应用推荐