对于空间曲线上的第二类曲线积分也有类似的公式。
Similar formulae hold for a line integral of the second type taken over a space curve.
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.
对第二类曲线积分的教学方法进行了探索，提出了自己的做法。
This article expounds the teaching method reformation of second curvilinear integral, bring upped the own way of doing.
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.
假定波浪为正弦波，利用曲线积分和幂级数展开推导出板形公差。
Provided that the buckle as the sine-wave, the shape tolerance can be deduced by using the power series expansion as well as the curve integration.
给出把一类二重积分化为曲线积分的一个定理，讨论定理的一些应用。
In this paper, a theorem which turns double integral into linear integral is given, and its application is discussed.
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.
讨论了第一类曲线积分中值定理“中间点”的渐近性质，得到了更具一般性的新结果。
This paper is devoted to studying the asymptotic behavior of the intermediate point in the mean value theorem for first form curve integrals. A general result is obtained.
微积分基本定理，不是曲线积分的，告诉我们，如果对函数的导数积分，就会得回原函数。
So, the fundamental theorem of calculus, not for line integrals, tells you if you integrate a derivative, then you get back the function.
文章结合代数曲线积分思想与活性边表技术，提出了一种新的任意多边形代数积分算法。
Based on the Curvilinear Integral and its sorted edge table, a new polygon fill algorithm is developed in the article.
同时还指出了几本高等数学参考书中关于不定积分、二重积分、曲线积分计算中出现的错误。
And it also points out the mistakes in some Higher Mathematics Reference Books concerning indefinite integral, double integral and curve integral calculations.
本文在复连域上引进势函数的概念，并给出其计算方法，以此来解决一些复杂的曲线积分计算问题。
This paper introduces the concept of potential function into the multiply-connected region and provides its method of computation in order to solve some complex problems concerning line integrals.
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.
对教材《高等数学》（同济大学数学教研室主编，第四版）中关于两类平面曲线积分关系证明中的一处疏漏给出补充证明。
This paper makes some supplement for the error in demonstrating the relations between two curve line integrals in the textbook advanced mathematics (edited by Tongji university 4th edition).
文章研究了第一型曲线积分中值定理“中间点”的渐近性，获得了一些重要结果，得出它也是定积分中值定理相应结果的推广。
This paper discusses the asymptotic property of the Mid-point of the mean theorem for first form curvilinear integral.
应该怎样沿着围绕这个区域的曲线，做线积分呢？
How do I compute the line integral along the curve that goes all around here?
在平面上的线积分中，有两个变量，可以通过了解曲线的形成规律，从而去掉一个变量。
In the line integral in the plane, you had two variables that you reduced to one by figuring out what the curve was.
那我想计算那条曲线上的线积分。
And so I want to compute for the line integral along that curve.
但仍然想要沿着封闭曲线的线积分计算。
And, I still want to compute the line integral along a closed curve.
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.
如果是一条闭曲线，也可以用二重积分来代替的。
If it is a closed curve, we should be able to replace it by a double integral.
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。
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.
如果无法对曲线参数化，那么就很难计算线积分了。
If you cannot parameterize the curve then it is really, really hard to evaluate the line integral.
对于沿曲线的线积分。
先看看简单些的曲线的情形，这样我们解决二重积分会简单许多。
So maybe we first want to look at curves that are simpler, that will actually allow us to set up the double integral easily.
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。
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.
只是提醒你，是在封闭曲线上做积分。
It just reminds you that you are doing it on a closed curve.
如果你喜欢也可以，但是沿着这条路径积分却不好计算，尤其是还没告诉你曲线的定义。
I mean if that is your favorite path then that is fine, but it is not very easy to compute the line integral along this, especially since I didn't tell you what the definition is.
如果你喜欢也可以，但是沿着这条路径积分却不好计算，尤其是还没告诉你曲线的定义。
I mean if that is your favorite path then that is fine, but it is not very easy to compute the line integral along this, especially since I didn't tell you what the definition is.
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