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.
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.
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.
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.
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?
如果给定一条封闭曲线，那么求所做功的线积分为零。
If we have a closed curve then the line integral for work is just zero.
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.
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.
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.
一个保守的向量场就是说,沿任意闭曲线的线积分的结果是。
So, to say that a vector field with conservative means 0 that the line integral is zero along any closed curve.
如果要计算沿这条曲线的线积分，我们不得不把它分解成3部分。
If we want to compute the line integral along this guy then we have to break it into a sum of three terms.
它们产生在一条曲线上，任何的积分都可以只用一个变量来表示，进行变量替换后，你得到一个，我们知道怎样计算的，单变量积分。
They take place in a curve.You express everything in terms of one variable and after substituting, you end up with a usual one variable integral that you know how to evaluate.
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.
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.
了解这两种表述后，我们不仅需要向量场，就是左边这里，这是曲线c上的线积分，向量场在曲线上有定义。
So, in both cases, we need the vector field to be defined not only, I mean, the left hand side makes sense if a vector field is just defined on the curve because it's just a line integral on c.
用格林公式计算…，只是计算…，让我们忘记…，应该是，算沿闭曲线的线积分值，可以通过二重积分来算。
To compute things, Green's theorem, let's just compute, well, let us forget, sorry, find the value of a line integral along the closed curve by reducing it to double integral.
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.
左边的这个积分定义在曲线，右侧的积分定义在整个内部区域。
This one on the left-hand side lives only on the curve, while the right-hand side lives everywhere in this region inside.
给出一条曲线，计算沿着这条曲线的积分。
Well, let's say I give you a curve and I ask you to compute this integral.
对我喜欢的曲线，计算其上的线积分，在这条线上所做的功。
And let's take my favorite curve and compute the line integral of that field, you know, the work done along the curve.
如果曲线c，起点为P0，终点为1，那么计算所做功的线积分，只与端点位置有关，而与我们选择的路径无关。
P1 If we have a curve c, from a point p0 to a point p1 then the line integral for work depends only on the end points and not on the actual path we chose.
实验结果表明，混合体系中各组分在核磁共振图谱中的积分曲线高度也具有加和性。
The result of experiment makes clear: in the mixed system the each component height of integral in the NMR spectrum has additivity.
MARC软件对捷达轿车连杆起裂过程进行了数值分析，得出了裂解力与J积分的关系曲线。
MARC. Jetta car's connecting rod was analyzed. Through analysis, the curve between J integral and splitting force was established.
微积分都是关于连续统的——变化的比率，曲线的面积，立体的体积。
Calculus is all about continuums - rates of change, areas under curves, volumes of solids.
演算，由牛顿和莱布尼茨的，是基于对衍生工具和积分的曲线。
Calculus, developed by Newton and Leibniz, is based on derivatives and integrals of curves.
积分是一段连续曲线的相加，所以那不会让你郁闷太久。
Integration is just a summation over a continuous section of a curve, so that won't stay scary for very long, either.
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.
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