So, line integrals we know how to evaluate.
我们知道如何计算线积分。
OK, anyway, let's move on to line integrals.
再转到线积分。
Some of them compute different line integrals.
它们中的一些在做不同的线积分。
I would just calculate three easy line integrals.
我只需计算三个简单的线积分。
What if I add all of the small line integrals?
那如果将所有线积分都加起来呢？
You don't have to set up all these line integrals.
不用建立这些线积分。
Let's start right away with line integrals in space.
现在开始学习空间线积分了。
So conceptually it is very similar to line integrals.
从概念上来看，这与线积分相似。
The other one is setting up and evaluating line integrals.
另一个是建立并计算线积分。
Unfortunately, it is not quite as simple as with line integrals.
很遗憾，它的过程不像线积分那样简单。
OK, so I proved that my two line integrals along C1 and C2 are equal.
我证出了分别沿着C1和C2的两个线积分是相等的。
We've learned about double integrals, and we've learned about line integrals.
我们已经学过，二重积分和线积分。
It is not a surprise that you will get the same answer for both line integrals.
对它们两个做线积分，得到相同结果，就不令人感到吃惊了。
OK, so that should give you overview of various ways to compute line integrals.
这向我们展示了，计算线积分的办法。
Line integrals are used extensively in the theory of functions of a complex variable.
在复变量函数理论中广泛使用线积分。
So, Green's theorem is another way to avoid calculating line integrals if we don't want to.
格林公式是另一种可以,避免计算线积分的方法。
Line Integrals, Potentials, Curl and Gradient in 3d (Omit: Electricity and magnetism).
三维空间的线积分，位势，旋度和梯度(省略：电磁场)。
So, these are special cases of what's called the fundamental theorem of calculus for line integrals.
这些都是反映了,线积分的微积分基本定理的特例。
Let me just switch gears completely and switch to today's topic, which is line integrals and work in 3D.
我们换一个话题,开始今天的内容，线积分和3维空间中的“功”
You have three line integrals to compute instead of two, but conceptually it remains exactly the same idea.
要计算三个线积分，而不是两个，但概念上是一样的。
That is the key to computing things in practice. It means, actually, you already know how to compute line integrals for flux.
这是计算过程的关键步骤，也就是说，你们已经知道如何用线积分去计算通量了。
So, the fundamental theorem of calculus, not for line integrals, tells you if you integrate a derivative, then you get back the function.
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。
But another place where this comes up, if you remember what we did in the plane, curl also came up when we tried to convert line integrals into double integrals.
除了这些，旋度还有一个用处，如果你还记得，我们讨论平面的时候,当把线积分转换成二重积分的时候。
And whether these line integrals or double integrals are representing work, flux, integral of a curve, whatever, the way that we actually compute them is the same.
不管是线积分或是二重积分，也不管它们表示的是功还是通量，计算它们的方法实际上是一样的。
Just the physical interpretations will be very different, but for a mathematician these are two line integrals that you set up and compute in pretty much the same way.
只不过物理上的解释大相径庭，但对于数学家来说，它们是两种建立过程和计算办法都一样的线积分。
When I sum all of the little line integrals together, all of the inner things cancel out, and the only ones that I go through only once are those that are at the outer most edges.
当将所有线积分加在一起时，所有内部的线积分都抵消了，只剩下仅需要一次计算的，就是外围边缘部分。
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).
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 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。
Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.
格林定理及其应用、三重积分、空间中的线积分和面积分、散度定理、斯托克斯定理应用。
Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.
格林定理及其应用、三重积分、空间中的线积分和面积分、散度定理、斯托克斯定理应用。
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