那么怎么在c1上计算积分？
两个最基本的操作就是：寻找峰值，计算积分。
Two basic operations: Peak picking and integration calculation.
为了计算积分，我们在边界上构造了背景网格。
A background cell structure is constructed for purposes of numerical integration.
这是我编的一个利用梯形法数值计算积分的程序。
This is for my use of a trapezoidal method of numerical integration procedures.
不过绝大多数情况下，都需要知道怎么去计算，建立与计算积分。
But, mostly you should know how to compute, set up and compute these things.
下面是一个基本定理，它给出了，无需计算积分就得到结果的办法。
And then we have a fundamental theorem that gives us a way to compute this without computing it.
计算积分的方法、复数方法、无线级数、奇殊函数、微分方程、向量及矩阵、群论。
Methods in evaluating integrals, some complex variable methods, infinite series, special function, ordinary differential equations, vector and materials, groups and group representations.
那用什么坐标系来计算这个积分呢？
我们需要计算通量的积分。
至此我已经得到了，用来计算二重积分的所有量。
那么，怎么计算这个二重积分呢？
如果我们必须计算线积分，就必须通过寻找一个参数，并建立起一切。
If we have to compute a line integral, we have to do it by finding a parameter and setting up everything.
就像做功一样，当计算这线积分时，通常不这样用几何方法来做。
Just as we do work, when we compute this line integral, usually we don't do it geometrically like this.
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.
我们知道如何计算线积分。
它实际上做的是计算线积分。
当然，还要学习如何去计算二重积分。
注意，大家需要知道，如何建立和计算这种形式的线积分。
Remember, you have to know how to set up and evaluate a line integral of this form.
为了求做功，我们要计算线积分。
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.
我不需要真的计算通量积分。
如果没有问题的话，我们就来算算它吧，如何计算这个线积分的值呢？
If there are no other questions then I guess we will need to figure out how to compute this guy and how to actually do this line integral.
格林公式是另一种可以,避免计算线积分的方法。
So, Green's theorem is another way to avoid calculating line integrals if we don't want to.
这向我们展示了，计算线积分的办法。
OK, so that should give you overview of various ways to compute line integrals.
如果不喜欢计算线积分，可以通过增加一条线积分让曲线封闭起来，然后就可以用格林公式来计算了。
Or, if you really don't like that line integral, you could close the path by adding some other line integral to it, and then compute using Green's theorem.
一个是建立并计算二重积分。
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.
另一个是建立并计算线积分。
如果想计算这个积分，先做内积分。
Now, if I want to compute this integral, so let's first do the inner integral.
如果想计算这个积分，先做内积分。
Now, if I want to compute this integral, so let's first do the inner integral.
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