求通量的线积分就变成这样了。
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.
我们已经学过，二重积分和线积分。
We've learned about double integrals, and we've learned about line integrals.
而且我们能用线积分的定义计算出来。
And this we can compute using the definition of the line integral.
它实际上做的是计算线积分。
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.
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.
得到c1上的线积分，大部分就消去了。
I get that the line integral on c1 — Well, a lot of stuff goes away.
对于沿曲线的线积分。
这就变成了Pd x +Qdy的线积分。
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?
你们需要记住，什么是线积分，什么是场的散度？
So, you should remember, what is this line integral, and what's the divergence of a field?
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.
那么我们只需要知道，线积分正是势函数值的变化。
Then, we just have to, well, the line integral is just the change in value of a potential.
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.
这些都是反映了,线积分的微积分基本定理的特例。
So, these are special cases of what's called the fundamental theorem of calculus for line integrals.
我证出了分别沿着C1和C2的两个线积分是相等的。
OK, so I proved that my two line integrals along C1 and C2 are equal.
要计算三个线积分，而不是两个，但概念上是一样的。
You have three line integrals to compute instead of two, but conceptually it remains exactly the same idea.
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.
对它们两个做线积分，得到相同结果，就不令人感到吃惊了。
It is not a surprise that you will get the same answer for both line integrals.
它说，如果你对一个函数的梯度做线积分，就能得到原函数。
It tells you, if you take the line integral of the gradient of a function, what you get back is the function.
基本上说是从下周末开始，学习三元积分，空间线积分等等。
I mean, basically starting at the end of next week, we are going to do triple integrals, line integrals in space and so on.
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.
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.
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”
Let me just switch gears completely and switch to today's topic, which is line integrals and work in 3D.
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.
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.
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.
那么，它的总的线积分,就等于沿着C’，我是指外面这个的线积分。
And, so the total line integral for this thing is equal to the integral along C prime, I guess the outer one.
那么对做功求线积分,就变成坐标积分,同时也有对通量求线积分的。
So, the line integral for work Mdx+Ndy becomes in coordinates integral of Mdx plus Ndy while we've also seen line integral for flux.
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