OK, so there's the line integral.
这就是线积分了。
We have a line integral along a curve.
对于沿曲线的线积分。
This side here is a usual line integral.
这边是普通的线积分。
A line integral for flux just becomes this.
求通量的线积分就变成这样了。
And we will evaluate the line integral for work.
为了求做功，我们要计算线积分。
We say that the line integral is path independent.
我们称之为线积分与积分路径无关。
So that will be the line integral of Pdx plus Qdy.
这就变成了Pd x +Qdy的线积分。
Well, we cannot really think of flux as a line integral.
那就无法把通量想成线积分了。
So, what it does actually is it computes a line integral.
它实际上做的是计算线积分。
The line integral was actually the work done by the force.
线积分的意义，就是这力所做的功。
And so I want to compute for the line integral along that curve.
那我想计算那条曲线上的线积分。
And this we can compute using the definition of the line integral.
而且我们能用线积分的定义计算出来。
And so it will be a surface integral, not a line integral anymore.
所以要用面积分，而非线积分。
What is flux? Well, flux is actually another kind of line integral.
通量是什么？，通量其实是又一种线积分。
And, I still want to compute the line integral along a closed curve.
但仍然想要沿着封闭曲线的线积分计算。
And then I add these together. That is what the line integral means.
把这些加到一起，这就是线积分了。
On the math it is a line integral of something dx plus something dy.
从数学意义上来说，这是dx部分加上dy部分的线积分。
I get that the line integral on c1 — Well, a lot of stuff goes away.
得到c1上的线积分，大部分就消去了。
If we have a closed curve then the line integral for work is just zero.
如果给定一条封闭曲线，那么求所做功的线积分为零。
How do I compute the line integral along the curve that goes all around here?
应该怎样沿着围绕这个区域的曲线，做线积分呢？
Remember, you have to know how to set up and evaluate a line integral of this form.
注意，大家需要知道，如何建立和计算这种形式的线积分。
This relates a line integral for one field to a surface integral from another field.
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。
And then we had to evaluate the line integral for the work done along this path.
然后计算，沿此路径所做的功。
Then, we just have to, well, the line integral is just the change in value of a potential.
那么我们只需要知道，线积分正是势函数值的变化。
So, you should remember, what is this line integral, and what's the divergence of a field?
你们需要记住，什么是线积分，什么是场的散度？
If you cannot parameterize the curve then it is really, really hard to evaluate the line integral.
如果无法对曲线参数化，那么就很难计算线积分了。
Just as we do work, when we compute this line integral, usually we don't do it geometrically like this.
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。
Then I can actually -- --replace the line integral for flux by a double integral over R of some function.
那么我就能名正言顺地,用R上的某个函数的二重积分来替代通量的线积分。
If we have to compute a line integral, we have to do it by finding a parameter and setting up everything.
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。
So, that's a really strange statement if you think about it because the left-hand side is a line integral.
那么，如果你仔细想，会有一种很奇特的结论,因为，左面的是一个线积分。
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