We have a line integral along a curve.
对于沿曲线的线积分。
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。
Graphics are included not merely as a means of making the information easier for the student to grasp, but as an integral part of the way social scientists think.
加入图形不仅是让学生更容易掌握信息的手段，也是社会科学家思考方式的一个组成部分。
A name is an integral part of anyone's personal and professional identity—just like the town you're born in and the place where you're raised.
名字是任何人个人和职业身份中不可分割的一部分——就像你出生的城镇和你长大的地方一样。
A line integral for flux just becomes this.
求通量的线积分就变成这样了。
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。
If you manage your project from a good plan — and an integral part of that plan is a good estimate — there is going to be significant cost savings.
如果您采用一个好的计划管理您的项目——其中一个完整的部分就是一个好的评估——那么您将能够节省大量的开销。
Flux looks quite different in the plane and in space because, in the plane, it is just another kind of line integral, while in space it is a surface integral.
平面中的通量和空间中的通量有很大区别，在平面中，通量仅仅是线积分的另一种形式，而在空间中，它表现为曲面积分。
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。
And, then I have a single variable integral.
然后就变成了一个单变量积分。
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。
Not a very big one, not a very complicated integral, but imagine it could get potentially much harder.
也不是太大的麻烦，也不是一个非常复杂的积分，但估计可能变得更加困难。
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。
Whenever you do a surface integral for flux you have two parts of the story.
每当做通量的曲面积分时，你要做两件事。
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。
So, what it does actually is it computes a line integral.
它实际上做的是计算线积分。
Once you have computed what this guy is, it's really just a triple integral of the function.
一旦需要计算这个积分，只需要计算这个函数的三重积分。
So, using Green's theorem, the way we'll do it is I will, instead, compute a double integral.
那么，使用格林公式，我们去计算二重积分。
And instead I will want to reduce things to a surface integral.
相反，我想要把它变成面积分。
No matter which form it is, it relates a line integral to a double integral Let's just try to see if we can reduce it to the one we had yesterday.
不管哪种形式，都把线积分和二重积分联系在一起，来看看，能不能通过化简得到昨天的公式。
What ends up happening is it gives you this integral times a certain number.
结果是这个积分，乘以某个数。
And so, this becomes the integral from a to b.
那么这也就成为从a至b的积分了。
Integral in going from a to B, integral in going from a to B.
从a到b的积分，从a到b的积分。
So, we need to set up the flux integral for a vector field dot ndS.
需要建立向量场点乘nds的通量积分。
Or, if you prefer, that's negative integral from a to b of M of dx.
或者你喜欢的话，那就是积分的负值,在a到b上对MdX的积分。
So, one of them says the line integral for the work done by a vector field along a closed curve counterclockwise is equal to the double integral of a curl of a field over the enclosed region.
其中一种说明了，在向量场上，沿逆时针方向，向量做的功等于，平面区域上旋度F的二重积分。
So, one of them says the line integral for the work done by a vector field along a closed curve counterclockwise is equal to the double integral of a curl of a field over the enclosed region.
其中一种说明了，在向量场上，沿逆时针方向，向量做的功等于，平面区域上旋度F的二重积分。
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