We have a line integral along a curve.
对于沿曲线的线积分。
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。
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。
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。
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。
Use geometry or you need to set up for double integral of a surface.
总之，就是用几何方法或是在曲面上建立二重积分。
You know how to compute a double integral of a function.
要懂得如何计算一个函数的二重积分。
All of these are integral to morphing a good brand into a great brand!
所有这些都是将一个好品牌变成一个优秀品牌的不可或缺的组成部分！
So, what it does actually is it computes a line integral.
它实际上做的是计算线积分。
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。
This is just your standard triple integral over a region in space.
这就是空间区域中的标准三重积分。
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。
You can make help a dynamic, integral part of your development environment.
您可以让帮助成为开发环境的一个动态的、有机的组成部分。
Business is complex and fast-moving, and designers need to make themselves an integral part of a collaborative process.
商业是一个复杂和快速的过程，设计师需要将自己变成其中一个有机部分。
Since a name was considered an integral part of a person, it could be an effective target for sorcery.
因为名字被认为是一个人的重要组成部分，就有可能成为巫术的有效攻击靶子。
One example that we did, in particular, was to compute the double integral of a quarter of a unit disk.
我们已经做过的一个例子是，计算四分之一单位圆上的二重积分。
That is what a double integral means.
这也就是二重积分的意义。
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。
How do you express the area as a double integral?
如何用二重积分来表示面积呢？
How do you express the area as a double integral?
如何用二重积分来表示面积呢？
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