二重对数是对数的积分。
那么这也就成为从a至b的积分了。
那么你要做一个以t为变量的积分。
我来计算ydx+xdy的积分。
事实上，这就变为t从0到1的积分。
That just becomes the integral from, well, I guess t goes from zero to one, actually.
我们需要计算通量的积分。
也就是在某个边界里算dxdy的积分。
So, we should get to something that will be the same integral dx dy.
给出一条曲线，计算沿着这条曲线的积分。
Well, let's say I give you a curve and I ask you to compute this integral.
从a到b的积分，从a到b的积分。
Integral in going from a to B, integral in going from a to B.
称之为在区域D上divdV的积分。
这就是- C3的积分，它沿着相反方向的路径。
That's the integral along minus C3, along the reversed path.
这里我说的积分常数，它是不含x的一项。
And here, when I say there is an integration constant, it just means the extra term does not depend on x.
我们继续吧，还有很多其他的积分类型要看。
OK. Let me move on a bit because we have a lot of other kinds of integrals to see.
对一个循环过程作d q除以t的积分，小于等于零。
Going around in a cycle the integral of dq over t is less than or equal to zero.
一般是,先考虑内积分，这里即f对dy的积分。
Remember the general method is that we first look at the inner integral, here integral of f dy.
还有一个相反的技巧，也就是给出了区域，如何去建立它的积分。
And then there's a converse skill which is given the region, how to set up the integral for it.
有两种选择，当你要建立通量的积分时,就必须取定曲面的正向。
There are two choices. Basically, whenever you want to set up a flux integral you have to choose one side of the surface.
那么，到目前为止，我们已经学了空间中的三个重要的积分。
So, at this point, we've learned about three kinds of beasts in space.
我们用一个参数来表示所有的项，最后得到一个通常的积分。
We express everything in terms of a single parameter, and that will give us a usual single integral.
请说，你想知道极坐标系下的积分边界，这是一个二重积分。
Yes? In case you want the bounds for this region in polar coordinates, indeed it would be double integral.
所要做的功,就等于A到B的,我的力的积分,这是个点积。
So the work that Walter Lewin has to do to bring it from A to B is the integral in going from A to B of my force, dx.
左边的这个积分定义在曲线，右侧的积分定义在整个内部区域。
This one on the left-hand side lives only on the curve, while the right-hand side lives everywhere in this region inside.
但我不知道有别的更简单的方法没，除了翻阅笔记中的积分表。
So, but I don't know any simpler solution except for one simpler solution, which is you have a table of integrals of this form inside the notes.
要是他的积分是680，则按揭利率会再低上0.5个百分点。
Had his score been 680, the rate he was offered would have been a half-percent lower.
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。
Not a very big one, not a very complicated integral, but imagine it could get potentially much harder.
内政部说到目前的积分的系统提供了一个“强有力和灵活的控制”。
The Home Office said the current points-based system provided "a powerful and flexible set of controls".
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。
Or, if you prefer, that's negative integral from a to b of M of dx.
假如有三个不同种类的积分，它们做积分的时候在某些方面是相同的。
OK, so now we have three different kinds of integrals, and really, well, they certainly have in common that they integrate things somehow.
需要指出的是，如果要分别计算两边的积分，这就是标准的通量积分。
But, what I want to point out is if you have to compute the two sides separately, well, this is just, you know, your standard flux integral.
需要指出的是，如果要分别计算两边的积分，这就是标准的通量积分。
But, what I want to point out is if you have to compute the two sides separately, well, this is just, you know, your standard flux integral.
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