二重对数是对数的积分。

The dilogarithm is an integral of the logarithm.

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实际上，要拆成两个不同的积分。

So, in fact, I have to break this into two different integrals.

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那就是，昨天学的那个万恶的积分。

And that is, again, the evil integral that we had yesterday.

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那么这也就成为从a至b的积分了。

And so, this becomes the integral from a to b.

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那么你要做一个以t为变量的积分。

And then, you'll do a one variable integral over t.

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我来计算ydx+xdy的积分。

OK, so I'm computing the integral of y dx x dy.

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由于对称性，y的积分值为。

The integral of y will be zero also by symmetry.

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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.

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我们需要计算通量的积分。

We have to compute the integral for flux.

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这里的积分常数是什么呢？

Now, what do we mean by integration constant.

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这个常数是，开始我们得到的积分常数。

That constant is just the integration constant that we had from the beginning.

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为了得到W现在要做,整个过程的积分。

Now I have to do the integral over the entire path to get W.

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我们可以做温度的积分。

We can do an integral over temperature.

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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.

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这就变成了C上的积分。

That will just become the integral over C.

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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.

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在这里，是三维区域上的积分。

Here, you integrate over a three-dimensional region.

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我不会给出一个完整的积分表的。

OK, I don't want to give you a complete list.

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简单三角函数的积分是什么意思呢？

What do I mean by easy trigonometrics?

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这是根据电流对，时间的积分。

This is on the basis of the integral of the current times the time.

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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.

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当然，所做的积分是不同的。

And, of course, what you're integrating is different.

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称之为在区域D上divdV的积分。

Let me call that region D of divergence, of curl F dV.

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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.

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下面应该如何，在这样的曲面上建立通量的积分。

Now, how would we actually set up a flux integral on such a surface.

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今天这个是用来求F的法向分量的积分的。

The one today is for integrating the normal component of F.

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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.

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无论如何，再说一次，从概念上说有三个不同种类的积分。

OK, so anyway, again, conceptually, we have, really, three different kinds of integrals.

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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.

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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.

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