他开始了大学生活，课程表上包括生物、微积分和政治学。
He began college with a timetable that included biology, calculus and political science.
他们可以在网页上把积分换成礼物。
They can change the points into some gifts from the website.
像英国和加拿大等国使用的是积分系统，它的目标是挑选那些有教育背景或有专业技能的(这些在经济上被认为是需要的。)
Countries like Britain and Canada use a "points" system, which aims to select migrants who have educational levels or specialised skills that are deemed economically desirable.
这就会变成…，如果俯视这个抛物面，所看到的就是单位圆盘，这就应该是单位圆上的二重积分了。
That is just going to be, if you look at this paraboloid from above, all you will see is the unit disk so it will be a double integral of the unit disk.
一个向量场,如果定义在单连通区域并且旋度为零，那么它就是一个梯度场，并且其上的线积分与路径无关。
OK, so, we've seen that if we have a vector field defined in a simply connected region, and its curl is zero, then it's a gradient field, and the line integral is path independent.
这些都是三重积分的例子，从概念上讲是相同的。
OK, so these are just formulas to remember for examples of triple integrals It doesn't change conceptually.
区域R的面积是函数1在R上的二重积分。
得到c1上的线积分，大部分就消去了。
I get that the line integral on c1 — Well, a lot of stuff goes away.
求S上zdxdy的二重积分。
不是变量，这是在一个圆上。，R，is，not，a，variable。，You，are，on，the，circle。,这个是二重积分，如果你们这么做的话，在圆盘上，如果你们用极坐标的话,就需要用到R和θ
R This one is a double integral. So, if you are doing it, say, on a disk, you would have both R and theta if you're using polar coordinates.
二重积分可以用来得到关于这个区域的某些信息，或者这个区域上的一个函数的平均值，等等。
We can use that to get information about maybe the region or about the average value of a function in that region and so on.
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。
Then I can actually -- --replace the line integral for flux by a double integral over R of some function.
在这之后是游戏的分配阶段，负责分配的这个孩子会获得他和他的搭档在这项任务上花费的时间以及他们获得的积分奖励。
Afterward, in the distribution phase of the game, the dictator was given information about his or her partner's time spent on the task and the number of points that were collected.
实际上，我们对二重积分的定义并不严谨，那应该怎样严格地定义它呢？
OK, so actually, how do we define it, that's not really much of a definition yet.
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。
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.
因此在决定选择单桅帆船还是双桅帆船时，决定因素并非帆面积，而是将这个帆面积分布在两面帆或是三面帆上带来的相应差异。
So it's not the sail area that matters when deciding between a sloop and a ketch, but differences resulting from having that area spread over three rather than two sails.
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。
One example that we did, in particular, was to compute the double integral of a quarter of a unit disk.
事实上，可以把求导放在积分里面。
在这里，是三维区域上的积分。
为了提醒我们是在封闭曲线上做积分，经常在积分符号上加个圆圈，告诉我们，这条曲线自我封闭。
To remind ourselves that we are doing it along a closed curve, very often we put just a circle for the integral to tell us this is a curve that closes on itself.
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.
它实际上做的是计算线积分。
要计算三个线积分，而不是两个，但概念上是一样的。
You have three line integrals to compute instead of two, but conceptually it remains exactly the same idea.
这就变成了C上的积分。
了解这两种表述后，我们不仅需要向量场，就是左边这里，这是曲线c上的线积分，向量场在曲线上有定义。
So, in both cases, we need the vector field to be defined not only, I mean, the left hand side makes sense if a vector field is just defined on the curve because it's just a line integral on c.
对我喜欢的曲线，计算其上的线积分，在这条线上所做的功。
And let's take my favorite curve and compute the line integral of that field, you know, the work done along the curve.
这让我们可以把精力集中到联赛上，去追回一些积分。
This allows us to concentrate on the league and recuperating some points.
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.
那么在一个空间上的区域的三重积分，对标量dV做积分。
OK, so triple integrals over a region in space, we integrate a scalar quantity, dV.
一般cp和，都是温度的函数，因此实际上，我们可以将这个积分计算出来。
Cv So, for Cp and Cv, these are often quantities that are measured as a function of temperature, and one could, in fact, calculate this integral.
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