我所要求的只是替代方案，而不是让我们所有人都走上学习微积分的道路。
All that I ask is that alternatives be offered instead of putting all of us on the road to calculus.
所以，只需带一张信用卡。可以携带一张积分或返现多的，或者一张不收取国际手续费的（我倾向于这点）。
Instead, stick to one credit card. Either one that gets you the most points or cashback or one that doesn't charge international fees (my preference).
我证出了分别沿着C1和C2的两个线积分是相等的。
OK, so I proved that my two line integrals along C1 and C2 are equal.
就我这一方面而言，我获得三站冠军，六次杆位和相同的最快圈速。相对我去年的积分有提升。
For my part, I won three RACES and took six pole positions and the same number of race fastest laps, improving on my points score from last year.
我对你们的要求越低，物理老师或其他科老师，就会抱怨的越厉害，他们会抱怨“这些家伙居然不知道怎么求积分
I should say, the more I tell you I don't need you to know, the more your physics and other teachers might complain that, oh, these guys don't know how to integrate.
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.
我要说的是取其一，积分得到包括另一变量的函数的结果，然后对结果求导，进行比较看得到什么。
But what I am saying is just take one of them, integrate, get an answer that involves function of the other variable, then differentiate that answer and compare and see what you get.
我要把我的黑板分成三部分，在这我将要写三重积分。
OK, so I'm going to divide my blackboard into three pieces, and here I will write triple integrals.
这里使用dS而不是dA，因为我习惯把dA看成是，坐标平面中的面积元，就像在二重积分中所做的那样，待会大家会了解。
I am using dS rather than dA because I still want to think of dA as maybe the area in one of the coordinate planes like the one we had in double integrals. You will see later where this comes in.
也就是，如果我需要计算一个线积分，无论我取怎样的路径,只要路径是从点a到点。
OK, so the claim is if I have a line integral to compute, that it doesn't matter which path I take b as long as it goes from point a to point b.
我将要证明，一个只有z分量的向量场的通量,等于一个三重积分，其中被积表达式为。
I'm going to prove that the flux of a vector field that has only a z component is actually equal to the triple integral of, RzdV well, the divergence of this is just R sub z dV.
我来计算ydx+xdy的积分。
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.
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.
你或许因此对自己说，趁着我的大脑还没退化，赶紧报名学习史瓦西里文和微积分，还有手风琴班吧！
So you may be saying to yourself, I have to sign up right now for Swahili and calculus and accordion lessons before my brain withers away!
但是在这段时间里，我更多使用的是离散数学而不是微积分，所以我需要复习才能够找出答案。
But, I use discrete mathematics much more than calculus these days, so I need to go back and look at how to find the answer.
重新分解积分域。。。,我的目标是，用另外一种形式重写这个积分。
The other way around, I have to, so my goal, now, is to rewrite this as an integral.
我的排放交易网站在自愿的碳市场上以你的名义出售该积分，现行价格为10～25美元，所有交易抽取20%的费用。
My Emissions Exchange sells the credit on your behalf on the voluntary carbon market at a going rate of between $10 and $25, taking a 20 percent cut on all sales.
当然我更希望你们记住啦，但是如果脑海中一片空白的话，如果可以写出积分边界和其他的东西，也是可以得到部分分数的。
Well, I mean, I prefer if you know, but if you have a complete blank in your memory, there will still be partial credit were setting up the bounds and everything else.
有些网站给好的答案送积分，但是有些积分系统比较随意，我至今也不知道积分到底有何价值。
Other sites award points for good answers, although some points systems may be arbitrary and I'm still not sure what value most of those points have.
那么，它的总的线积分,就等于沿着C’，我是指外面这个的线积分。
And, so the total line integral for this thing is equal to the integral along C prime, I guess the outer one.
我指的是，在这种形式下，它和一元微积分的表述是一样的。
I mean, in this form, actually it's the same statement as in single variable calculus.
他补充道：“我很高兴球队现在处在积分榜第四的位置，对小伙子们的表现非常满意。”
He added: "I would be happy with fourth place." I am happy with the play of the team.
如果有一个梯度场，然后想计算线积分，我知道积分值是,起点的函数值减去终点的函数值。
Well, if I have a gradient field, then if I try to compute this line integral, I know it will be the value of the function at the end point minus the value at the starting point.
节后(高中最后一学期初)，我去找微积分老师交给他厚厚的一摞纸张。
After the holiday break (at the start of my final semester of high school), I went up to my calculus teacher and handed him a big stack of papers.
这个等式的理由是,我们在内部积分了两次，我要说的是？
OK and the reason for this equality here is because we go twice through the inner part. What do I want to say?
对于剩下的愿意忍受我的人，我们继续，现在我们需要积分。
For the rest of you who are willing to bear with me, let's continue. So now let's just integrate.
如今我们积分已经挤进前四，我认为保持目前的状态是很重要的。
We're involved in the top four now and I think it's important that we stay in where we are.
我强烈建议,如果你忘记了单变量积分的知识，现在就是一个很好的复习机会。
OK, so, yeah that's a strong suggestion that if you've forgotten everything about single variable calculus, now would be a good time to actually brush up on integrals.
我强烈建议,如果你忘记了单变量积分的知识，现在就是一个很好的复习机会。
OK, so, yeah that's a strong suggestion that if you've forgotten everything about single variable calculus, now would be a good time to actually brush up on integrals.
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