牛顿发展了现代微积分的基础数学概念和技术。
Newton developed mathematical concepts and techniques that are fundamental to modern calculus.
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.
而且我们能用线积分的定义计算出来。
And this we can compute using the definition of the line integral.
这向我们展示了，计算线积分的办法。
OK, so that should give you overview of various ways to compute line integrals.
因为这个变量，和积分的变量不一样。
And it's because this variable here is not the same as the variables on which we are integrating.
下载学习资料并参加评估会得到积分的奖励。
Downloading the studying material and taking the assessment is rewarded with points.
这意味着，在积分的时候会遇到一点小麻烦。
That means you will have a little bit of trig to do in the integral.
来看看更多的取通量积分的方法。
第一个是积分的对象，称为积分要素。
格林公式是另一种可以,避免计算线积分的方法。
So, Green's theorem is another way to avoid calculating line integrals if we don't want to.
他会掌握微积分的，因为他一直在温习这门课程。
让我们考虑关于S1和S2的通量积分的比较。
Well, let's think about comparing the flux integral for S1 and the flux integral for S2.
他也是一位杰出的数学家，微积分的发明者之一。
He was a brilliant mathematician; he was one of the inventers of Calculus.
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.
这就是需要记住的公式，以上这些公式可以作为积分的例子。
That is the formula I have in mind. But, see, all these formulas just give you examples of things to integrate.
这些都是反映了,线积分的微积分基本定理的特例。
So, these are special cases of what's called the fundamental theorem of calculus for line integrals.
一个保守的向量场就是说,沿任意闭曲线的线积分的结果是。
So, to say that a vector field with conservative means 0 that the line integral is zero along any closed curve.
计算这个曲面积分的方法，和其他任何曲面积分的一样。
And, the way in which you would compute the surface integral is just as with any surface integral.
如果没有问题的话，我们就来算算它吧，如何计算这个线积分的值呢？
If there are no other questions then I guess we will need to figure out how to compute this guy and how to actually do this line integral.
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.
但是，我想强调的是，这两种建立积分的方法其实是相同的。
But, what I want to emphasize in this way is that both of these you set up pretty much in the same way.
也就是二重和三重积分的内容，以及平面和空间中的向量积分。
And so that was stuff about double and triple integrals and vector calculus in the plane and in space.
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.
那么对做功求线积分,就变成坐标积分,同时也有对通量求线积分的。
So, the line integral for work Mdx+Ndy becomes in coordinates integral of Mdx plus Ndy while we've also seen line integral for flux.
另一件关于二重积分的是，我们已经讲过了，如何做更复杂的变量变换。
OK, now another thing we've seen with double integrals is how to do more complicated changes of variables.
排名的计算以四年为一周期，即便是友谊赛，对积分的影响也微乎其微。
Calculations are made on the basis of results over a four-year period and even friendlies have a minor place in the formula.
所以，x的范围是从0到，来看看内积分的积分范围，是怎样取决于x的。
So, x goes from zero to one. OK, and now, see how in the inner integral, the bounds depend on x.
所以，x的范围是从0到，来看看内积分的积分范围，是怎样取决于x的。
So, x goes from zero to one. OK, and now, see how in the inner integral, the bounds depend on x.
应用推荐