This relates a line integral for one field to a surface integral from another field.
这把一个向量场的线积分，和另外一个向量场的曲面积分联系起来。
Flux looks quite different in the plane and in space because, in the plane, it is just another kind of line integral, while in space it is a surface integral.
平面中的通量和空间中的通量有很大区别，在平面中，通量仅仅是线积分的另一种形式，而在空间中，它表现为曲面积分。
Then I can actually -- --replace the line integral for flux by a double integral over R of some function.
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。
But, you know, it gives you an example where you can turn are really hard line integral into an easier double integral.
但是你知道，它给了你一个例证，其中你可以，把复杂的线积分化成简单些的二重积分。
So, now, if I compare my double integral and, sorry, my triple integral and my flux integral, I get that they are, indeed, the same.
比较这个二重积分的话，抱歉。。。，比较这个三重积分和通量积分，就可以看到，它们是一样的。
Or, if you really don't like that line integral, you could close the path by adding some other line integral to it, and then compute using Green's theorem.
如果不喜欢计算线积分，可以通过增加一条线积分让曲线封闭起来，然后就可以用格林公式来计算了。
And, the way in which you would compute the surface integral is just as with any surface integral.
计算这个曲面积分的方法，和其他任何曲面积分的一样。
And, so the total line integral for this thing is equal to the integral along C prime, I guess the outer one.
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。
But it is still a line integral so it is still going to turn into a single integral when you plug in the correct values.
但是这仍然是一个线积分，代入计算就会发现，这仍然会变为单变量积分。
Part of it does seem definitely new — an integral semiotics, as well as an integral calculus, a form of mathematics that replaces variables with perspectives.
其中很多内容无疑是非常前沿——整合语言学，还有整合积分学，这是一种取代变量的数学视角。
To compute things, Green's theorem, let's just compute, well, let us forget, sorry, find the value of a line integral along the closed curve by reducing it to double integral.
用格林公式计算…，只是计算…，让我们忘记…，应该是，算沿闭曲线的线积分值，可以通过二重积分来算。
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.
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。
No matter which form it is, it relates a line integral to a double integral Let's just try to see if we can reduce it to the one we had yesterday.
不管哪种形式，都把线积分和二重积分联系在一起，来看看，能不能通过化简得到昨天的公式。
So, if we want to compare them, we should probably subtract them from each other. OK, so let's do the flux integral for S1 minus the flux integral for S2 of the same thing.
那么，如果我们想要比较它们，我们应该把它们相减，用S1的通量积分减掉s2的通量积分。
The first one that I will mention is actually something you thought maybe you could do with a single integral, but it is useful very often to do it as a double integral.
第一点就是，有些你以为是用一重积分来做的，但却通常是用二重积分来完成的。
Now, if I want to compute this integral, so let's first do the inner integral.
如果想计算这个积分，先做内积分。
Well, let's think about comparing the flux integral for S1 and the flux integral for S2.
让我们考虑关于S1和S2的通量积分的比较。
Integral in going from a to B, integral in going from a to B.
从a到b的积分，从a到b的积分。
So, one of them says the line integral for the work done by a vector field along a closed curve counterclockwise is equal to the double integral of a curl of a field over the enclosed region.
其中一种说明了，在向量场上，沿逆时针方向，向量做的功等于，平面区域上旋度F的二重积分。
We have the integral from minus infinity to plus infinity of F(x)*x*dx, and that's really--you see it's the same thing because an integral is analogous to a summation.
我们用从负无穷到正无穷的积分公式F(x)*x*dx,你看到的，这是相同的事，因为积分是类似的求和过程。
So, that means that the double integral for flux through the top of R vector field dot ndS becomes double integral of the top of R dxdy.
这就是说通量的二重积分,顶部R•ndS的二重积分,变成了Rdxdy的二重积分。
Remember the general method is that we first look at the inner integral, here integral of f dy.
一般是,先考虑内积分，这里即f对dy的积分。
S if we have a closed surface with a normal vector pointing outwards, and we want to find a flux integral for it, well, we can replace that with a triple integral.
的法向量似乎是指向外部。，And, the,normal, vector, to,s, seems, to, be, pointing, out wards, everywhere。，如果我们有一个法向量指向外部的封闭曲面，并且我们要求出它的一个通量积分，那么我们可以用一个三重积分来代替这个通量积分。
On a 64-bit system, integral types are converted to 64-bit integral types, and single precision floating point types are promoted to double precision.
在 64位系统上，整型被转换成 64 位的整型值，单精度的浮点类型被转换成双精度的浮点类型。
Finally, simplicity of aiming was achieved by use of integral reflex type collimating sight and integral laser aiming module.
最后，旨在实现了由简单的积分使用反射式准直激光视力和整体目标模块。
Similarly, human is an integral variety and each person is also an integral individual.
同理，人类是一个完整性的物种，每个人也是一个完整的个体。
Butt sometimes it is useful. Sometimes it is also useful backwardS because , actually, you have asurface integral that you would like to turn into a line integral.
但这招经常很有用，这个公式有时候反过来也可以用，比如你有一个面积分，但是你想，把它变成线积分的时候也可以用的。 。
Than the structure details contained in integral joint are analyzed, and the fatigue property of steel bridge integral joint is evaluated.
然后分析了钢桥整体节点所包含的构造细节，评价了钢桥整体节点的疲劳性能。
The integral upper limit function strengthened between the differential and the integral relation, is the definite integral fundamental formula rationale.
积分上限函数加强了微分和积分之间的联系，是定积分基本公式的理论基础。
An ordinary definite integral is defined over a line segment, whereas a line integral may use a more general path, such as a parabola or a circle.
普通定义的积分是定义在一段线段上的，而线积分则用更一般的路径，像抛物线或圆。
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