一旦我们做出了选择，同时也就可以定义通量积分。
Now, once we have made a choice then we can define the flux integral.
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.
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.
如果旋度在原点有定义，你就可以试试了，计算二重积分。
So, if a curl was well defined at the origin, you would try to, then, take the double integral.
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.
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.
如果你喜欢也可以，但是沿着这条路径积分却不好计算，尤其是还没告诉你曲线的定义。
I mean if that is your favorite path then that is fine, but it is not very easy to compute the line integral along this, especially since I didn't tell you what the definition is.
这就是为什么，这个线积分，有着完美的定义，但却不能对它使用格林公式的原因。
And so that's why you have this line integral that makes perfect sense, but you can't apply Green's theorem to it.
这就是问题所在，如果你想积分，我们需要它处处有定义，旋度是0，那么你的积分也是。
OK, so that's the problem. I mean, if you try to integrate, we've said everywhere where it's defined, 0 the curl is zero. So, what you would be integrating would be zero.
这就是二重积分的基本定义了。
左边的这个积分定义在曲线，右侧的积分定义在整个内部区域。
This one on the left-hand side lives only on the curve, while the right-hand side lives everywhere in this region inside.
那么我们将不会建立带上下限的积分，和积分dx,dy, dy, dx或者在极坐标做线积分，需要说，让我们回忆一下质心的定义。
So, instead of setting up the integral with bounds and integrating dx dy or dy dx or in polar coordinates, I'm just going to say, well, let's remember the definition of a center of mass.
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。
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.
用积分和的极限定义的黎曼积分对于初学者来说是一个很难理解的概念。
It is difficult for learner to understand the concept of Riemann integral which is defined by using the limit of Riemann sum.
给出了积分滑模控制中切换函数的定义方法，以及非线性微分控制中微分系数的非线性函数表达式。
The switching function of integral sliding mode control is defined, and the nonlinear expression of derivative coefficient is presented.
对X光晶体学直接法中的毗邻理论联合概率标准化定积分的定义做了必要的讨论和扩充。
In this paper, the necessary discussion and extension are made on the standardized definite integral definition of joint probability of neighborhood theory in direct methods of X-ray crystallography.
本文就非正常积分的定义提出了一些看法。
Viewpoint which are about definition of the non-normal integration are given in this paper.
职能的综合在理查页范曼的1942哲学博士长篇论文成功进入物理学作为道路积分，但是它确实没有意义作为一个数学定义。
Functional integration successfully entered physics as path integrals in the 1942 Ph. D. dissertation of Richard P. Feynman, but it made no sense at all as a mathematical definition.
本网站允许用户创建自定义的纹理和动画，玩游戏，赢积分三维头像的生物。
This site allows users to create 3-dimensional avatar creatures with custom textures and animations, play games and win points.
用路径积分定义宇宙量子态的理论可以解释宇宙结构的起源。
The theory that cosmological quantum state can be described by the route integral can explain the origin of the universe.
本文定义了象差积分函数，导出了形式简洁的全部二级象差函数。
By defining integration function, all second-order aberration coefficients have been expressed in compact form.
定义了区间值函数的RS积分，并利用它来定义模糊值函数的RS积分，给出了这种积分的一些性质。
RS integral of interval valued functions and Fuzzy valued functions are defined. Some properties of this kind of integral are obtained.
由涡的定义导出的向量泊桑方程的积分形式的解析解，被用于低速风洞壁引起的干扰速度的计算。
The analytical solution of Poisson's equation, derived from the definition of vortex, has been applied to the calculations of interference velocities due to the presence of wind tunnel walls.
本文基于“F”与“V”类积分方程序的基本形式，抽象出积分方程序“核”的基本定义。
This article discusses the basic form of "f" and "V" type integral equations, abstracts the basic definition of integral equation "nucleus".
本文给出了线、面、体积分定义的一般描述，并导出了利用对称性简化积分计算的定理。
This paper gives the general description of the definition of curve, curved surface and triple integrals, and derives a theorem on simplified integration by using their symmetry property.
微积分研究的对象是函数，研究工具是极限，微积分概念的定义方式与以往学习的初等数学概念有本质的区别。
Functions are research objects of the calculus and limit is its research tool. The way to define the calculus concept differs distinctly from that of elementary mathematics.
通过引入新空间及其范数，逐步定义了有界双参过程的积分和双参过程的积分，探讨了收敛性。
From a new space H introduced and its norm, integrations of bounded two parameter process and two parameter process are gradually defined, and convergence is investigated as well.
文章先介绍了黎曼积分的产生以及黎曼积分的定义性质与应用。
The article first introduced the production of Riemann Integration and by the way tell the nature, definition and application of Riemann Integration.
文章先介绍了黎曼积分的产生以及黎曼积分的定义性质与应用。
The article first introduced the production of Riemann Integration and by the way tell the nature, definition and application of Riemann Integration.
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