二重积分

利用线性规划论的最优性原理和图解法解决一些特殊二重积分的估值问题。

This paper solves some special estimating value problems of double integral with the help of the optimality principle and graphical solution of the linear programming .

为更好地指导教学，文章还对如何准确、有效地利用对称性简化二重积分的计算作了进一步的探讨。

To give better advice on teaching , futher discussion , is made on how to simplify the double integral operation with symmetry , precisely and effectively .

区域R的面积是函数1在R上的二重积分。

The area R is the double integral over R of a function one .

而这里是关于x，y的函数的二重积分。

Here , it 's a double integral of some function of x and y.

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 .

这是封闭曲面上F•ndS的二重积分。

Double integral of f dot n dS of a closed surface S.

那就是一个在区域U上的二重积分，我还没有给出对区域U的描述。

That means it 's a double integral over this region , U , which I haven 't described to you .

我们已经定义了，平面上区域R的fdA的二重积分。

We have defined the double integral over a region R and plane of a function f of x , y dA .

我会计算，在这个区域内旋度FdA的二重积分。

So , I will & compute the double integral over the region inside of curl F dA .

接下来要证明的是，这是等于dA的二重积分。

And we want to show that this is equal to the double integral of P sub x Q sub y dA .

运用复合梯形公式作为求二重积分数值解的基本算法，采用Matlab软件编写程序，求解二重积分方程组。

Demand the use of double integral composite trapezoidal formula as the basic numerical solution algorithm , using Matlab software , programming , solve the double integral equations .

在柱面坐标系下，点电荷的相互作用能是z和ρ的具有极点的二重积分。

The interaction energy of two point charged particles is the double integral of z and ρ with poles in the cylindrical coordinate system .

关于在xy坐标系里建立二重积分有问题吗？

OK , any questions about how to set up double integrals in xy coordinates ?

下面我将怎么做呢？，我已经把给定积分，变换成了对adS的二重积分。

Now what do I do with that ? Well , I have turned my integral into the double integral of a dS .

有时要计算的体积处于x，y平面，和某函数图象之间，则可以直接当作二重积分计算。

Sometimes if it 's the volume between the x , y plane and the graph of some function , you can just set it up directly as a double integral .

举个例子，区域R的面积是dA的二重积分，便于理解，在这里写成。

So , for example , the area of region is the double integral of just dA , 1dA or if it helps you , one dA if you want .

我只需要计算xdA在R上的二重积分，看上去确实简单许多。

I will just get double integral over R of x dA , which looks certainly a lot more pleasant .

它就是曲面上对FdS的二重积分。

It will just be the double integral over a surface of F dot n dS .

格林公式内容是如果有一条封闭曲线，那么对F的线积分就等于，区域内F旋度的二重积分。

So , Green 's theorem says that if I have a closed curve , then the line integral of F is equal to the double integral of curl on the region inside .

称之为区域R上fdA的二重积分，会向大家解释这些符号的含义的。

So , we 'll call that the double integral of our region , R , of f of xy dA and I will have to explain what the notation means .

在RCS的计算中是将散射体表面的物理光学积分通过参数变换为NURBS参数域上的二重积分，利用驻定相位法得出组合NURBS表面的散射场。

During the computation of RCS we transform the physical optics integral over the surfaces of the scatterer into the dual integral of nurbs parameter field .

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。

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 .

有一个和散度定理很像的东西，当然，对于通量和div，f的二重积分，都可以使用类似的理论。

There is a similar thing with the divergence theorem , of course , with flux and double integral of div f , you can apply exactly the same argument .

要用一般方法来求这些二重积分，也就是，有一个关于x和y的函数，在区域上，找出积分边界。

The way you would evaluate these double integrals is just the usual way . Namely , you have a function of x and y , you have a region and you set up the bounds for the isolated integral .

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。

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 .

就是做F·dS或是F·ndS的二重积分，为了能建立积分，需要用到曲面的几何性质，这与该曲面的类型有关。

Double integral of F.dS or F.ndS if you want , and to set this up , of course , I need to use the geometry of the surface depending on what the surface is .

本文则采用线性基函数，详细推导了在线性单元上二重积分第一重积分的解析公式，第二重积分使用Gauss数值积分。

For calculation of the double integration on the boundary , the analytical formula of the first integral is deduced in detail with linear basic function , and a Gauss integration formula is used in the second integral .

我要做的就是把它划成小区域，这些小区域有清晰的上下边界，那么就能轻松地建立关于dy，dx的二重积分。

OK , so what I will do is I will cut this into smaller regions for which I have a well-defined lower and upper boundary so that I will be able to set up a double integral , dy dx , easily .

milch定理及适当变换，使二重积分方程化为一维的Fredholm方程，并获得严格的形式解，还提出一个判别法，并用以证明解的唯一性。

According to Schlomilch theorem , the 2-dimensional problem is reduced to be a one dimensional Fredholm equation . An exact formal solution is obtained and a criterion is suggested to prove the uniqueness of the formal solution .

当我们建立积分时，记住，我们将在这建立二重积分，当我们对dy，dx积分时，表示我们将平面区域用垂线分成小片。

Yeah ? Well , so , when we set up remember , we are setting up a double integral dy dx here So , when we do it dy dx it means we slice this region of a plane by vertical line segments .