Let's think about various USES of double integrals.
来想想二重积分的其他用途吧。
Anyway, that is double integrals in polar coordinates.
这是极坐标系下的二重积分。
One is setting up and evaluating double integrals.
一个是建立并计算二重积分。
So, the first topic will be setting up double integrals.
先来说说关于建立二重积分的事。
That is because we are only doing double integrals so far.
这是由于我们才刚开始做二重积分的缘故。
But most of the time we need to learn how to set up double integrals.
但在大多数情况下，我们需要了解怎样建立二重积分。
The claim is we are able, to do double integrals in polar coordinates.
这也就说明了，可以用极坐标做二重积分。
We've learned about double integrals, and we've learned about line integrals.
我们已经学过，二重积分和线积分。
OK, any questions about how to set up double integrals in xy coordinates?
关于在xy坐标系里建立二重积分有问题吗？
The double integrals in these expressions are reduced to single integrals.
同时把这两种方法中含有二重积分的公式简化为一重积分形式。
OK, now another thing we've seen with double integrals is how to do more complicated changes of variables.
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, if that's not completely clear to you, then I encourage you to go over how we set up double integrals again.
如果你还不是完全清楚，建议你去复习一下，如何建立二重积分。
But another place where this comes up, if you remember what we did in the plane, curl also came up when we tried to convert line integrals into double integrals.
除了这些，旋度还有一个用处，如果你还记得，我们讨论平面的时候,当把线积分转换成二重积分的时候。
And whether these line integrals or double integrals are representing work, flux, integral of a curve, whatever, the way that we actually compute them is the same.
不管是线积分或是二重积分，也不管它们表示的是功还是通量，计算它们的方法实际上是一样的。
Coupled IFFT algorithm and adaptive quadrature algorithm was employed to compute double integrals in the analytical solutions. All the subsonic, transonic and supersonic cases were computed.
然后采用IFFT算法和自适应数值积分算法计算二维积分，得到了低音速、跨音速和超音速移动荷载作用下动力响应的数值结果。
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.
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。
And this is finally where I have left the world of surface integrals to go back to a usual double integral.
也就是最终要摆脱曲面积分，回到常规的二重积分。
And so that was stuff about double and triple integrals and vector calculus in the plane and in space.
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。
In this paper, we calculate the double chord-power integrals on circular disk, and arrive at some important properties and inequalities of the double chord-power integrals.
给出圆盘上的一些双弦幂积分公式，并由此获得一些重要的性质及双弦幂积分的一些不等式。
In this paper, some relationships between improper double integral and improper iterated are discussed, and the corresponding results are generalized into the improper triple integrals.
本文通过讨论广义重积分与广义逐次积分之间的关系，得出一些结论，并将相应结果推广到广义三重积分与广义三次积分中。
In this paper, some relationships between improper double integral and improper iterated are discussed, and the corresponding results are generalized into the improper triple integrals.
本文通过讨论广义重积分与广义逐次积分之间的关系，得出一些结论，并将相应结果推广到广义三重积分与广义三次积分中。
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