这就是空间区域中的标准三重积分。
This is just your standard triple integral over a region in space.
我们学过的另一种积分是三重积分。
几个星期后，我们会学习三重积分。
这是建立三重积分的绝佳练习。
See, this is actually good practice to remember how we set up triple integrals.
现在，在球坐标中进行三重积分。
So now we're going to triple integrals in spherical coordinates.
这等价于在这个区域内部的三重积分。
这和计算其他三重积分的方法是相同的。
It's just the same way that you would compute any other triple integral.
先看看怎么，在球坐标中建立三重积分。
Well, we have to figure out how to set up our triple integral in spherical coordinates.
我们学过了三重积分。
这与三重积分不同。
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.
我要把我的黑板分成三部分，在这我将要写三重积分。
OK, so I'm going to divide my blackboard into three pieces, and here I will write triple integrals.
做三重积分，和二重一样，当然，我们会有更多的坐标系。
When we do triple integrals in space, well, it is the same kind of story, except now we have, of course, more coordinate systems.
一旦需要计算这个积分，只需要计算这个函数的三重积分。
Once you have computed what this guy is, it's really just a triple integral of the function.
那么在一个空间上的区域的三重积分，对标量dV做积分。
OK, so triple integrals over a region in space, we integrate a scalar quantity, dV.
本文还给出了计算三重积分的MATLAB程序。
In addition, the MATLAB program of estimating multivariable integral is given.
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.
运用对称性简化计算柱面坐标三重积分，并给予证明。
This paper indicates and proves the method of making use of symmetry to predigest calculation of cylindrical coordinates triple integral.
我们目前已经学习了三重积分，以及如何在各种坐标系中建立它们。
We have been working with triple integrals and seeing how to set them up in all sorts of coordinate systems.
指出了椭球形区域上三重积分的一科变量替换方法，并说明了其应用。
This paper points out the variable substitution method of triple integral of elliptical volume and examples its applications.
指出了椭球形区域上三重积分的一科变量替换方法，并说明了其应用。
We establish an equivalence between the generalized strongly complementarity problems and the Wiener Hopf equations by using a change of variables technique.
指出了椭球形区域上三重积分的一科变量替换方法，并说明了其应用。
Two techniques of chaos synchronization, variable substitution and variable feedback, are applied to Lorenz systems.
计算质量可看成二重或三重积分，这取决于空间维数，取决于密度函数，dA还是。
And mass will be double or triple integral, depending on how many dimensions you have, dV of whatever density function you have, dA or dV.
我将要证明，一个只有z分量的向量场的通量,等于一个三重积分，其中被积表达式为。
I'm going to prove that the flux of a vector field that has only a z component is actually equal to the triple integral of, RzdV well, the divergence of this is just R sub z dV.
那么，那将是。。。，抱歉，这和区域中,烟雾总量关于t的导数一样，由u的三重积分给出。
So, that should be, sorry, that's the same thing as the derivative with respect to t of the total amount of smoke in the region, which is given by the triple integral of u.
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。
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.
格林定理及其应用、三重积分、空间中的线积分和面积分、散度定理、斯托克斯定理应用。
Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.
将空间区域的对称性应用于三重积分的计算之中，归纳出了利用对称性计算三重积分的方法。
This essay deduces a way of calculating triple integral by the use of symmetry, by defining the symmetry of 3-degree area, and applying it to the triple integral computation.
利用高斯数值积分方法，推导了方形活塞辐射阻抗的三重积分表达式，计算了方形活塞声源的辐射阻抗。
Triple integral expression of the radiation impedance of square piston is deduced and the numerical results of the radiation impedance are calculated with the method of Gaussian numerical integral.
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.
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