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。
现在，在球坐标中进行三重积分。
So now we're going to triple integrals in spherical coordinates.
我们已经学过，二重积分和线积分。
We've learned about double integrals, and we've learned about line integrals.
这是极坐标系下的二重积分。
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.
要计算二重积分，要做的就是要利用切面。
So, to compute this integral, what we do is actually we take slices.
我们学过了三重积分。
那么，使用格林公式，我们去计算二重积分。
So, using Green's theorem, the way we'll do it is I will, instead, compute a double integral.
然后观察二重积分,看看能不能使两式相等。
Next, I should try to look at my double integral and see if I can make it equal to that.
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?
这与三重积分不同。
就二重积分来讲,它是对区域里函数值求总和。
The way we actually think of the double integral is really as summing the values of a function all around this region.
如果是一条闭曲线，也可以用二重积分来代替的。
If it is a closed curve, we should be able to replace it by a double integral.
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.
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.
也就是最终要摆脱曲面积分，回到常规的二重积分。
And this is finally where I have left the world of surface integrals to go back to a usual double integral.
当计算二重积分时，要多了解这些符号的具体含义。
OK, we'll come up with more concrete notations when we see how to actually compute these things.
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.
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。
OK, if that's not completely clear to you, then I encourage you to go over how we set up double integrals again.
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.
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.
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.
请说，你想知道极坐标系下的积分边界，这是一个二重积分。
Yes? In case you want the bounds for this region in polar coordinates, indeed it would be double integral.
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, now another thing we've seen with double integrals is how to do more complicated changes of variables.
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.
我们已经做过的一个例子是，计算四分之一单位圆上的二重积分。
One example that we did, in particular, was to compute the double integral of a quarter of a unit disk.
先看看简单些的曲线的情形，这样我们解决二重积分会简单许多。
So maybe we first want to look at curves that are simpler, that will actually allow us to set up the double integral easily.
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.
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.
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