And, of course, we know also how to set up these integrals in polar coordinates.
当然，我们也知道，怎样用极坐标来计算这些积分。
Anyway, that is double integrals in polar coordinates.
这是极坐标系下的二重积分。
OK, the right way to do this will be to integrate it in polar coordinates.
这道题最合适的方法，应该在极坐标系里面计算。
So, in a way, you can think of these as a space analog of polar coordinates because you just use distance to the origin, and then you have to use angles to determine in which direction you're going.
也可以把这种坐标，看成是空间中的极坐标，它其实使用了，距离原点的距离，然后用角度这种标尺，来确定了方向。
Yes? In case you want the bounds for this region in polar coordinates, indeed it would be double integral.
请说，你想知道极坐标系下的积分边界，这是一个二重积分。
If my solid is actually just going to be round then I might want to use polar coordinates.
如果实体是圆形状的话，那就可以考虑用极坐标系。
Or I can use polar coordinates,and it works a little bit better that way.
或者还是用极坐标系，这比直角坐标系更好。
And I would like to switch that to polar coordinates.
把它转换到极坐标系下。
That's in line with the idea that we are just doing again, polar coordinates in the rz directions.
球坐标中的平面也差不多是这样了，只要把rz看做极坐标就行了。
But still we could set this up and then switch to polar coordinates to evaluate this integral.
但还是可以建立这个积分，然后转换到极坐标系下去求结果。
But somehow maybe if you suspect that polar coordinates will be involved somehow in the answer then maybe it makes sense to choose different paths.
但是你可能推测，如果结果中包含极坐标，那么选择不同的路径都行得通。
I mean usually you will switch to polar coordinates either because the region is easier to set up.
通常你转换到极坐标系下，有可能积分区域更容易建立。
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。
But, if it's like a circle or a half circle, or things like that, then even if a problem doesn't tell you to do it in polar coordinates you might want to seriously consider it.
但是如果它是一个圆或者半圆，或者是类似的，即使题目没有提示你在极坐标里做，你也应该认真考虑一下。
You have a function that you want to switch between rectangular and polar coordinates.
把一个函数，在直角坐标和极坐标中转换。
Let's check what we had for polar coordinates.
我们来回顾一下极坐标。
Now, of course, is the region is really looking like this, then you're not going to do it in polar coordinates.
当然，如果区域是这样的，就没有必要用极坐标来做了。
I am going to use polar coordinates.
我将用极坐标。
So the idea of spherical coordinate is you're going to polar coordinates again in the rz plane.
所以说球坐标系,其实就是在rz平面再建立一个极坐标。
So, the bad news is we have to be able to do it not only in xy coordinates, but also in polar coordinates.
接下来有个坏消息就是，不仅要会在xy坐标系里做，还要会在极坐标系里做。
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.
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, when you go to polar coordinates, basically all you have to remember on the side of integrate is that x becomes r cosine theta. y becomes r sine theta.
在极坐标系里，要记住在积分一侧,只是把x变成rcosθ，y变成rsinθ
With the plot command, gnuplot can operate in rectangular or polar coordinates.
使用plot命令，gnuplot可以在直角二维坐标系中进行操作。
Say, polar coordinates. Let's say that I have a function but is defined in terms of the polar coordinate variables on theta.
比如说极坐标，有一个函数,是根据极坐标θ定义的。
We could do it dy dx. Or, maybe we will want to actually make a change of variables to first shift this to the origin, x-2 you know, change x to x minus two and then switch to polar coordinates.
也可以dy，dx来划分，或者,对变量做一下变化，移动到原点，就是把x变成,然后再转化成极坐标。
Or, the other way around, I have a function of x and y and I want to express it in terms of the polar coordinates r and theta.
反过来，有一个关于x和y的函数,也可以表示成极坐标r和θ
R This one is a double integral. So, if you are doing it, say, on a disk, you would have both R and theta if you're using polar coordinates.
不是变量，这是在一个圆上。，R，is，not，a，variable。，You，are，on，the，circle。,这个是二重积分，如果你们这么做的话，在圆盘上，如果你们用极坐标的话,就需要用到R和θ
Remember that polar coordinates are about replacing x and y as coordinates for a point on a plane by instead r, r and theta,which is the angle measured counterclockwise from the positive x-axis.
极坐标是关于r，θ的,用r，θ代替平面上一点的x，y坐标，是从原点到那一点的距离，，which，is，the，distance，from，the，origin，to，a，point，θ是由x轴逆时针旋转,而得到的夹角。
Allocate random positions within a disc according to a given distribution for the polar coordinates of each node with respect to the provided center of the disc.
在按照一个给定分布极光盘随机地位坐标分派方面的光盘供给中间各节点。
According to this process polar coordinates can be extrapolated by means of data in changeable length.
根据这方法能够用可变长度的资料对地极坐标进行外推。
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