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。
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。
dy x dt That tells us dx is x prime of t dt.
是y’dt，dy，becomes，y，prime，of，t，dt。,是x，So，，we，know，x，is，x，of，t。,它告诉我们dx是x
And, once you have a formula for ndS, that will relate ndS to maybe dx dy, or something else.
一旦有一个ndS的公式，那将会把ndS和dx,dy联系起来，或者其他的一些东西。
If you end up with a dx, dy, dz in the surface integral, something is seriously wrong.
如果在这个曲面积分中是以dx,dy,dz结尾的，则表明出现了很大的问题。
Once you have found the ratio between du dv and dx dy, then it works one way or it works the other way.
你一旦找到了dudv和dxdy之间的比率，它在这个情况下行得通，在相反的情况下行得通。
And, for that, we'll have to figure out a way to slice this region nicely. We could do it dx dy.
为了计算二重积分，我们必须想出一个办法，来适当地划分这个区域，我们可以按dx,dy来划分。
Or, it tells you actually anything you might want to know about the relations between these variables so, for example, you can move dz to that side, and then express dz in terms of dx and dy.
事实上它告诉了我们，任何的关于这些变量之间的关系，例如，可以把dz移到另一边，用dx和dy来表示。
OK, and so the coefficients of dx and dy are the partials.
那么dx和dy前面的系数就是偏导数。
While the function that you are integrating goes before the dx dy and not into the bounds or anything like that.
而积分函数就是dxdy前的这个函数，它与积分边界无关。
So we found an expression for dz in terms of dx and dy.
我们得到了dz关于dx和dy的表达式。
Or, you can move dy to that side and express dy in terms of dx and dz, and so on.
或者把dy移到另一边，用dx和dz来表示dy，诸如此类。
Well, we can do that in rectangular coordinates dz dx dy where dV becomes something like, maybe, dz dx dy, or any permutation of these.
我们在直角坐标系中处理，在这里dV有可能变成，或者它们的其他排列。
So, that means du dv is worth five times dx dy.
所以，这也意味着dudv比dxdy大五倍。
So, if we want to set it up instead dx dy, then it means we are going to actually look at the converse question which is, for a given value of y, what is the range of values of x?
如果要用dxdy来替换，就是说,转化为与之相反的问题来看，给定y的一个值，那么x的取值范围是什么呢？
So, the area elements are actually related by the absolute -10 value of this guy. OK, so if you find -10 as your 10*dxdy answer, then du dv is still ten times dx dy.
所以说，面积其实是跟这个东西的绝对值有关的,如果你们得到的答案是,那么dudv依旧是。
dxdy Last time I told you if we have dx dy we could switch it to r dr d theta.
上一次课提到，如果给出,我们可以把它转换为rdrdθ
So, we should get to something that will be the same integral dx dy.
也就是在某个边界里算dxdy的积分。
dxdy So that's just x. So, that means that du dv is x dx dy.
这个就是x，这就意味着dudv是x倍的。
the general notation for vector dr-- y equals dx, x roof, plus dy, y roof, z plus dz, z roof.
总的向量dr的符号-,=dr向量x+dy向量，+dz向量。
So, we know that we want to integrate just the area element, let's say, dx dy over the origin inside the ellipse.
我们知道现在要积分面积元素，那就对dxdy积分。
OK, so I'm computing the integral of y dx x dy.
我来计算ydx+xdy的积分。
所以Tds就是。
Namely, let's say that you have a function maybe of three variables, x, y, z, df = fxdx+fydy +fzdz then you would write df equals f sub x dx plus f sub y dy plus f sub z dz.
即是说，如果你有一个,含自变量，x，y，z的函数，那么。
Namely, let's say that you have a function maybe of three variables, x, y, z, df = fxdx+fydy +fzdz then you would write df equals f sub x dx plus f sub y dy plus f sub z dz.
即是说，如果你有一个,含自变量，x，y，z的函数，那么。
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