the ax has enlarged the borders of our settlements, and the mines, as well of iron and coal as of the precious metals, have yielded even more abundantly than heretofore.

VOA: special.2009.11.23

Not to harp on the mathematical features of this, but cubing, AX*X*X you know, if you're starting to do AX star, X star, X, every time you want to cube some value in a program, it just feels like this is going to get a little messy looking, if nothing else.

不要总是说这个的数学特性，但是体积，你们懂的，如果你开始做，在一个程序中，每次你想算几个数值的体积，感觉它就变得,有一点凌乱的，如果没有其他的。

哈佛公开课 - 计算机科学课程节选

So there is a harp, making its melodious harmonious sounds, and then you take an ax to the harp, bang bang bang, chop chop chop, or a hammer whatever.

假设这有一个竖琴,发出悦耳和谐的声音，然后你拿起一把斧子，梆梆邦，砍砍砍，或者拿一把锤子。

耶鲁公开课 - 死亡课程节选

Therefore, the vector A that you gave me, I have managed to write as i times Ax plus j times Ay.

这样,你给我的矢量 A,我已把它写成 i ? Ax + j ? Ay的形式

耶鲁公开课 - 基础物理课程节选

You can ask yourself, "If you gave me a particular vector, what do I use for Ax and Ay?"

你可以问问自己,"如果给定一个矢量,该怎么确定 Ax 和 Ay"

耶鲁公开课 - 基础物理课程节选

But then, I combine i times Ax with i times Bx, because that's the vector parallel to i, with the length Ax.

然后我把 i ? Ax 和 i ? Bx 加起来,因为这是平行于 i,模长为 Ax 的矢量

耶鲁公开课 - 基础物理课程节选

That's clear. If I add them, I'll get a vector parallel to i with lengths Ax + Bx.

这很明显 如果把他们加起来,就得到一个平行于 i,模长为 Ax + Bx 的矢量

耶鲁公开课 - 基础物理课程节选

But this is the same vector we are calling i times Ax plus j times Ay.

这和矢量 i ? Ax + j ? Ay 是一样的

耶鲁公开课 - 基础物理课程节选

The point is the arrow A, somebody has chosen to write in terms of i prime and j prime as Ax prime and Ay prime.

解决问题的关键在于矢量 A,可以用这样的形式来描述,i' ? Ax' + j' ? Ay'

耶鲁公开课 - 基础物理课程节选

In practice, most of the time we work with these two numbers, Ax and Ay.

在实际过程中,大多数时间我们就用 Ax 和 Ay 来计算

耶鲁公开课 - 基础物理课程节选

If the Ax and Ay, some are positive and some are negative, this is the way by which we have learned we should combine multiples of i.

如果 Ax 和 Ay 有正有负,这就要用到我们所学过的方法,将所有 i 的倍数加起来

耶鲁公开课 - 基础物理课程节选

But Ax prime and Ay prime will continue to be the coefficients.

但系数仍然是 Ax' 和 Ay'

耶鲁公开课 - 基础物理课程节选

So whatever number it takes, Ax times i is this part.

因此无论系数是多少,Ax 乘以 i 表示这一部分

耶鲁公开课 - 基础物理课程节选

If you give me a pair of numbers, Ax and Ay, that's as good as giving me this arrow, because I can find the length of the arrow by Pythagoras' theorem.

如果给我一组数字,Ax 和 Ay,就相当于给了我这个箭头示意图,因为我可以利用毕达哥拉斯定理定理求出模长

耶鲁公开课 - 基础物理课程节选

That would come by saying, "I'm taking i times Ax + j times Ay + i times Bx + j times By, " and I'm trying to add all these guys.

那就这样做,"i ? Ax + j ? Ay + i ? Bx + j ? By",然后把它们都加起来

耶鲁公开课 - 基础物理课程节选

satisfies the condition tan is Ay over Ax..

满足其正切等于 Ay 除以 Ax

耶鲁公开课 - 基础物理课程节选

Now, when you work with components, Ax and Ay, if I didn't mention it, they are the components of the vector, you can do all your bookkeeping in terms of Ax and Ay.

当你们在计算分量 Ax 和 Ay 的时候,即使我没有说明,你们也要记得它们是矢量的分量,你们可以都用 Ax 和 Ay 的形式来表示

耶鲁公开课 - 基础物理课程节选

A very important result is that if two vectors are equal, if A = B, the only way it can happen is if separately Ax is equal to Bx and Ay is equal to By.

这里有一个相当重要的结论,如果两个矢量相等,例如 A = B,那么当且仅当,Ax = Bx 和 Ay = By 分别成立

耶鲁公开课 - 基础物理课程节选

They won't invert the relation.

不是简单地交换 Ax 和 Ax'

耶鲁公开课 - 基础物理课程节选

the Ax prime will drop off.

x 就被消去了

耶鲁公开课 - 基础物理课程节选

I gave you a law of transformation of the components; namely, if the vector has components Ax and Ay in one reference frame and Ax prime and Ay prime in another reference frame, how are the two related?

我介绍过分量变换的法则,即如果矢量在一个坐标系的分量为 Ax 和 Ay,在另一坐标系中的分量为 Ax' 与 Ay',它们有着什么样的联系

耶鲁公开课 - 基础物理课程节选