let me make a trip in the xy plane

我们在 xy 坐标系里走一段
made up of a whole bunch of little segments

由一个个小片段连成的路程
in each one of which I calculate this, and I add them all up.

每一段都可以那样计算然后相加
On the left-hand side, the ΔKs,

等式左边是一些 ΔK
this whole thing was defined so that it's equal to ΔK, right?

这个东西被定义为等于 ΔK
ΔK was equal to this, that's equal to all that.

ΔK 等于它等于后面这些
So, if you add all the ΔKs, it's very clear what you will get.

如果把所有的 ΔK 加起来结果很清楚
You will get the kinetic energy at the end

可以得到最后的动能
minus kinetic energy at the beginning.

减去开始时的动能
On the right-hand side,

在等式右边
you are told to add F·dr for every tiny segment.

需要把每一小段的 F·dr 相加
Well, that is written symbolically as this.

用这样的符号表示