用动力学在 3.14 分钟内理解 e^(iπ) | DE5：英语精读讲义

One way to think about the function e to the t is to ask what properties does it have? 表達脈絡

思考函数 e^t 的一种方式是问：它有什么性质？

Probably the most important one,

也许最重要的一点，

and from some points of view the defining property, 語氣意圖

从某些角度来看也是它的决定性性质，

is that it is its own derivative. 語法

就是它的导数等于它本身。

Together with the added condition that inputting zero returns one, 用詞

再加上输入为零时返回一这个附加条件，

it's actually the only function with this property.

它实际上是唯一具有此性质的函数。

And you can illustrate what this means with a physical model. 短語搭配

你可以用一个物理模型来说明这意味着什么。

If e to the t describes your position on a number line as a function of time, 用詞

如果 e^t 描述了你在数轴上的位置随时间变化的函数，

then you start at the number one,

那么你从数字 1 开始，

and what this equation is saying is that your velocity, 語法

这个等式表达的是你的速度，

the derivative of position,

也就是位置的导数，

is always equal to that position.

总是等于你所处的位置。

The farther away from zero you are, 語法

你离零越远，

the faster you move.

你移动得就越快。

So even before knowing how to compute e to the t exactly, 語法

所以，甚至在确切知道如何计算 e^t 之前，

going from a specific time to a specific position,

也就是从特定时间得出特定位置之前，

this ability to associate each position with a velocity 短語搭配

这种将每个位置与速度联系起来的能力，

paints a very strong intuitive picture of how the function must grow. 習語

已经非常直观地描绘出了这个函数的增长方式。

You know that you'll be accelerating,

你知道你将不断加速，

and at an accelerating rate, 短語搭配

并且以递增的速率加速，

with an all-around feeling of things getting out of hand quickly. 習語

会有一种事情很快失控的整体感觉。

And if you add a constant to that exponent,

如果你给指数乘上一个常数，

like e to the 2 times t,

比如 e 的 2t 次方，

the chain rule tells us that the derivative is now 2 times itself.

链式法则告诉我们，现在的导数是它本身的 2 倍。

So at every point on the number line,

所以在数轴上的每一个点，

rather than attaching a vector corresponding to the number itself, 語法

与其附加上与数字本身相对应的向量，

first double the magnitude of the position,

不如先将位置的大小翻倍，

then attach it.

然后再附加上去。

Moving so that your position is always e to the 2t

让你的位置始终是 e^(2t) 来移动，

is the same thing as moving in such a way 語法

其实就等同于这样一种移动方式：

that your velocity is always twice your position.

你的速度始终是你位置的两倍。

The implication of that too

这同时也意味着，

is that our runaway growth feels all the more out of control. 短語搭配

我们那种失控的增长会让人感觉更加无法掌控。

If that constant was negative,

如果那个常数是负数，

say negative 0.5,

比如 -0.5，

then your velocity vector is always negative 0.5 times your position vector,

那么你的速度向量就始终是位置向量的 -0.5 倍，

meaning you flip it around 180 degrees

意味着你将它翻转 180 度，

and scale its length by a half. 用詞

并将其长度缩小一半。

Moving in such a way

按照这样的方式移动，

that your velocity always matches this flipped and squished copy of your position vector,

也就是让你的速度始终与这个翻转且缩小的位置向量副本相匹配，

you go the other direction,

你就会朝着相反的方向前进，

slowing down in an exponential decay towards zero. 用詞

以指数衰减的方式减速向零靠近。

But what about if that constant was i, 語氣意圖

但如果那个常数是 i 呢，

the square root of negative one?

也就是负一的平方根？

If your position was always e to the i t,

如果你的位置始终是 e^(it)，

how would you move as the time t ticks forward?

随着时间 t 的流逝，你会如何移动？

Well, now the derivative of your position will always be i times itself,

嗯，现在你位置的导数将始终是它本身的 i 倍，

and multiplying by i has the effect of rotating numbers 90 degrees. 短語搭配

而乘以 i 会产生将数字旋转 90 度的效果。

So, as you might expect, 表達脈絡

所以，正如你所料，

things only make sense here if we start thinking beyond the number line 習語

只有当我们开始跳出数轴的限制，

and in the complex plane.

在复平面上思考时，这一切才说得通。

So, even before you know how to compute e to the i times t,

这样，甚至在你知道如何计算 e^(it) 之前，

you know that for any position this might give for some value of time,

你就知道，无论某个时间值对应的是什么位置，

the velocity at that time will be a 90 degree rotation of that position.

在那个时间的速度都会是该位置旋转 90 度的结果。

Drawing this for all possible positions you might come across, 短語動詞

把你可能遇到的所有位置都画出来，

you get a vector field,

你会得到一个向量场，

where as usual with vector fields,

在这里，和通常处理向量场一样，

you shrink things down to avoid clutter. 短語動詞

你把它们缩小以避免看起来太杂乱。

At time t equals zero,

当时间 t 等于零时，

e to the i t will be one,

e^(it) 等于一，

that's our initial condition,

这是我们的初始条件，

and there's only one trajectory starting from that position

并且从该位置开始只有一条轨迹，

where your velocity is always matching the vector that it's passing through,

能让你的速度始终与它经过的向量相匹配，

a 90 degree rotation of the position.

也就是位置旋转 90 度的向量。

It's when you go around a circle of radius one

那就是当你沿着半径为一的圆，

at a speed of one unit per second. 用詞

以每秒一个单位的速度运动时。

So, after pi seconds,

所以，经过 pi 秒后，

you've traced a distance of pi around.

你就在圆上走过了 pi 的距离。

So, e to the i times pi should be negative one. 語法

因此，e^(iπ) 应该等于负一。

After tau seconds,

经过 tau 秒后，

you've gone full circle.

你刚好走完整整一圈。

E to the i times tau equals one.

e^(iτ) 等于一。

And more generally, 表達脈絡

更一般地说，

e to the i times t equals a number

e^(it) 等于一个数字，

that's t radians around this unit circle in the complex plane.

它在复平面的单位圆上转过了 t 个弧度。

Nevertheless, 表達脈絡

尽管如此，

something might still feel immoral about putting an imaginary number up in that exponent. 習語

把一个虚数放在指数的位置，还是会让人觉得有些不合情理。

And you would be right to question that. 語氣意圖

你的这种质疑是对的。

What we write as e to the t is a bit of a notational disaster, 習語

我们写成 e^t 其实在记号上是个灾难，

giving the number e and the idea of repeated multiplication

它给了数字 e 以及重复相乘的概念，

way more emphasis than they deserve. 語法

远远超出它们应得的强调。

But my time is up, 習語

但我的时间到了，

so I'll spare you the full rant until the next video. 語氣意圖

所以我会在下一个视频里接着完整地吐槽。