用動力學在 3.14 分鐘內理解 e^(iπ) | DE5：英語精讀講義

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

思考函數 e^t 的一種方式是問：它有什麼性質？

Probably the most important one,

也許最重要的一點，

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

從某些角度來看也是它的決定性性質，

is that it is its own derivative. 文法

就是它的導數等於它本身。

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

再加上輸入為零時返回一這個附加條件，

it's actually the only function with this property.

它實際上是唯一具有此性質的函數。

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

你可以用一個物理模型來說明這意味著什麼。

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

如果 e^t 描述了你在數線上位置隨時間變化的函數，

then you start at the number one,

那麼你從數字 1 開始，

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

這個方程式表達的是你的速度，

the derivative of position,

也就是位置的導數，

is always equal to that position.

總是等於你所處的位置。

The farther away from zero you are, 文法

你離零越遠，

the faster you move.

你移動得就越快。

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

所以，甚至在確切知道如何計算 e^t 之前，

going from a specific time to a specific position,

也就是從特定時間得出特定位置之前，

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

這種將每個位置與速度聯繫起來的能力，

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

已經非常直觀地描繪出了這個函數的增長方式。

You know that you'll be accelerating,

你知道你將不斷加速，

and at an accelerating rate, 片語搭配

並且以遞增的速率加速，

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

會有一種事情很快失控的整體感覺。

And if you add a constant to that exponent,

如果你給指數乘上一個常數，

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.

連鎖律告訴我們，現在的導數是它本身的 2 倍。

So at every point on the number line,

所以在數線上的每一個點，

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

與其附加上與數字本身相對應的向量，

first double the magnitude of the position,

不如先將位置的大小翻倍，

then attach it.

然後再附加上去。

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

讓你的位置始終是 e^(2t) 來移動，

is the same thing as moving in such a way 文法

其實就等同於這樣一種移動方式：

that your velocity is always twice your position.

你的速度始終是你位置的兩倍。

The implication of that too

這同時也意味著，

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

我們那種失控的增長會讓人感覺更加無法掌控。

If that constant was negative,

如果那個常數是負數，

say negative 0.5,

比如 -0.5，

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

那麼你的速度向量就始終是位置向量的 -0.5 倍，

meaning you flip it around 180 degrees

意味著你將它翻轉 180 度，

and scale its length by a half. 用詞

並將其長度縮小一半。

Moving in such a way

按照這樣的方式移動，

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

也就是讓你的速度始終與這個翻轉且縮小的位置向量副本相匹配，

you go the other direction,

你就會朝著相反的方向前進，

slowing down in an exponential decay towards zero. 用詞

以指數衰減的方式減速向零靠近。

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

但如果那個常數是 i 呢，

the square root of negative one?

也就是負一的平方根？

If your position was always e to the i t,

如果你的位置始終是 e^(it)，

how would you move as the time t ticks forward?

隨著時間 t 的流逝，你會如何移動？

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

嗯，現在你位置的導數將始終是它本身的 i 倍，

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

而乘以 i 會產生將數字旋轉 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 慣用語

只有當我們開始跳出數線的限制，

and in the complex plane.

在複數平面上思考時，這一切才說得通。

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

這樣，甚至在你知道如何計算 e^(it) 之前，

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

你就知道，無論某個時間值對應的是什麼位置，

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

在那個時間的速度都會是該位置旋轉 90 度的結果。

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

把你可能遇到的所有位置都畫出來，

you get a vector field,

你會得到一個向量場，

where as usual with vector fields,

在這裡，和通常處理向量場一樣，

you shrink things down to avoid clutter. 片語動詞

你把它們縮小以避免看起來太雜亂。

At time t equals zero,

當時間 t 等於零時，

e to the i t will be one,

e^(it) 等於一，

that's our initial condition,

這是我們的初始條件，

and there's only one trajectory starting from that position

並且從該位置開始只有一條軌跡，

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

能讓你的速度始終與它經過的向量相匹配，

a 90 degree rotation of the position.

也就是位置旋轉 90 度的向量。

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

那就是當你沿著半徑為一的圓，

at a speed of one unit per second. 用詞

以每秒一個單位的速度運動時。

So, after pi seconds,

所以，經過 pi 秒後，

you've traced a distance of pi around.

你就在圓上走過了 pi 的距離。

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

因此，e^(iπ) 應該等於負一。

After tau seconds,

經過 tau 秒後，

you've gone full circle.

你剛好走完整整一圈。

E to the i times tau equals one.

e^(iτ) 等於一。

And more generally, 表達脈絡

更一般地說，

e to the i times t equals a number

e^(it) 等於一個數字，

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

它在複數平面的單位圓上轉過了 t 個弧度。

Nevertheless, 表達脈絡

儘管如此，

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

把一個虛數放在指數的位置，還是會讓人覺得有些不合情理。

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

你的這種質疑是對的。

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

我們寫成 e^t 其實在記號上是個災難，

giving the number e and the idea of repeated multiplication

它給了數字 e 以及重複相乘的概念，

way more emphasis than they deserve. 文法

遠遠超出它們應得的強調。

But my time is up, 慣用語

但我的時間到了，

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

所以我會把完整的吐槽留到下一個影片。