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微积分课上学不到的秘籍

What they won't teach you in calculus

三蓝一棕
3Blue1Brown
想像下 你是刚入门微积分的学生
Picture yourself as an early calculus student
正准备开始第一门课
about to begin your first course.
接下来的几个月
The months ahead of you hold within
你要接受一连串考验
them a lot of hard work:
你将见到简洁清晰的公式
Some neat examples,
复杂晦涩的公式
some not so neat examples,
有的呈现物理之美
beautiful connections to physics,
更多的则冗长枯燥而难以记忆
not so beautiful piles of formulas to memorise,
很多时候你陷入困境
plenty of moments of getting stuck and
甚至想去撞墙
banging your head into a wall,
也有一些恍然大悟的瞬间
a few nice ‘aha’ moments sprinkled in as well,
有些精妙的图象化工具
and some genuinely lovely graphical intuition
可以帮你从直觉上准确理解
to help guide you through it all.
不过 如果你即将学习的课程
But if the course ahead of you is anything
和我上学时的微积分入门课
like my first introduction to calculus or any of
或者和我后来见过的所有入门课一样
the first courses that I’ve seen in the years since,
今天我要说的内容从来没在课堂出现过
there’s one topic that you will not see,
我相信它会加速你对微积分的理解
but which I believe stands to greatly accelerate your learning.
第一年你对微积分形成的图形直觉
You see almost all of the visual intuitions
都建立在函数图形基础上
from that first year are based on graphs –
导数是函数图形的斜率
the derivative is the slope of a graph,
积分是函数图形下方的那部分面积
the integral is a certain area under that graph,
但如果将函数的取值和结果
but as you generalize calculus
推广到不只是数字的时候
beyond functions whose inputs and outputs are simply numbers,
有些函数就无法画出对应的图形
it’s not always possible to graph the function that you’re analyzing.
不过想直观理解函数
There’s all sorts of different ways that
方法有很多
you’d be visualizing these things
如果你对导数之类的基础概念
so if all your intuitions for the fundamental ideas, like derivatives,
过于依赖图形直觉进行理解
are rooted too rigidly in graphs,
那么当你学到更深入的知识时
it can make for a very tall and largely unnecessary
理解概念时就会遇到不必要障碍
conceptual hurdle between you and the more”advanced topics”,
比如多元微积分 复分析 微分几何
like multivariable calculus, and complex analysis, differential geometry…
现在 我想向你分享的是
Now, what I want to share with you
另一种导数的思考方式
is a way to think about derivatives
我把它叫做“变换的视角”
which I’ll refer to as the transformational view,
它可以更自然地推广为更一般的微积分
that generalizes more seamlessly into some of those more general context where calculus comes up
一会我们就用“变换的视角”
And then we’ll use this alternate view
试着做一道有趣的连分数题目
to analyze a certain fun puzzle about repeated fractions.
但是首先 我和你要形成统一
But first off, I just want to make sure
我们理解导数的一般角度是一致的
that we’re all on the same page about what the standard visual is.
假如画一个函数的图形
If you were to graph a function,
输入一个数字 得出一个数字
which simply takes real numbers as inputs and outputs,
你在微积分的第一节课学到的
one of the first things you learn in a calculus course
导数是函数图形的斜率
is that the derivative gives you the slope of this graph.
这里面的深层含义是
Where what we mean by that is that
函数的导函数是新的函数
the derivative of the function is a new function
对每个输入的x 给出原函数对应的斜率
which for every input x returns that slope.
这里 我建议你先别把“导数是斜率”
Now I’d encourage you not to think of this derivative
当作导数的定义
as slope idea as being the definition of a derivative
你可以先把导数看成
instead think of it as being more fundamentally
对输入值微小变化的“精确反映”
about how sensitive the function is to tiny little nudges around the input and
斜率只是用图形分析函数时
the slope is just one way to think about that sensitivity relevant
“精确反映”的一种体现而已
only to this particular way of viewing functions.
这方面的内容
I have not just another video,
我做了整套视频
but a full series on this topic
你感兴趣的话 可以看一下
if it’s something you want to learn more about.
还有一种看待导数的方法
Now the basic idea behind the alternate visual
基本思路是把这个函数看成
for the derivative is to think of this function
输入到数轴上的所有的点
as mapping all of the input points
在另一条输出数轴上对应的映射
on the number line to their corresponding outputs on a different number line.
在这种解释里
In this context what the derivative gives you
导数就是输入数轴的空间
is a measure of how much the input space
在各个区间内拉伸或压缩的程度
gets stretched or squished in various regions.
如果你放大某一段具体的输入值
That is if you were to zoom in around a specific input
观察它周围均匀分布的点
and take a look at some evenly spaced points around it,
那段区域中
the derivative of the function of that input
函数的导数就代表
is going to tell you how spread out or contracted
映射后 这些点变得多密集或多分散
those points become after the mapping.
我举个具体的例子帮你理解
Here a specific example helps
函数f(x)=x²
take the function x squared
1映射到1 2映射到4
it maps 1 to 1 and 2 to 4
3映射到9 依此类推
3 to 9 and so on
中间区域的各个点也是这样的
and you could also see how it acts on all of the points in between
如果你放大观察数字1周围的点
and if you were to zoom in on a little cluster of points around the input 1
在输出轴上找到它们的映射
and then see where they land around the relevant output
这个函数里 1的输出值恰好也是1
which for this function also happens to be 1
你会发现 输出值趋向于被分散
you’d notice that they tend to get stretched out.
事实上
In fact,
看上去被拉伸了2倍
it roughly looks like stretching out by a factor of 2
你在输入轴放大的程度越大
and the closer you zoom in
映射出的输出 就越接近输入值乘2
the more this local behavior Looks just like multiplying by a factor of 2.
这就是函数x²在x=1时
This is what it means for the derivative of x squared
导数为2的含义
at the input x equals 1 to be 2.
这就是用“变换的视角”解释出的道理
It’s what that fact looks like in the context of transformations.
如果你观察输入值3附近的点
If you looked at a neighborhood of points around the input 3,
你会发现它们大约被拉伸了6倍
they would get roughly stretched out by a factor of 6.
意思就是 该函数在x=3时导数为6
This is what it means for the derivative of this function at the input 3 to equal 6.
在输入值¼附近
Around the input 1/4 a small
这片区域趋向于被压缩
a small region actually tends to get contracted,
具体来说 是压缩成原来的½
specifically by a factor of 1/2
导数小于1时 效果都是类似这样的压缩
and that’s what it looks like for a derivative to be smaller than 1.
当输入为0时就有意思了
Now the input 0 is interesting,
我们放大10倍来看
zooming in by a factor of 10
能看出它不是以常数倍拉伸或压缩的
It doesn’t really look like a constant stretching or squishing,
所有的输出都在正数区域
for one thing all of the outputs end up on the right positive side of things
如果你逐渐放大 到100倍或1000倍来观察数轴
and as you zoom in closer and closer by 100x or by 1000 X
在0附近有一群点
It looks more and more like a small neighborhood of points around zero
看起来像要掉进0点一样
just gets collapsed into zero itself.
这就是导数为0时的直观表现
And this is what it looks like for the derivative to be zero,
局部区域表现得
the local behavior looks
趋向于整个数轴乘以0的样子
more and more like multiplying the whole number line by zero.
在特定缩放比例下
It doesn’t have to completely collapse everything to a point
附近点并不会真的缩到一点
at a particular zoom level.
在0的邻域上无限放大时
Instead it’s a matter of what the limiting behavior is
你会发现点与点的间距越来越小
as you zoom in closer and closer.
观察下x为负的情况来增进理解吧
It’s also instructive to take a look at the negative inputs here.
视频里的点有些拥挤
Things start to feel a little cramped
毕竟(-x)²等于x²
since they collide with where all the positive input values go,
这里运用“变换的视角”确实有些不便
and this is one of the downsides of thinking of functions as transformations,
但是对导数来说
but for derivatives,
我们只需关心局部的变化
we only really care about the local behavior anyway.
即给定输入点附近的映射会如何变化
what happens in a small range around a given input.
比如-2附近的一小片区域
Here, notice that the inputs in a little neighborhood around say negative two.
它们不仅拉伸了 还发生了正负反转
They don’t just get stretched out – they also get flipped around.
具体来说 随着放大 这段的映射变化
Specifically, the action on such a neighborhood
越来越像给这段区域乘了-4的结果
looks more and more like multiplying by negative four the closer you zoom in
这就是函数的导数是负数时的现象
this is what it looks like for the derivative of a function to be negative
我认为你已经领悟到了要点
and I think you get the point.
这很好
This is all well and good,
那么在解题时应该怎么用呢
but let’s see how this is actually useful in solving a problem a
最近一位朋友问了我一个有趣的问题
Friend of mine recently asked me a pretty fun
有关连分数
question about the infinite fraction one plus one divided
1+(1÷(1+(1÷(1+(1÷(..……))))))
by one plus one divided by one plus one divided by one and on and on…
会在网上看数学视频的你 也许见过这道题
Clearly you watch math videos online So maybe you’ve seen this before
但这个朋友的问题
but my friend’s question actually cuts to something
你之前可能从没想过
that you might not have thought about before
这个问题有关我们刚刚说得导数视角
Relevant to the view of derivatives that we’re looking at here
你可能会用典型的答题方法
the typical way that you might evaluate an expression like this
把结果设为X
is to set it equal to X
然后就得出了 分数线下方的值等于整个分数本身
and then notice that there’s a copy of the full fraction inside itself
所以就可以用x代替整个分母
So you can replace that copy with another X
就求出了x的值
and then just solve for X
我们的答案就是
That is what you want is to find a fixed point
函数f(x)=1+1/x的不动点
of the function 1 plus 1 divided by X
但是问题在于
But here’s the thing
1+1/x=x有两个解
there are actually two solutions for X two special numbers
有两个特殊数值
were one plus one divided by that number
对x=1+1/x来说都成立
Gives you back the same thing
一个答案是黄金分割率Φ 约等于1.618
One is the golden ratio phi Φ around 1.618
另一个答案是-1/Φ 约等于-0.618
and the other is negative 0.618 which happens to be -1/φ.
我喜欢叫这个值“φ的小弟”
I like to call this other number phi’s little brother
因为Φ有的性质 它基本都有
since just about any property that phi has, this number also has
那么问题来了
And this raises the question:
Φ的小弟-0.618
Would it be valid to say that that infinite fraction that we saw,
是不是刚才无限叠加的连分数题的答案?
is somehow also equal to phi’s little brother: -0.618?
也许你一开始会说 当然不是了!
Maybe you initially say, obviously not!
等式的左边都是正数
Everything on the left hand side is positive.
结果怎么可能得出负数呢?
So how could it possibly equal a negative number?”
好吧 我们首先要清楚
Well first we should be clear about
这个的表达式的真正含义
what we actually mean by an expression like this.
有一种思路是
One way that you could think about it,
可选思路很多 你不必拘泥哪一种
and it’s not the only way there’s freedom for choice here,
你可以选择先代入一个常数 比如1
is to imagine starting with some constant like 1
再把上个式子的结果代到1+1/x中
and then repeatedly applying the function 1+1/x
不断代入 最终计算结果接近哪个值
then asking what is this approach as you keep going?
单从式子来看
I mean certainly symbolically
这个数字是越来越接近无限分数的
what you get looks more and more like our infinite fraction
如果你想找到收敛值
so maybe if you wanted to equal a number
可以思考下 这个数列最终会逼近哪个值
you should ask what this series of numbers approaches
顺着这个思路
And if that’s your view of things,
想象如果开始代入的是负数 结果会怎样
maybe you start off with a negative number
所以最后结果也有可能是负数的
So it’s not so crazy for the whole expression to end up negative.
毕竟 如果你一开始把-1/Φ
After all If you start with -1/φ
代入到函数f(x)=1+1/x中
then applying this function 1+1/x
你还会得到原来的-1/Φ
You get back the same number -1/φ.
无论你代入多少次
So no matter how many times you apply it
得出的永远是同样的结果-0.618
you’re staying fixed at this value.
但是 尽管如此
But even then
标准答案是黄金分割率Φ 还是有理由的
there is one reason that you should probably view phi as the favorite brother in this pair,
这么试一下:
here try this:
随便拿一个科学计算器 随便从一个数开始
pull up a calculator of some kind then start with any random number
代入函数f(x)=1+1/x中
and then plug it into this function 1+1/x
然后把得数再代入1+1/x
and then plug that number into 1 + 1/x
然后一直重复这个过程
and then again and again and again…
无论你输入哪个初始值 最终结果总是1.618
No matter what constant you start with you eventually end up at 1.618
即使你开始输入的是负数
Even if you start with a negative number
即使是非常很接近Φ的小弟-0.618
even one that’s really really close to phi’s little brother
最终你得到的结果 总会回到Φ这个值
Eventually it shys away from that value and jumps back over to phi
这到底是怎么回事?
So what’s going on here?
同样是不动点 为什么结果只能是Φ?
Why is one of these fixed points favored above the other one?
也许你已经感觉出来了
Maybe you can already see
如果用变换的视角看待导数
how the transformational understanding of derivatives
对理解这个问题有所帮助
is going to be helpful for understanding this set up,
但是为了形成对比的效果
but for the sake of having a point of contrast,
我给你展示下这种问题
I want to show you how a problem like this
用图形法来做是什么样
is often taught using graphs.
输入一个随机值x
If you were to plug in some random input to this function,
对应的输出结果就是y 对吧?
the y-value tells you the corresponding output, right?
接着 把结果y再代入函数中
So to think about plugging that output back into the function,
你要先沿x轴方向移动 找到y=x的交点
you might first move horizontally until you hit the line y equals x
这样就得到
and that’s going to give you a position where the x-value
x对应的那个y值 对吧?
corresponds to your previous y-value, right?
然后再沿y轴方向移动 就得到此时x对应输出的y值
So then from there you can move vertically to see what output this new x-value has
然后重复之前的过程
And then you repeat
沿x轴方向移动到y=x的线上 找到此时y对应的x
you move horizontally to the line y = x, to find a point whose x-value
把这一结果作为x值 再沿y轴方向移动
is the same as the output that you just got and then you move vertically
重新计算函数结果
to apply the function again.
我个人认为 像这样反复代入函数
Now personally, I think this is kind of an awkward way to think about repeatedly
实在太麻烦了 你说呢?
applying a function, don’t you?
虽然符合函数图形原理
I mean it makes sense,
但是你不能每算一步
but you can’t have to pause and think
都停下来想下一步往哪画
about it to remember which way to draw the lines,
你真想这么做也行
and you can if you want
找到最终不动点
think through what conditions make this spiderweb process
收敛在蜘蛛网线的哪里 或者发散到哪里
narrow in on a fixed point versus propagating away from it
实际上 不妨把这当做练习画一下
And in fact, go ahead pause right now
然后思考一下 被斜率影响的收敛性
and try to think it through as an exercise. It has to do with slopes
或者你想要跳过这个
Or if you want to skip the exercise for something that
换成我觉得更好的理解方式
I think gives a much more satisfying understanding
把这个函数看成是映射的变换
think about how this function acts as a transformation.
我要在这画很多箭头
So I’m gon na go ahead and start here by drawing a whole bunch of arrows
每个箭头都从输入值 指向输出值
to indicate where the various sample the input points will go,
插句题外话
and side note:
画出来的图案真是美妙绝伦 是不是?
Don’t you think this gives a really neat emergent pattern?
我也很惊讶这个效果
I wasn’t expecting this,
它突然出现的一刻真让人惊艳
but it was cool to see it pop up when animating.
我想1/x画出来的应该是完美的圆形
I guess the action of 1 divided by x gives this nice emergent circle
我只是向右平移了1而已
and then we’re just shifting things over by 1.
综上 我想让你思考下
Anyway, I want you to think about
无限重复地代入某些函数的意义是什么
what it means to repeatedly apply some function
就像1+1/x意味着什么
like 1 + 1/x in this context.
当画出所有输入输出的映射关系
Well after letting it map all of the inputs to the outputs,
这张图可以这样理解
you could consider those as the new inputs
这样做其实就是同一个步骤的首尾相接
and then just apply the same process again
不断重复 随便多少次
and then again and do it however many times you want
注意动态过程中 被标出的那些点
Notice in animating this with a few dots representing the sample points,
这些点只经历几次重复 就聚集到1.618附近
it doesn’t take many iterations at all before all of those dots kind of clump in around 1.618.
复习一下
Now remember,
刚才我们学到了1.618 和它小弟-0.618
we know that 1.618… and its little brother -0.618…
在反复代入过程中 都会越来越靠近某个位置
on and on stay fixed in place during each iteration of this process,
但是在Φ附近放大
but zoom in on a neighborhood around phi
在映射过程中 Φ附近的点 会不断聚集到Φ上
during the map points in that region get contracted around phi
这意味着 在函数f(x)=1+1/x中
meaning that the function 1 + 1/x
这一点的导数绝对值小于1
has a derivative with a magnitude that’s less than 1 at this input in
这个导数实际算出来大约是-0.38
Fact this derivative works out to be around -0.38.
这就意味着 每次代入函数时
So what that means, is that each repeated application
Φ附近的点像受到吸引一样
scrunches the neighborhood around this number smaller and smaller
越来越趋近Φ点
like a gravitational pull towards phi.
那么你可以猜一下
So now tell me what you think
Φ的小弟附近的点会如何变化
happens in the neighborhood of phi’s little brother.
Φ的小弟附近的点 导数的绝对值大于1
Over there the derivative actually has a magnitude larger than one,
所以该定点附近的点被推开了
so points near the fixed point are repelled away from it
如果进行计算
and when you work it out, you can
你就能发现它们在每一次迭代中 被拉伸了不止两倍
see that they get stretched by more than a factor of two in each iteration.
而且因为导数为负 这些点也是正负翻转的
They also get flipped around because the derivative is negative here,
但是决定映射是否稳定的 还是导数的绝对值
but the salient fact for the sake of stability is just the magnitude.
数学家称右边的1.618为“稳定不动点”
Mathematicians would call this right value a stable
左边的-0.618为“不稳定不动点”
fixed point and the left one is an unstable fixed point
“稳定”是指 当受到轻微扰动
Something is considered stable if when you perturb it just a little bit,
它倾向于回到原处 而不是远离原处
it tends to come back towards where it started rather than going away from it.
这是个有用的知识点
So what we’re seeing is a very useful little fact:
不动点的稳定性
that the stability of a fixed point
由此时导数绝对值是否大于1决定
is determined by whether or not the magnitude of its derivative is bigger or smaller than one
这就解释了
And this explains why phi
为什么用计算器算无限分数式时
always shows up in the numerical play
结果中总是会出现Φ
where you’re just hitting enter on your calculator over and over
但它的小弟不会出现如此的结果
but phi’s little brother never does.
那么Φ的小弟是不是无限分数式的答案呢
Now as to whether or not you want to consider phi’s little brother a valid value of the infinite fraction
其实这个要看你了
Well, that’s really up to you.
我们刚刚演示过
Everything we just showed suggests that
如果你认为这个分数式是在求极限
if you think of this expression as representing a limiting process
正因为除了Φ的小弟之外
then because every possible seed value other
所有值最终都会回到Φ
than phi’s little brother gives you a series converting to φ
那么这两个值的确不是同一个
It does feel kind of silly to put them on equal footing with each other.
但如果你不觉得这个式子是在求极限
But maybe you don’t think of it as a limit
可能只把它当成是对代数式的计算
Maybe the kind of math you’re doing lends itself to treating this as a purely algebraic object
像多项式的零点一样 结果可以是多个取值
like the solutions of a polynomial, which simply has multiple values.
今天不说多项式
Anyway, that’s beside the point
我的意思不是说 要用点的密度变化表示导数
and my point here is not that viewing derivatives as this change in density
比用直观图形表达函数更好
is somehow better than the graphical intuition on the whole.
事实上 和图形法相比
In fact picturing an entire function this way
用变换法描述函数有时非常麻烦
can be kind of clunky and impractical as compared to graphs.
我的重点在于
My point is that it deserves more of a mention
微积分入门课应该加入这个内容
in most of the introductory calculus courses,
因为它能使学生对导数的理解更加灵活
because it can help make a student’s understanding of the derivative a little bit more flexible.
像我之前提到的
Like I mentioned the real reason that I’d recommend
这个方法不止限于一元微积分
you carry this perspective with you as you learn new topics
你学习后续知识时
is not so much for what it does with your understanding of single variable calculus
也能帮到你
it’s for what comes after
大学数学还有很多关于微积分的高等知识
after there are many topics typically taught in a college math department which…
我该……怎么解释呢?
How shall I put this lightly?
想理解透彻可不容易
don’t exactly have a reputation of being super accessible.
下一个视频中 我将向你展示
So in the next video I’m gonna show you how a
高等微积分中的一些有趣的问题
few ideas from these subjects with fancy sounding
比如“正则函数”和“雅克比行列式”
names like holomorphic functions and the Jacobian determinant
都只是本次内容的延伸而已
are really just extensions of the idea shown here.
这些概念都很优美
They really are some beautiful ideas,
无论你有多深厚的数学基础
which I think can be appreciated
你都会爱上它们
from a really wide range of mathematical backgrounds
而且它们可以和看似无关的概念产生联系
and they’re relevant to a surprising number of seemingly unrelated ideas.
所以做好准备吧
So stay tuned for that.
视频的最后
Now for the final animation
我想向你分享之前做的 动态向量场的动画
I just want to show you a little more of that time-dependent vector field I flashed earlier,
但是首先让我们看看从本视频赞助商
but first let’s look at some of the principles of learning
Brilliant.org那里学到的一些学习准则
from this video sponsor: Brilliant.org
列表中有很多好的建议
There’s a lot of good stuff on this list,
但我想让你看看第二条
but I want you to look at number two
“数学和科学的持续学习可以培养好奇心”
effective math and science learning cultivates curiosity.
我喜欢这里的措辞
I love the word choice here.
你不能虎头蛇尾
It’s not just that you should be curious in one moment
而是要持之以恒的培养好奇心
It means creating a context where that curiosity is constantly growing.
回头再看看那个无限嵌套分数的例子
Just look at the infinite fraction example here
好奇为什么结果趋向于一个值
It would be one thing if you were curious about
固然很好
why the numbers bounce around the way that they do,
但我希望的是 你不仅仅学会这一个例子
but hopefully the conclusion is not just to understand this one example
我希望你可以多找一些这样的式子
I would want you to start looking at all sorts of other infinite expressions
试着求解当中的不动点
and wonder if there’s some fixed point phenomenon in them,
或者想想这个导数视角
or wonder where else this view of derivatives
能不能用于求解其它问题
can be conceptually helpful
brilliant.org是一个通过互动学习
Brilliant.org is a site where you can learn math and science topics
学习数学和科学学科的网站
through active problem-solving
如果你愿意点进去看看
and if you go take a look I think you’ll agree
就会明白这个网站怎样实践它的准则
that they really do adhere to these learning principles,
从这个视频中
coming from this video
你可能就爱上网站里的“微积分应该这样学”课程
you would probably enjoy their”Calculus Done Right,” lessons
网站还提供更多数学和自然学科课程
and they also have many other courses in various math and science topics.
大部分都是免费的
Much of it you can check out for free,
但网站提供了付费订阅
but they also have a subscription service
付费会员拥有专属答疑功能
that gives you access to all sorts of nice guided problems.
登录Brilliant.org/3B1B
Going to Brilliant.org/3B1B
让他们知道你是从我这来的
Lets them know that you came from this channel
你就能以8折价格获得年度订阅
and it can also get you 20 % off of their annual subscription.

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视频概述

导数不仅仅是斜率

听录译者

收集自网络

翻译译者

短尾龙

审核员

审核员 EM

视频来源

https://www.youtube.com/watch?v=CfW845LNObM

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