Hey everyone, Grant here.This is the first video in a series on the essence of calculus.And I'll be publishing the following videos once per day for the next 10 days.The goal here, as the name suggests, is to really get the heart of the subject out in one binge watchable set.
大家好，我是格兰特。这是《微积分的本质》系列视频的第一讲。从今天往后的十天内，我将会每天发布一个新的视频。而本课程的学习目标，就是从长的系列课程中学到微积分的本质。
But with a topic that's as broad as calculus.There's a lot of things that can mean. So, here's what I've in my mind specifically.
正如微积分一样，该话题涵盖了很多的内容，所以有很多的知识去理解学习。下面就是我脑海中的具体内容。
Calculus has a lot of rules and formulas which are often presented as
things to be memorised. Lots of derivative formulas, the product rule, the chain rule, implicit differentiation, the fact that integrals and derivatives are opposite, Taylor series;
在微积分中有很多的规定和公式需要去记忆，如很多的求导公式：乘积法则、链式法则、隐函数的求导、积分和导数是逆运算、泰勒级数等。
Just a lot of things like that. And my goal is for you to come away feeling like you could have invented calculus yourself.
很多知识需要去死记硬背。而我的目标是当你完成学习时就像是你自己创造了微积分。
That is, cover all those core ideas, but in a way that makes clear where they actually come from and what they really mean using an all-around visual approach.
这是如何做到的呢？是以清晰而且用可视化的方式深刻的阐释了微积分的原理。
Inventing math is no joke, and there is a difference between being told why something's true and actually generating it from scratch. But at all points I want you to think to yourself if you were an early mathematician, pondering these ideas and drawing out the right diagrams, does it feel reasonable that you could have stumbled across these truths yourself?
创造数学并不是开玩笑，理解一件事是正确的和理解它的起源是两码事。而我的想法是说，如果你是一个早期的数学家，思考一些点子并且画出正确的图形，偶然发现这些真相，这是合理的嘛？
In this initial video, I want to show how you might stumble into the core
ideas of calculus by thinking very deeply about one specific bit of
geometry: the area of a circle. Maybe you know that this is pi times its radius squared, but why? Is there a nice way to think about where this formula comes from?
在第一个视频中，我想通过思考一个特定几何图形（圆）来思索微积分的核心概念。也许你知道圆的面积是π*r^2，但这是为什么呢？是否存在一个漂亮的方式来推导出它的结果？
Well, contemplating this problem and leaving yourself open to exploring the interesting thoughts that come about can actually lead you to a glimpse of three big ideas in calculus: integrals, derivatives, and the fact that they're opposites. But the story starts more simply—just you and a circle; let's say with radius three.
思考该问题并且发散自己的思维，实际上可以一瞥微积分的三个核心概念：积分、微分和两者可逆。但这个故事很简单，即从你和一个圆开始，假设圆的半径为3。
You're trying to figure out its area, and after going through a lot of paper trying different ways to chop up and rearrange the pieces of that area, many of which might lead to their own interesting observations, maybe you try out the idea of slicing up the circle into many concentric rings.
你试图求出它的面积，在大量的纸张切割并且重新排列碎片，有很多方式可以构建出有趣的结果，其中有一种方式就是把圆分割成不同大小的同心环。
This should seem promising because it respects the symmetry of the circle, and math has a tendency to reward you when you respect its symmetries. Let's take one of those rings which has some inner radius R that's between 0 and 3.
这种方式看起来很有希望，因为它保留了圆的对称性，在数学学科中，如果保留了对称性，数学往往会给你意想不到的奖励。我们从半径从0到3（左闭右闭）的圆环从拿出其中一个来分析。
If we can find a nice expression for the area of each ring like this one, and if we have a nice way to add them all up, it might lead us to an understanding of the full circle's area.
如果你对其中一个能够用表达式来表示的话，那我们就可以把他们加进来，从而得到圆的面积。
Maybe you start by imagining straightening out this ring. And you could try thinking through exactly what this new shape is and what its area should be, but for simplicity let's just approximate it as a rectangle.The width of that rectangle is the circumference of the original ring, which is two pi times R.
也许你开始想象拉直这个圆环。你开始思考它的形状究竟是什么，如何去求解它的面积，我们近似它是个矩形。这个矩形的宽是圆环的周长，也就是2πR。
