Incidentally, to a good approximation wehave another law, which says that the change in distance of a moving point isthe velocity times the time interval, or Δs=vΔt . This statement is true only if the velocity is not changing duringthat time interval, and this condition is true only in the limit as Δtgoes to 0 . Physicists like to write it ds=vdt , because by dt they mean Δt in circumstances in which it is very small; with this understanding,the expression is valid to a close approximation. If Δt is too long, the velocity might change during the interval, and theapproximation would become less accurate. For a time dt , approaching zero, ds=vdt precisely. In this notation we can write (8.5)as
顺便说一句，我们有另外一条规律，可以有一个好的近似，该规律说，对于一个移动的点，其距离的变化，是速度乘以时间间隔，或 Δs=vΔt。这一说法，只有当速度在这段时间内不变时，才为真，而不变这个条件，只有当Δt取极限、趋于零时，才为真。物理学家喜欢把它写为ds=vdt，因为通过dt，他们意味着Δt处在一个它很小的情况中；用这种理解，这个表达式，对于近似来说，就是有效的。如果Δt太长，那么，在这段时间内，速度就可能变化，而近似就会变得不准确。当时间dt趋于零时，ds=vdt就是精确的。用这套符号，可以把（8.5）写为：

The quantity ds/dt which we found above is called the “derivative of s with respect to t ” (this language helps to keep track of what was changed), and thecomplicated process of finding it is called finding a derivative, ordifferentiating. The ds ’s and dt ’s which appear separately are called differentials. Tofamiliarize you with the words, we say we found the derivative of thefunction 16t2 , or the derivative (with respect to t ) of 16t2 is 32t . When we get used to the words, the ideas are more easily understood.For practice, let us find the derivative of a more complicated function. We shallconsider the formula s=At3+Bt+C , which might describe the motion of a point. The letters A , B , and C represent constant numbers, as in the familiar general form of a quadraticequation. Starting from the formula for the motion, we wish to find thevelocity at any time. To find the velocity in the more elegant manner, wechange t to t+Δt and note that s is then changed to s+some Δs ; then we find the Δs in terms of Δt . That is to say,
s+Δs=A(t+Δt)3+B(t+Δt)+C=At3+Bt+C+3At2Δt+BΔt+3At(Δt)2+A(Δt)3,
but since
s=At3+Bt+C,
we find that
Δs=3At2Δt+BΔt+3At(Δt)2+A(Δt)3.
But we do not want Δs —we want Δs divided by Δt . We divide the preceding equation by Δt , getting
ΔsΔt=3At2+B+3At(Δt)+A(Δt)2.
As Δt goes toward 0 the limit of Δs/Δt is ds/dt and is equal to
ds/dt=3At2+B.
我们上面发现的量ds/dt，被称为s对t的导数（这个语言有助于我们追踪什么改变了），而发现它的这一复杂过程，被称为寻找一个导数，或微分。分别出现的ds和dt，被称为微分。为了熟悉这些词，我们说，我们找到函数16t2 的导数、或16t2的导数，是32t。当我们习惯了这些词以后，这个想法，就是更容易理解了。为了练习，让我们来找一个更复杂的方程的导数。我们看公式s=At3+Bt+C，它可能描述了一个点的运动。这里字母A、B和C，代表常数，正如在熟悉的二次方程的一般形式中的字母一样。从这个运动的方程开始，我们希望找到任何时间的速度。要以更优美的方式，找到速度，我们把t变成t+Δt，并注意s也就变成了s+某个 Δs；然后，我们发现依据 Δt的Δs。也就是说：
s+Δs=A(t+Δt)3+B(t+Δt)+C=At3+Bt+C+3At2Δt+BΔt+3At(Δt)2+A(Δt)3,
但是，由于
s=At3+Bt+C,
我们得到
Δs=3At2Δt+BΔt+3At(Δt)2+A(Δt)3。
但我们要的不是Δs，而是Δs/Δt。我们让上面的方程除以Δt，就得到：
ΔsΔt=3At2+B+3At(Δt)+A(Δt)2.
由于 Δt 趋于0，Δs/Δt 的极限就是 ds/dt，且等于：
ds/dt=3At2+B.

This is the fundamental process of calculus, differentiatingfunctions. The process is even more simple than it appears. Observe that whenthese expansions contain any term with a square or a cube or any higher power of Δt, such terms may be dropped at once, since they will go to 0 when the limit is taken. After a little practice the process getseasier because one knows what to leave out. There are many rules or formulasfor differentiating various types of functions. These can be memorized, or canbe found in tables. A short list is found in Table 8–3.
这就是微积分的基本过程，求函数的微分。这个过程，甚至比它看上去，还要简单。注意观察，当展开式中，包含着Δt的平方、立方或任何更高次方的项时，则这些项，可立即抛弃，因为，取极限时，它们会变成零。稍经练习，这个过程，就会变得更容易，因为，人们将知道，该省略什么。对不同类型的函数，求微分，有很多规则或公式。这些规则可被记忆，或者可以在表中找到。表8-3，就是一个简短的列表。
表8-3 一个简短的导数表

8–4Distance as an integral 8-4 距离作为积分
Table 8–4Velocity of a Falling Ball 表8-4 一个下落球的速度
t (sec) v (ft/sec)
0 000
1 032
2 064
3 096
4 128
Now we have to discuss the inverse problem.Suppose that instead of a table of distances, we have a table of speeds atdifferent times, starting from zero. For the falling ball, such speeds andtimes are shown in Table 8–4. Asimilar table could be constructed for the velocity of the car, by recordingthe speedometer reading every minute or half-minute. If we know how fast thecar is going at any time, can we determine how far it goes? This problem isjust the inverse of the one solved above; we are given the velocity and askedto find the distance. How can we find the distance if we know the speed? If thespeed of the car is not constant, and the lady goes sixty miles an hour for amoment, then slows down, speeds up, and so on, how can we determine how far shehas gone? That is easy. We use the same idea, and express the distance in termsof infinitesimals. Let us say, “In the first second her speed was such andsuch, and from the formula Δs=vΔt we can calculate how far the car went the first second at that speed.”Now in the next second her speed is nearly the same, but slightly different; wecan calculate how far she went in the next second by taking the new speed timesthe time. We proceed similarly for each second, to the end of the run. We nowhave a number of little distances, and the total distance will be the sum ofall these little pieces. That is, the distance will be the sum of the velocitiestimes the times, or s=∑vΔt , where the Greek letter ∑ (sigma) is used to denote addition. To be more precise, it is the sumof the velocity at a certain time, let us say the i -th time, multiplied by Δt .
现在，我们要讨论相反的问题。假设我们的表，不是距离的，而是从零开始的、不同时间的速度的表。对于一个下降的球，这样一个速度和时间的表，如表8-4所示。也可以为汽车的速度，建类似的表，即通过速度计，读取每一分钟或没半分钟的数。如果我们知道了汽车在任何时间走的有多快，我们能知道它走了多远吗？这个问题，与我们上面讨论的问题，正好相反。我们有速度，要去求距离。如果我们只知道速度，如何才能得到距离呢？如果汽车的速度不是常数，此女士用60英里1小时开了一会儿，然后，减速、加速等等，她究竟走了多远，我们如何得到呢？这很容易。我们用同样的想法，用无限小量，来表现距离。我们说：“在第一秒钟，她的速度，如此如此，根据公式Δs=vΔt，我们可以计算出她的汽车，在第一秒、以此速度，走了多远。”现在，在下一秒，她的速度，基本一样，但稍有不同，我们通过用新的速度，乘以时间，计算出它在这一秒走了多远。对每一秒，都如此处理，直到结束。我们现在有了一系列小的距离，而总的距离，就是这些小段之和。也就是说，距离将是速度乘以时间之和，或 s=∑vΔt，这里希腊字母西格玛∑ (sigma)，被用来说明加。为了更精确，距离就是：某一时间的速度（比如说第i次时间）乘以Δt之和：
(8.6)

The rule for the times is that ti+1=ti+Δt . However, the distance we obtain by this method will not be correct,because the velocity changes during the time interval Δt . If we take the times short enough, the sum is precise, so we takethem smaller and smaller until we obtain the desired accuracy. The true sis
对时间的规则是ti+1=ti+Δt。然而，通过这种方法所得到的距离，并不正确，因为，在时间Δt内，速度会变化。如果我们让时间足够短，那么，这个总和就会变精确，所以，我们让时间的取值，越来越小，直到我们得到所期待的精度。所以真正的s就是： (8.7)

The mathematicians have invented a symbol for this limit, analogous tothe symbol for the differential. The Δ turns into a d to remind us that the time is as small as it can be; the velocity isthen called v at the time t , and the addition is written as a sum with a great “s ,” ∫ (from the Latin summa), which has become distorted and is nowunfortunately just called an integral sign. Thus we write
s=∫v(t)dt. (8.8)
数学家为这种极限，发明了个一个符号，一个类似于积分的符号。Δ变成了d，提醒我们，时间要尽可能地小；速度被称为在时间t时的v，而加法，则被写作一个总和，用一个大写的“s,”、即 ∫ (来自拉丁文 summa)，其意义，已被扭曲，现在，只是很不幸地被称为一个积分符号。这样我们就有：
s=∫v(t)dt. (8.8)

This process of adding all these terms together is called integration,and it is the opposite process to differentiation. The derivative of thisintegral is v , so one operator (d ) undoes the other (∫ ). One can get formulas for integrals by taking the formulas for derivativesand running them backwards, because they are related to each other inversely.Thus one can work out his own table of integrals by differentiating all sortsof functions. For every formula with a differential, we get an integral formulaif we turn it around.
这个过程，把所有的项加在一起，被称为积分，且它是微分的反过程。这个积分的导数就是 v，于是，一个操作符(d ) ，取消且复原另一个(∫ )。你可以把导数的公式拿来，反向运行它们，就可以得到积分的公式，因为它们是反向相关的。这样，你就可以通过所有函数进行微分，得到自己的积分表。对于每个有微分项的公式，我们都可以通过回转，得到其积分公式。

Acceleration is defined as the time rate ofchange of velocity. From the preceding discussion we know enough already towrite the acceleration as the derivative dv/dt , in the same way that the velocity is the derivative of the distance.If we now differentiate the formula v=32t we obtain, for a falling body,
a=dv/dt=32. (8.9)
加速度被定义为：速度的变化，对时间的比率。从前面的讨论，我们足以知道，可把加速度写作导数dv/dt，与速度是距离的导数，是一样的。如果我们对公式v=32t求导，就得到一个下落物体的加速度为：
a=dv/dt=32. (8.9)

For reference, we state two very useful formulas,which can be obtained by integration. If a body starts from rest and moves witha constant acceleration, g , its velocity v at any time t is given by
v=gt.
The distance it covers in the same time is
s=1/2gt2.
备考，我们声明了两个非常有用的公式，它们可以通过积分得到。如果一个物体，从静止开始，以恒定的加速度g运动，那么，它在任何时间t的速度v就是：
v=gt。
在同一时间，它所走的距离就是：
s=1/2gt2。

Various mathematical notations are used in writing derivatives. Sincevelocity is ds/dt and acceleration is the time derivative of the velocity, we can alsowrite

(8.10)
which are common ways of writing a second derivative.
在写导数时，使用了各种不同的数学符号。由于速度是ds/dt，而加速度是速度对时间的导数，我们也可以这样写：

(8.10)
这就是写二次导数的常见方法。

We have another law that the velocity isequal to the integral of the acceleration. This is just the opposite of a=dv/dt; we have already seen that distance is the integral of the velocity,so distance can be found by twice integrating the acceleration.
我们还有另外一条规律，即速度等于对加速度的积分。这正与a=dv/dt相反；我们已经看到，距离是速度的积分，所以，距离就可以通过对加速度的两次积分来得到。

In the foregoing discussion the motion wasin only one dimension, and space permits only a brief discussion of motion inthree dimensions. Consider a particle P which moves in three dimensions in any manner whatsoever. At thebeginning of this chapter, we opened our discussion of the one-dimensional caseof a moving car by observing the distance of the car from its starting point atvarious times. We then discussed velocity in terms of changes of thesedistances with time, and acceleration in terms of changes in velocity. We cantreat three-dimensional motion analogously. It will be simpler to illustratethe motion on a two-dimensional diagram, and then extend the ideas to three dimensions.We establish a pair of axes at right angles to each other, and determine the positionof the particle at any moment by measuring how far it is from each of the two axes.Thus each position is given in terms of an x -distance and a y -distance, and the motion can be described by constructing a table inwhich both these distances are given as functions of time. (Extension of thisprocess to three dimensions requires only another axis, at right angles to thefirst two, and measuring a third distance, the z -distance. The distances are now measured from coordinate planesinstead of lines.) Having constructed a table with x - and y -distances, how can we determine the velocity? We first find the componentsof velocity in each direction. The horizontal part of the velocity, or x-component, is the derivative of the x -distance with respect to the time, or
vx=dx/dt.(8.11)
Similarly, the vertical part of the velocity, or y -component, is
vy=dy/dt.(8.12)
In the third dimension,
vz=dz/dt.(8.13)
在前面的讨论中，运动只是在一个维度，而空间，则允许对运动在三个维度上，做出一种简洁的讨论。考虑一个粒子P，在三维中，以任意方式运动。在本章的开始，我们是这样开始讨论的：一辆正在移动的汽车，是一维的，我们观察汽车在不同的时间、距起点的距离。然后，我们用距离对时间的变化，来讨论速度，用速度对时间的变化，来讨论加速度。对于三维运动，可类似处理。在一个二维图表上，例示这种运动，相对比较简单，然后，我们可以把这些观念，扩展到三维。我们建立了一对坐标轴，它们相互正交，然后，通过测量任一瞬间，粒子距每个轴有多远，来决定粒子的位置。这样，每个位置，就是用x距离和y距离，来给出。而运动，可通过建一个表，来描述，在此表中，这些距离，都是作为时间的函数，而被给予。（把此过程，扩展到三维，只需要另外一个轴，它与前两个正交，测量第三个距离：z距离。此距离，现在是从坐标平面开始测量，而不是从线开始）。构造了一个有x距离和y距离的表之后，我们如何规定速度呢？我们首先找出每个方向上的速度分量。速度的水平方向的部分、或x分量，是x距离，对时间的导数，或者：
vx=dx/dt.(8.11)
类似地，速度的垂直方向的部分，或者y分量，就是：
vy=dy/dt.(8.12)
在第三个维度，就是：
vz=dz/dt.(8.13)

Fig. 8–3.Description of the motion of abody in two dimensions and the computation of its velocity. 图8-3 在二维中一个物体的运动的描述，及其速度的计算。
Now, given the components of velocity, how can we find the velocityalong the actual path of motion? In the two-dimensional case, consider twosuccessive positions of the particle, separated by a short distance Δsand a short time interval t2−t1=Δt. In the time Δt the particle moves horizontally a distance Δx≈vxΔt, and vertically a distance Δy≈vyΔt . (The symbol “≈ ” is read “is approximately.”) The actual distance moved isapproximately
Δs≈(Δx)2+(Δy)2−−−−−−−−−−−−√,(8.14)
as shown in Fig. 8–3. Theapproximate velocity during this interval can be obtained by dividing by Δtand by letting Δt go to 0 , as at the beginning of the chapter. We then get the velocity as
v=ds/dt=(dx/dt)2+(dy/dt)2−−−−−−−−−−−−−−−−√=v2x+v2y−−−−−−√.(8.15)
For three dimensions the result is
v=v2x+v2y+v2z−−−−−−−−−−√.(8.16)
现在，有了速度的分量，我们如何才能得到：实际运动路径上的速度呢？考虑在二维的情况下，考虑粒子的两个相继的位置，由一个短的距离Δs和一个短的时间间隔t2−t1=Δt分开，在时间Δt内，粒子水平移动的距离为Δx≈vxΔt，垂直移动的距离为Δy≈vyΔt。（这里 “≈ ”读作“约等于”）实际移动的距离就约等于：
（8.14）

如图8-3所示。在这段间隔内的近似速度，可以通过除以Δt并让Δt趋于零而得到。就如在本章的开始那样。因此我们就得到速度为：
（8.15）
三维结果则是：
（8.16）

In the same way as we defined velocities, we can define accelerations:we have an x -component of acceleration ax , which is the derivative of vx , the x -component of the velocity (that is, ax=d2x/dt2, the second derivative of x with respect to t ), and so on.
我们可以用定义速度的方式，来定义加速度：我们有一个加速度的x分量ax，它是vx的导数，vx是速度的x分量（也就是说，ax=d2x/dt2，x对时间的二级导数），如此等等。