Therefore, in that special coordinate system, we have proved that a⋅b is the length of a times the length of b times cosθ . But if it is true in one coordinate system, it is true in all,because a⋅b is independent of the coordinate system; that is our argument.
因此，在这个特殊的坐标系中，我们已经证明了，a⋅b就是a的长度，乘以b的长度乘以cosθ。但是，如果它在一个坐标系中为真，那么它在所有坐标系中就都为真，因为a⋅b是独立于坐标系的；这就是我们的论证。

What good is the dot product? Are there anycases in physics where we need it? Yes, we need it all the time. For instance,in Chapter 4 the kinetic energy was called (1/2)mv2, but if the object is moving in space it should be the velocitysquared in the x -direction, the y -direction, and the z -direction, and so the formula for kinetic energy according to vectoranalysis is
K.E.=(1/2)m(v⋅v)=(1/2)m(v2x+v2y+v2z). (11.22)
Energy does not have direction. Momentum has direction; it is avector, and it is the mass times the velocity vector.
点积有什么用处呢？在物理学中，我们有需要它的情况吗？是的，我们一直需要它。例如，在第4章中，动能被称为(1/2)mv2，但是，如果对象在空间中移动，那么，它就应该是x方向、y方向、和z方向的速度的平方，于是，根据矢量分析，动能的公式就是：
K.E.=(1/2)m(v⋅v)=(1/2)m(v2x+v2y+v2z). (11.22)
能量没有方向。动量有方向；它是一个矢量，它是质量乘以矢量速度。

Another example of a dot product is thework done by a force when something is pushed from one place to the other. Wehave not yet defined work, but it is equivalent to the energy change, theweights lifted, when a force F acts through a distance s :
Work=F⋅s. (11.23)
点积的另一个例子，就是功，它由力所做，把某物被从一个地方推到另一个地方。我们还没有定义功，但是，它与能量变化、重量被提升，是相当的，它是一个力F，作用一段距离s：
Work=F⋅s. (11.23)

It is sometimes very convenient to talk about the component of avector in a certain direction (say the vertical direction because that is thedirection of gravity). For such purposes, it is useful to invent what we call aunit vector in the direction that we want to study. By a unit vector wemean one whose dot product with itself is equal to unity. Let us call this unitvector i ; then i⋅i=1 . Then, if we want the component of some vector in the directionof i , we see that the dot product a⋅i will be acosθ , i.e., the component of a in the direction of i . This is a nice way to get the component; in fact, it permits us toget all the components and to write a rather amusing formula. Supposethat in a given system of coordinates, x , y , and z , we invent three vectors: i , a unit vector in the direction x ; j , a unit vector in the direction y ; and k , a unit vector in the direction z . Note first that i⋅i=1 . What is i⋅j ? When two vectors are at right angles, their dot product is zero.Thus
i⋅ i =1
i⋅j =0 j⋅j=1
i⋅k=0 j⋅k=0 k⋅k=1 (11.24)
Now with these definitions, any vector whatsoever can be written thisway:
a=axi+ayj+azk. (11.25)
By this means we can go from the components of a vector to the vectoritself.
有时，谈论矢量在某一方向的分量，会很方便（比如垂直方向，因为它是重力的方向）。为了这种目的，在我们想研究的方向上，发明我们称为单位矢量的东西，是很有用的。通过单位矢量，我们的意思是，该矢量与自己的点积，等于一个单位。假设这个单位矢量是 i ; 那么i⋅i=1。因此，如果我们想得到某一矢量在 i方向的分量，那么我们看到，点积a⋅i将是 acosθ，亦即，a 在i方向上的分量。这是一种得到分量的好方法；事实上，它允许我们得到所有的分量，且可以写出一个相当有趣的公式。假设在一个被给予的坐标系x、y和z中，我们发明了三个矢量：i，j，k，它们分别是x、y、z方向的单位矢量。首先注意 i⋅i=1 。那么 i⋅j是什么呢?当两个矢量垂直时，其点积为零。这样：
i⋅ i =1
i⋅j =0 j⋅j=1
i⋅k=0 j⋅k=0 k⋅k=1 (11.24)
现在，用这些定义，任何矢量都可写为：
a=axi+ayj+azk. (11.25)
通过这种方式，我们就可从一个矢量的分量，得到矢量本身。

This discussion of vectors is by no meanscomplete. However, rather than try to go more deeply into the subject now, weshall first learn to use in physical situations some of the ideas so fardiscussed. Then, when we have properly mastered this basic material, we shallfind it easier to penetrate more deeply into the subject without getting tooconfused. We shall later find that it is useful to define another kind of productof two vectors, called the vector product, and written as a×b. However, we shall undertake a discussion of such matters in a laterchapter.
1. In type, vectors are represented by boldface; in handwritten form anarrow is used: r⃗ .
这个关于向量的讨论，绝不完整，然而，与其现在进一步深入这个话题，不如我们先学习在一些物理情况中，使用迄今为止所讨论过的一些想法。因此，当我们大概确实掌握了这个基本的材料之后，我们将发现，深入这个主题，将更容易，且不会陷入混淆。稍后，我们将发现，定义两个矢量的另外一个积，即矢量积a×b，非常有用。然而，我们将在稍后的一章中，讨论这个事情。
脚注1、在排版中，矢量通过粗体、手写的形式来表示，且用到了一个箭头：r⃗。

Chapter12. Characteristicsof Force第12章 力的特点
12–1What is a force? 12-1 什么是力？
Although it is interesting and worthwhileto study the physical laws simply because they help us to understand and to usenature, one ought to stop every once in a while and think, “What do they reallymean?” The meaning of any statement is a subject that has interested andtroubled philosophers from time immemorial, and the meaning of physical laws iseven more interesting, because it is generally believed that these lawsrepresent some kind of real knowledge. The meaning of knowledge is a deep problemin philosophy, and it is always important to ask, “What does it mean?”
物理规律，能够帮助我们理解、及利用自然，虽然学习它们，有趣且值得，但是，人们时不时地，还是会停下来思考：“它们的意义，究竟为何“？自古以来，任何陈述的意义，都让哲学家们感兴趣，并困扰着他们，而物理规律的意义，则更有趣，因为，一般认为，这些规律，代表着某种真正的知识。在哲学中，知识的意义，是一个很深的问题，所以，提问：“它有什么意义？”总是很重要的。

Chapter12. Characteristicsof Force第12章 力的特点
12–1What is a force? 12-1 什么是力？
Although it is interesting and worthwhileto study the physical laws simply because they help us to understand and to usenature, one ought to stop every once in a while and think, “What do they reallymean?” The meaning of any statement is a subject that has interested andtroubled philosophers from time immemorial, and the meaning of physical laws iseven more interesting, because it is generally believed that these lawsrepresent some kind of real knowledge. The meaning of knowledge is a deep problemin philosophy, and it is always important to ask, “What does it mean?”
物理规律，能够帮助我们理解、及利用自然，虽然学习它们，有趣且值得，但是，人们时不时地，还是会停下来思考：“它们的意义，究竟为何“？自古以来，任何陈述的意义，都让哲学家们感兴趣，并困扰着他们，而物理规律的意义，则更有趣，因为，一般认为，这些规律，代表着某种真正的知识。在哲学中，知识的意义，是一个很深的问题，所以，提问：“它有什么意义？”，总是很重要。
Let us ask, “What is the meaning of thephysical laws of Newton, which we write as F=ma ? What is the meaning of force, mass, and acceleration?”我们问：“牛顿规律，我们写作F=ma，它有什么意义呢？力、质量、和加速度，又有什么意义呢？” Well, we can intuitively sense the meaning of mass, and we can defineacceleration if we know the meaning of position and time. We shall not discussthose meanings, but shall concentrate on the new concept of force. Theanswer is equally simple: “If a body is accelerating, then there is a force onit.” That is what Newton’s laws say, so the most precise and beautifuldefinition of force imaginable might simply be to say that force is the mass ofan object times the acceleration. 当然，我们可以直观地感觉到质量的意义，且如果我们知道位置和时间的意义的话，那么，我们可以定义加速度。我们不讨论这些的意义，将集中在力这一新概念上。答案是同样地简单：“如果一个物体正在加速，那么有力作用于其上。”这就是牛顿规律要讲的，所以，关于力，可以想象出的、最精确和最漂亮的定义，或许就是：力是一个对象的质量，乘以加速度。Suppose we have a law which says that the conservation of momentumis valid if the sum of all the external forces is zero; then the questionarises, “What does it mean, that the sum of all the external forces iszero?”假设我们有一条规律说，如果所有外部的力的总和为零，那么动量守恒就是有效的；因此，问题就是：“所有外部的力的总和为零，这句话意义为何？” A pleasant way to define that statement would be: “When the totalmomentum is a constant, then the sum of the external forces is zero.” 定义这个陈述，有一种愉快的方式，就是：“当总的动量为一个常数时，外部力的总和就是零。”There must be something wrong with that, because it is just notsaying anything new. 这里肯定出了什么问题，因为并没有讲出什么新的东西。If we have discovered a fundamental law, which asserts that the forceis equal to the mass times the acceleration, and then define the forceto be the mass times the acceleration, we have found out nothing. 如果我们发现了一条基本规律，它断言力等于质量乘以加速度，然后，定义力为质量乘以加速度，那么，我们实际上什么也没发现。We could also define force to mean that a moving object with noforce acting on it continues to move with constant velocity in a straight line.我们也可以把力定义为：一个正在运动的对象，没有力作用于其上的话，将继续做匀速直线运动。If wethen observe an object not moving in a straight line with a constant velocity,we might say that there is a force on it. 因此，如果我们观察到一个对象，在做匀速直线运动，我们就可以说，没有力作用于其上。Now such things certainly cannot be the content of physics, becausethey are definitions going in a circle. 现在，这种事情，肯定不能是物理学的内容，因为属于循环定义。The Newtonian statement above, however, seems to be a most precisedefinition of force, and one that appeals to the mathematician; nevertheless,it is completely useless, because no prediction whatsoever can be made from adefinition. 上面牛顿流的陈述，似乎是力的最精确的定义，且是对数学家有吸引力的一个；尽管如此，它是完全无用的，因为，从一个定义，不能做出任何预告。One might sit in an armchair all day long and define words at will,but to find out what happens when two balls push against each other, or when aweight is hung on a spring, is another matter altogether, because the way thebodies behave is something completely outside any choice of definitions.一个人可以整天坐在扶手椅上，随意定义词语，但是，当两个球被推撞在一起，或者当一个重量被挂在弹簧上时，要找出究竟发生了什么，则完全是另一回事，因为各种定义加起来，有一个范围，而物体表现的方式，可能完全超出了这个范围。

For example, if we were to choose to say thatan object left to itself keeps its position and does not move, then when we seesomething drifting, we could say that must be due to a “gorce”—a gorce is therate of change of position. Now we have a wonderful new law, everything standsstill except when a gorce is acting. You see, that would be analogous to the abovedefinition of force, and it would contain no information. The real content ofNewton’s laws is this: that the force is supposed to have some independentproperties, in addition to the law F=ma ; but the specific independent properties that the force haswere not completely described by Newton or by anybody else, and therefore thephysical law F=ma is an incomplete law. It implies that if we study the mass times theacceleration and call the product the force, i.e., if we study the characteristicsof force as a program of interest, then we shall find that forces have somesimplicity; the law is a good program for analyzing nature, it is a suggestionthat the forces will be simple.
例如，如果我们要说，一个对象，被放在那里，原地不动，那么，当我们看到某物在漂移时，我们可以说，这肯定要归于“格斯”，格斯就是位置的变化率。现在，我们有了一个精彩的新规律，所有的东西，都静止着，除了当格斯开始起作用。你看，它可类比于上面的关于力的定义，且它不包含任何信息。牛顿规律的真正内容，是这个：除了规律 F=ma外，力被认为，还有一些独立的属性；但是，力所拥有的这些独立的属性，究竟具体有哪些，牛顿或任何其他人，都没有说明，所以，物理规律F=ma，是一个不完整的规律。它暗示了，如果我们研究‘质量乘以加速度’，并称其乘积为力，亦即…，如果我们把力的特性，作为一种感兴趣的程序来研究，那么我们将发现，力有一些朴素特性；此规律对于分析自然来说，是一个好的程序，它提议说：力将是简单的。

Now the first example of such forces wasthe complete law of gravitation, which was given by Newton, and in stating thelaw he answered the question, “What is the force?” If there were nothing butgravitation, then the combination of this law and the force law (second law ofmotion) would be a complete theory, but there is much more than gravitation,and we want to use Newton’s laws in many different situations. Therefore inorder to proceed we have to tell something about the properties of force.
现在，这种力的第一个例子，就是万有引力的完整规律，它是牛顿提出的，在陈述这个规律时，他回答了问题“什么是力？”如果只有万有引力，而没有其他的东西，那么，这个规律与力的规律（第二运动规律）的联合，将是一个完整的理论，但是，除了万有引力之外，还有其他很多东西，且我们想在很多不同的情况下，使用牛顿规律。因此，为了继续前行，我们必须讲一些力的属性。

For example, in dealing with force thetacit assumption is always made that the force is equal to zero unless somephysical body is present, that if we find a force that is not equal to zero wealso find something in the neighborhood that is a source of the force. This assumptionis entirely different from the case of the “gorce” that we introduced above.One of the most important characteristics of force is that it has a materialorigin, and this is not just a definition.
例如，在处理力的时候，有一个心照不宣的假定，就是除非有某个物理物体在场，否则，力就等于零，如果我们发现，力不等于零，那么我们也就发现了，附近肯定有某物，它是力的源泉。这个假设，与我们上面所介绍的“格斯”情况，完全不同。力的最重要的特性之一，就是它有一种材料，是起源性的，而这就不仅仅是一个定义了。

Newton also gave one rule about the force:that the forces between interacting bodies are equal and opposite—action equalsreaction; that rule, it turns out, is not exactly true. In fact, the law F=mais not exactly true; if it were a definition we should have to saythat it is always exactly true; but it is not.
关于力，牛顿也曾给出一条规则：两个相互作用的物体之间的力，是相等且相反的，即作用力等于反作用力；结果就是，这条规则，并不完全为真。事实上，规律F=ma并不完全为真；如果它是一个定义，那么我们可以说，它总是完全为真；但是，它并不是一个定义。

The student may object, “I do not like thisimprecision, I should like to have everything defined exactly; in fact, it saysin some books that any science is an exact subject, in which everythingis defined.” If you insist upon a precise definition of force, you will neverget it! First, because Newton’s Second Law is not exact, and second, because inorder to understand physical laws you must understand that they are all somekind of approximation.
学生可能会反对说：“我不喜欢这种不精确性，我喜欢每件事情，都被准确地定义了；事实上，有些书说，任何科学，都是一个准确的课题，在其中，每件事情，都被定义了。”如果你坚持一个力的精确定义，那么你永远也得不到！首先，因为牛顿第二规律，并不够准确，其次因为，为了理解物理，你应该理解：它们都是某种近似。

Any simple idea is approximate; as anillustration, consider an object, … what is an object? Philosophersare always saying, “Well, just take a chair for example.” The moment they saythat, you know that they do not know what they are talking about any more. Whatis a chair? Well, a chair is a certain thing over there … certain?,how certain? The atoms are evaporating from it from time to time—not manyatoms, but a few—dirt falls on it and gets dissolved in the paint; so to definea chair precisely, to say exactly which atoms are chair, and which atoms areair, or which atoms are dirt, or which atoms are paint that belongs to thechair is impossible. So the mass of a chair can be defined only approximately.In the same way, to define the mass of a single object is impossible, becausethere are not any single, left-alone objects in the world—every object is a mixtureof a lot of things, so we can deal with it only as a series of approximationsand idealizations.
任何简单的想法，都是近似的；举个例子，考虑一个对象，…对了，什么是一个对象呢？哲学家总是说：“好，以一个椅子作为例。你知道，他们这样说的时候，就是他们已经不知道自己在说什么了。什么是一个椅子呢？好，一个椅子，就是那里放着的一个确定的事物，，，确定？有多确定？原子从它上面不断地蒸发—当然不是很多，但是有一些--，灰尘不断地落在其上，溶解在油漆里；所以，要精确地定义一个椅子，要精确地说哪个原子是椅子，哪个原子是空气，或哪个原子是灰尘，或哪个原子属于椅子的油漆的，是不可能的。所以，一个椅子的质量，只能大约地定义。同样，要定义一个单独对象的质量，是不可能的，因为，在这个世界上，没有任何单独的、独处的对象—每个对象都是很多事物的混合体，所以，我们只能把对象，当做一系列近似的和理想化了的东西来对待。

The trick is the idealizations. To anexcellent approximation of perhaps one part in 1010 , the number of atoms in the chair does not change in a minute, and ifwe are not too precise we may idealize the chair as a definite thing; in thesame way we shall learn about the characteristics of force, in an ideal fashion,if we are not too precise. One may be dissatisfied with the approximate view ofnature that physics tries to obtain (the attempt is always to increase theaccuracy of the approximation), and may prefer a mathematical definition; butmathematical definitions can never work in the real world. A mathematicaldefinition will be good for mathematics, in which all the logic can be followedout completely, but the physical world is complex, as we have indicated in anumber of examples, such as those of the ocean waves and a glass of wine. Whenwe try to isolate pieces of it, to talk about one mass, the wine and the glass,how can we know which is which, when one dissolves in the other? The forces ona single thing already involve approximation, and if we have a system ofdiscourse about the real world, then that system, at least for the present day,must involve approximations of some kind.
这里的诀窍是理想化。要达到1010之一这种杰出的近似，椅子中的原子数，在一分钟内，并不变化，如果我们不是太精确的话，我们可以把椅子，理想化为一个确定的事物；以同样的方式，如果不是太精确的话，我们将在一种理想的风格中，学习力的特性。对于自然，物理学尝试得到的，是一个近似视图（我们总是努力提高这种近似的准确性），有人对此，可能不满意，更喜欢一个数学的定义；但数学的定义，永远也不能在一个真实的世界中行得通。一个数学的定义，对于数学家来说，可能很好，在其中，所有的逻辑，可以完全跟着走，但是，物理世界，非常复杂，正如我们在一系列例子中所指出，例如，海洋波浪和一杯红酒的复杂性。当我们尝试孤立其中的一部分，来谈论某一个的质量，对于红酒和玻璃杯，当一个溶解到另外一个之中以后，我们如何知道：谁是谁呢？在一个单独的事物上的力，已经包含着近似了，如果我们有一个关于真实世界的演说的系统，那么，这个系统，至少对于今天来说，必然包含着某种近似。{？}

This system is quite unlike the case ofmathematics, in which everything can be defined, and then we do not knowwhat we are talking about. In fact, the glory of mathematics is that we donot have to say what we are talking about. The glory is that the laws, thearguments, and the logic are independent of what “it” is. If we have any otherset of objects that obey the same system of axioms as Euclid’s geometry, thenif we make new definitions and follow them out with correct logic, all theconsequences will be correct, and it makes no difference what the subject was.In nature, however, when we draw a line or establish a line by using a lightbeam and a theodolite, as we do in surveying, are we measuring a line in the senseof Euclid?No, we are making an approximation; the cross hair has some width,but a geometrical line has no width, and so, whether Euclidean geometry can beused for surveying or not is a physical question, not a mathematical question. However,from an experimental standpoint, not a mathematical standpoint, we need to knowwhether the laws of Euclid apply to the kind of geometry that we use in measuringland; so we make a hypothesis that it does, and it works pretty well; but it isnot precise, because our surveying lines are not really geometrical lines. Whetheror not those lines of Euclid, which are really abstract, apply to the lines ofexperience is a question for experience; it is not a question that can beanswered by sheer reason.
这个系统，与数学系统完全不同，在数学中，所有的事情都可被定义，然后，我们并不知道我们在谈论什么。事实上，数学的荣耀就是，我们并不需要说出：我们谈论的具体对象是什么。这种荣耀就是规律、论证、和逻辑，是独立于“它”是什么的。如果我们有任何其他组的对象，它们也遵循着与欧几里德几何相同的公理系统，那么，如果我们给出新的定义，并用正确的逻辑遵循它们，那么，所有的后果，都将是正确的，不论对象是什么，都不会什么区别。然而，在自然中，当我们画一条线，或通过使用光线和经纬仪来建立一条线，如我们在测绘中所做那样，我们是在用欧几里得的意义来测量一条线吗？不，我们是在做一种近似；十字线有宽度，但几何线没有宽度，于是，欧几里得几何是否可被用于测绘，就是一个物理问题，而不是一个数学问题。然而，从一个实验的立场、而不是数学的立场看，我们需要知道，欧几里得规律，是否可应用于我们测量陆地所用的那种几何学；于是，我们假定它可以，且它工作的也很好；但是，它并不精确，因为我们的测绘线，并不真正就是几何线。那些欧几里得线，是抽象的，它们是否可被应用于经验中的线，是一个由经验来回答的问题；而不是一个可由纯粹理性来回答的问题。