We digress at this point to note that insuch cases as this one may get the wrong sign for some quantity if thecoordinates are not handled in the right way. Why not write τyz=zFy−yFz? The problem arises from the fact that a coordinate system may beeither “right-handed” or “left-handed.” Having chosen (arbitrarily) a sign for,say τxy , then the correct expressions for the other two quantities may alwaysbe found by interchanging the letters xyz in either order
在这里，我们先岔开话题，去注意，如果坐标系没有掌握好，那么，在类似这种的情况下，对于某些量，可能会得到错误的符号。为什么不写成τyz=zFy−yFz呢？这个问题，产生于以下事实，即一个坐标系，可以是“右手的”，也可是“左手的”。比如，在为τxy选（任意地）了一个符号之后，那么，关于另外两个量的正确的表达，总是可以通过以下面两种方式之一，交换字母xyz的顺序，来得到：

Moe now calculates the torques in hissystem:
现在Moe，在他的坐标系中，计算扭力：
(20.2)

Now we suppose that one coordinate system is rotated by a fixedangle θ , such that the z - and z′ -axes are the same. (This angle θ has nothing to do with rotating objects or what is going on inside thecoordinate system. It is merely the relationship between the axes used by one manand the axes used by the other, and is supposedly constant.) Thus the coordinatesof the two systems are related by
现在，我们假设一个坐标系，旋转了一个固定的角度θ，且满足z和 z′是同样的（这个角度θ，与旋转对象、或者在这个坐标系中所发生的事情，无关。它只是一个人所用的坐标轴，与另外一个人所用的轴之间的关系，且它被假定为常数）。这样，这两个系统之间的坐标，通过下式相关：
(20.3)

Now we can find out how the torque transforms by merely substitutingfor x′ , y′ , and z′ the expressions (20.3),and for Fx′ , Fy′ , Fz′ those given by (20.4),all into (20.2).So, we have a rather long string of terms for τx′y′and (rather surprisingly at first) it turns out that it comes rightdown to xFy−yFx , which we recognize to be the torque in the xy -plane:
现在，我们可以找出，扭力是如何变换的，只需把表达式（20.3）中的x′ , y′ , 和 z′ ，和（20.4）中的Fx′ , Fy′ , Fz′，带入（20.2），做替换就行。于是，对于τx′y′，我们会有一个相当长的项串（开始会相当惊人），但结果则是，它正好变成xFy−yFx ，这我们认识，正是xy平面的扭力：
(20.5)

That result is clear, for if we only turn our axes in the plane,the twist around z in that plane is no different than it was before, because it is thesame plane! What will be more interesting is the expression for τy′z′, because that is a new plane. We now do exactly the same thing withthe y′z′ -plane, and it comes out as follows:
这个结果，很清楚，因为，如果我们只是在平面中，转动我们的轴，那么，在那个平面上绕着z轴的转动，与之前并无任何区别，因为是同一个平面！对于τy′z′，表达式会更有趣，因为，它是一个新平面。现在，我们对y′z′ 平面，做同样的事情，结果如下：
(20.6)
Finally, we do it for z′x′ :
最后，我们对z′x′ ，做变换：
(20.7)

We wanted to get a rule for finding torques in new axes in terms oftorques in old axes, and now we have the rule. How can we ever remember thatrule? If we look carefully at (20.5), (20.6),and (20.7),we see that there is a close relationship between these equations and theequations for x , y , and z . If, somehow, we could call τxy the z -component of something, let us call it the z -component of τ , then it would be all right; we would understand (20.5)as a vector transformation, since the z -component would be unchanged, as it should be. Likewise, if weassociate with the yz -plane the x -component of our newly invented vector, and with the zx -plane, the y -component, then these transformation expressions would read (20.8)
which is just the rule for vectors!
我们的目的，是想找一条规则，以用旧坐标轴中的扭力，表示新坐标轴中的扭力，现在，我们有了这个规则。我们如何记住这个规则呢？如果我们仔细查看（20.5）、（20.6）、和（20.7），我们将会看到，这些方程，与关于x , y , 和 z的方程，有着非常密切的关系。如果，以某种方式，我们可以把τxy，称为某物的z分量，比如说是τ的z分量，那么，这将是完全正确的；我们将把（20.5），理解为一个矢量变换，由于z分量并未变化，它本应如此。同样，对于我们新发明的矢量，如果我们把其x分量，与yz平面相联，把其y分量，与zx平面相联，那么，这些变换表达式将是：
(20.8)
这正是矢量的规则。

Therefore we have proved that we mayidentify the combination of xFy−yFx with what we ordinarily call the z -component of a certain artificially invented vector. Although atorque is a twist on a plane, and it has no a priori vector character,mathematically it does behave like a vector. This vector is at right angles tothe plane of the twist, and its length is proportional to the strength of thetwist. The three components of such a quantity will transform like a realvector.
因此，我们就有了一个人工发明的矢量；且已经证明，我们可以用该矢量的z分量，来标明组合xFy−yFx 。虽然扭力是一个平面上的扭转{twist}，且它没有天生的矢量特性，但数学上，它的表现，像一个矢量。这个矢量，与扭转的平面，呈直角，且其长度，正比于扭转的强度。这样一个量的三个分量，将会像一个真正的矢量那样变换。

So we represent torques by vectors; witheach plane on which the torque is supposed to be acting, we associate a line atright angles, by a rule. But “at right angles” leaves the sign unspecified. Toget the sign right, we must adopt a rule which will tell us that if the torquewere in a certain sense on the xy -plane, then the axis that we want to associate with it is in the “up”z -direction. That is, somebody has to define “right” and “left” for us.Supposing that the coordinate system is x , y , z in a right-hand system, then the rule will be the following: if wethink of the twist as if we were turning a screw having a right-hand thread,then the direction of the vector that we will associate with that twist is inthe direction that the screw would advance.
于是，我们就通过矢量，表示了扭力；对于一个平面，如果我们认为，扭力是作用于其上的，那么，我们将通过一条规则，让一条线，以直角，与此平面相关。但是，“以直角”，并未说明符号。要得到正确的符号，我们必须采取一条规则，它将告诉我们，如果扭力，在某种意义上，是在xy平面，那么，我们想与它相关的轴，就是在“向上”的z方向。也就是说，有人必须为我们定义“右”和“左”。假设坐标系是x , y , z，是一个右手系，那么，规则将如下：如果我们考虑一个扭转，就好像我们是在拧一个螺丝，它有一个右手的螺线，那么，我们将与这个扭转相关联的矢量方向，就是那个螺丝的前进方向。

Why is torque a vector? It is a miracle ofgood luck that we can associate a single axis with a plane, and therefore thatwe can associate a vector with the torque; it is a special property ofthree-dimensional space. In two dimensions, the torque is an ordinary scalar,and there need be no direction associated with it. In three dimensions, it is avector. If we had four dimensions, we would be in great difficulty, because (ifwe had time, for example, as the fourth dimension) we would not only haveplanes like xy , yz , and zx , we would also have tx -, ty -, and tz -planes. There would be six of them, and one cannot representsix quantities as one vector in four dimensions.
为什么扭力是一个矢量？我们能把一个轴，与一个平面相关，这是一个奇迹，运气好，因此，我们就可以把一个矢量，与一个扭力相关，它是三维空间中的一个特殊属性。在两维空间中，扭力是一个普通的标量，不需要让它，与一个方向相关。在三维中，它是一个矢量。如果我们有四维，那么，我们将会有很大困难，因为（例如，如果我们把时间作为呢？第四维），我们不仅要有xy , yz , 和zx这样的平面，我们还将有tx , ty ,和tz这样的平面。这样的平面，将有六个，在四维中，无法用一个量，来表示六个量。

We will be living in three dimensions for along time, so it is well to notice that the foregoing mathematical treatmentdid not depend upon the fact that x was position and F was force; it only depended on the transformation laws for vectors.Therefore if, instead of x , we used the x -component of some other vector, it is not going to make anydifference. In other words, if we were to calculate axby−aybx, where a and b are vectors, and call it the z -component of some new quantity c , then these new quantities form a vector c . We need a mathematical notation for the relationship of the newvector, with its three components, to the vectors a and b . The notation that has been devised for this is c=a×b. We have then, in addition to the ordinary scalar product in thetheory of vector analysis, a new kind of product, called the vector product.Thus, if c=a×b , this is the same as writing
(20.9)
If we reverse the order of a and b , calling a , b and b , a , we would have the sign of c reversed, because cz would be bxay−byax. Therefore the cross product is unlike ordinary multiplication, whereab=ba ; for the cross product, b×a=−a×b. From this, we can prove at once that if a=b, the cross product is zero. Thus, a×a=0 .
我们将在三维中，生活很长时间，所以，应该注意，前面的数学处理，并不依赖于事实：x是位置、F是力；它只依赖于矢量的变换规律。因此，如果我们不是使用x，而是使用的某个其他矢量的x分量，那么，将不会有任何区别。换句话说，如果我们要计算axby−aybx，这里a和b是矢量，那么，我们将把结果，称为某个新量c的z分量，这些新量，将构成矢量c。这个新矢量及其三个分量，与矢量a和b，有一种关系，我们需要一种数学表示法，来表示之。此表示法，已被发明，它就是c=a×b。因此，除了在矢量分析理论中，我们所谈到的标量积外，我们还有一种新的积，被称为矢量积。这样，如果c=a×b，那么，它与下式同：
(20.9)
如果我们改变 a和b的顺序, 把a变成 b，把b 变成 a , 那么，我们将会反转c的符号, 因为cz将会是bxay−byax。因此，叉积与普通乘法不一样，那里 ab=ba ; 而对于叉积, b×a=−a×b。由此，我们可以立即证明，如果 a=b ，那么，叉积为零。这样, a×a=0。

In order to complete the mathematicalproperties of vectors, we should know all the rules for their multiplication,using dot and cross products. In our applications at the moment, we will needvery little of this, but for the sake of completeness we shall write down allof the rules for vector multiplication so that we can use the results later.These are
为了使得矢量的数学特性，变得完整，对于其乘法，无论是点积还是叉积，所有规则，我们都应知道。在当前的应用中，我们所需很少，但是，为了完整，我们将把所有矢量乘法的规则，都写出来，这样，以后，我们就可以使用这个结果了。
(20.10)

20–2The rotation equations using cross products20-2 使用叉积的旋转方程
Now let us ask whether any equations inphysics can be written using the cross product. The answer, of course, is thata great many equations can be so written. For instance, we see immediately thatthe torque is equal to the position vector cross the force:
现在，我们要问，物理学中的任何方程，是否都可以用叉积来写。当然，答案就是，很多方程，都可以这样写。例如，我们立即看到，扭力就等于，位置矢量，叉积力：
τ=r×F. (20.11)
This is a vector summary of the three equations τx=yFz−zFy, etc. By the same token, the angular momentum vector, if there isonly one particle present, is the distance from the origin multiplied by thevector momentum:
这是τx=yFz−zFy等三个方程的一个矢量总结。通过同样的符号，如果只有一个粒子在场的话，角动量矢量，就是到原点的距离，乘以矢量动量：
L=r×p. (20.12)
For three-dimensional space rotation, the dynamical law analogous tothe law F=dp/dt of Newton, is that the torque vector is the rate of change with timeof the angular momentum vector:
对于三维空间中的旋转，有一个动力学规律，可类比于牛顿规律中的 F=dp/dt，它就是：扭力矢量，是角动量矢量，随着时间的变化率：
τ=dL/dt. (20.13)
If we sum (20.13)over many particles, the external torque on a system is the rate of change ofthe total angular momentum:
如果我们用（20.13）对很多粒子用13求和，那么，作用于一个系统上的外部的扭力，就是总角动量的变化率：
τext=dLtot/dt. (20.14)

Another theorem: If the total external torque is zero, then the totalvector angular momentum of the system is a constant. This is called the law of conservationof angular momentum. If there is no torque on a given system, its angularmomentum cannot change.
另外一个定理：对于一个系统，如果整个外部扭力为零，那么，其总的矢量角动量，就是一个常数。这被称为角动量守恒规律。给定一个系统，如果其上，没有扭力，那么，其角动量，不能改变。

666

What about angular velocity? Is it a vector?We have already discussed turning a solid object about a fixed axis, but for amoment suppose that we are turning it simultaneously about two axes. Itmight be turning about an axis inside a box, while the box is turning aboutsome other axis. The net result of such combined motions is that the objectsimply turns about some new axis! The wonderful thing about this new axis isthat it can be figured out this way. If the rate of turning in the xy -plane is written as a vector in the z -direction whose length is equal to the rate of rotation in the plane,and if another vector is drawn in the y -direction, say, which is the rate of rotation in the zx -plane, then if we add these together as a vector, the magnitude ofthe result tells us how fast the object is turning, and the direction tells usin what plane, by the rule of the parallelogram. That is to say, simply,angular velocity is a vector, where we draw the magnitudes of therotations in the three planes as projections at right angles to those planes.1
角速度如何？它是一个矢量吗？一个固体对象，绕一个固定轴旋转，我们已经讨论过了，但等一下，假设我们同时绕两个轴旋转。比如，在一个盒子里，绕一个轴转，而盒子，绕着另一个轴转。这种组合运动的净结果，就是对象绕着某个新轴转。关于这个新轴，有个奇妙的事情，可以下面这种方式，搞清楚。如果在xy平面的转动速率，被写成一个z方向的矢量，其长度，等于在那个平面内的旋转速率，如果另外一个矢量，沿着y方向画，说它是zx平面的旋转速率，那么，如果我们把这两个量，通过平行四边形的规则{？}，加在一起，作为一个结果矢量，那么，此结果的大小，就会告诉我们，这个对象旋转的有多快，而其方向，则会告诉我们，它是在哪个平面内。简单地说，角速度是一个矢量；把它在三个平面内，做直角投影，则旋转的大小，就是投影的大小。