1610912322-b551b095a53deaf3d3fbd1ed05ae9b84 (824701), страница 39
Текст из файла (страница 39)
Find an ideal in thering of germs of continuous functions at a.15. An ideal in a ring is maximal if it is not contained in any larger ideal exceptthe ring itself. The set C[a, b] of functions continuous on a closed interval forms aring under the usual operations of addition and multiplication of numerical-valuedfunctions. Find the maximal ideals of this ring.Chapter 5Differential Calculus5.1 Differentiable Functions5.1.1 Statement of the Problem and Introductory ConsiderationsSuppose, following Newton,1 we wish to solve the Kepler problem2 of two bodies,that is, we wish to explain the law of motion of one celestial body m (a planet) relative to another body M (a star).
We take a Cartesian coordinate system in the planeof motion with origin at M (Fig. 5.1). Then the position of m at time t can be characterized numerically by the coordinates (x(t), y(t)) of the point in that coordinatesystem. We wish to find the functions x(t) and y(t).The motion of m relative to M is governed by Newton’s two famous laws: thegeneral law of motionma = F,(5.1)connecting the force vector with the acceleration vector that it produces via thecoefficient of proportionality m – the inertial mass of the body,3 and the law ofuniversal gravitation, which makes it possible to find the gravitational action of thebodies m and M on each other according to the formulaF=GmMr,|r|3(5.2)1 I.
Newton (1642–1727) – British physicist, astronomer, and mathematician, an outstandingscholar, who stated the basic laws of classical mechanics, discovered the law of universal gravitation, and developed (along with Leibniz) the foundations of differential and integral calculus.He was appreciated even by his contemporaries, who inscribed on his tombstone: “Hic depositumest, quod mortale fuit Isaaci Newtoni” (here lies what was mortal of Isaac Newton).2 J. Kepler (1571–1630) – famous German astronomer who discovered the laws of motion of theplanets (Kepler’s laws).3 We have denoted the mass by the same symbol we used for the body itself, but this will not leadto any confusion.
We remark also that if m $ M, the coordinate system chosen can be consideredinertial.© Springer-Verlag Berlin Heidelberg 2015V.A. Zorich, Mathematical Analysis I, Universitext,DOI 10.1007/978-3-662-48792-1_51711725Differential CalculusFig. 5.1where r is a vector with its initial point in the body to which the force is applied andits terminal point in the other body and |r| is the length of the vector r, that is, thedistance between m and M.Knowing the masses m and M, we can easily use Eq. (5.2) to express the righthand side of Eq.
(5.1) in terms of the coordinates x(t) and y(t) of the body m attime t, and thereby take account of all the data for the given motion.To obtain the relations on x(t) and y(t) contained in Eq. (5.1), we must learnhow to express the left-hand side of Eq. (5.1) in terms of x(t) and y(t).Acceleration is a characteristic of a change in velocity v(t). More precisely, it issimply the rate at which the velocity changes.
Therefore, to solve the problem wemust first of all learn how to compute the velocity v(t) at time t possessed by a bodywhose motion is described by the radius-vector r(t) = (x(t), y(t)).Thus we wish to define and learn how to compute the instantaneous velocity of abody that is implicit in the law of motion (5.1).To measure a thing is to compare it to a standard.
In the present case, what canserve as a standard for determining the instantaneous velocity of motion?The simplest kind of motion is that of a free body moving under inertia. This isa motion under which equal displacements of the body in space (as vectors) occurin equal intervals of time. It is the so-called uniform (rectilinear) motion. If a pointis moving uniformly, and r(0) and r(1) are its radius-vectors relative to an inertialcoordinate system at times t = 0 and t = 1 respectively, then at any time t we shallhaver(t) − r(0) = v · t,(5.3)where v = r(1) − r(0).
Thus the displacement r(t) − r(0) turns out to be a linearfunction of time in this simplest case, where the role of the constant of proportionality between the displacement r(t) − r(0) and the time t is played by the vector v thatis the displacement in unit time. It is this vector that we call the velocity of uniformmotion. The fact that the motion is rectilinear can be seen from the parametric representation of the trajectory: r(t) = r(0) + v · t, which is the equation of a straightline, as you will recall from analytic geometry.We thus know the velocity v of uniform rectilinear motion given by Eq.
(5.3). Bythe law of inertia, if no external forces are acting on a body, it moves uniformly in astraight line. Hence if the action of M on m were to cease at time t, the latter wouldcontinue its motion, in a straight line at a certain velocity from that time on. It isnatural to regard that velocity as the instantaneous velocity of the body at time t.5.1 Differentiable Functions173Fig. 5.2However, such a definition of instantaneous velocity would remain a pure abstraction, giving us no guidance for explicit computation of the quantity, if not forthe circumstance of primary importance that we are about to discuss.While remaining within the circle we have entered (logicians would call it a “vicious” circle) when we wrote down the equation of motion (5.1) and then undertookto determine what is meant by instantaneous velocity and acceleration, we nevertheless remark that, even with the most general ideas about these concepts, one candraw the following heuristic conclusions from Eq.
(5.1). If there is no force, that is,F ≡ 0, then the acceleration is also zero. But if the rate of change a(t) of the velocityv(t) is zero, then the velocity v(t) itself must not vary over time. In that way, wearrive at the law of inertia, according to which the body indeed moves in space witha velocity that is constant in time.From this same Eq. (5.1) we can see that forces of bounded magnitude are capable of creating only accelerations of bounded magnitude.
But if the absolute magnitude of the rate of change of a quantity P (t) over a time interval [0, t] does not exceed some constant c, then, in our picture of the situation, the change |P (t) − P (0)|in the quantity P over time t cannot exceed c · t, that is, in this situation, the quantitychanges by very little in a small interval of time. (In any case, the function P (t) turnsout to be continuous.) Thus, in a real mechanical system the parameters change bysmall amounts over a small time interval.In particular, at all times t close to some time t0 the velocity v(t) of the bodym must be close to the value v(t0 ) that we wish to determine.
But in that case, ina small neighborhood of the time t0 the motion itself must differ by only a smallamount from uniform motion at velocity v(t0 ), and the closer to t0 , the less it differs.If we photographed the trajectory of the body m through a telescope, depending on the power of the telescope, we would see approximately what is shown inFig. 5.2.The portion of the trajectory shown in Fig.
5.2c corresponds to a time interval sosmall that it is difficult to distinguish the actual trajectory from a straight line, sincethis portion of the trajectory really does resemble a straight line, and the motionresembles uniform rectilinear motion. From this observation, as it happens, we canconclude that by solving the problem of determining the instantaneous velocity (velocity being a vector quantity) we will at the same time solve the purely geometricproblem of defining and finding the tangent to a curve (in the present case the curveis the trajectory of motion).Thus we have observed that in this problem we must have v(t) ≈ v(t0 ) for tclose to t0 , that is, v(t) → v(t0 ) as t → t0 , or, what is the same, v(t) = v(t0 ) + o(1)1745Differential Calculusas t → t0 .
Then we must also haver(t) − r(t0 ) ≈ v(t0 ) · (t − t0 )for t close to t0 . More precisely, the value of the displacement r(t) − r(t0 ) is equivalent to v(t0 )(t − t0 ) as t → t0 , orr(t) − r(t0 ) = v(t0 )(t − t0 ) + o v(t0 )(t − t0 ) ,(5.4)where o(v(t0 )(t − t0 )) is a correction vector whose magnitude tends to zero fasterthan the magnitude of the vector v(t0 )(t − t0 ) as t → t0 . Here, naturally, we mustexcept the case when v(t0 ) = 0. So as not to exclude this case from consideration ingeneral, it is useful to observe that4 |v(t0 )(t − t0 )| = |v(t0 )||t − t0 |.
Thus, if |v(t0 )| =0, then the quantity |v(t0 )(t − t0 )| is of the same order as |t − t0 |, and thereforeo(v(t0 )(t − t0 )) = o(t − t0 ). Hence, instead of (5.4) we can write the relationr(t) − r(t0 ) = v(t0 )(t − t0 ) + o(t − t0 ),(5.5)which does not exclude the case v(t0 ) = 0.Thus, starting from the most general, and perhaps vague ideas about velocity, wehave arrived at Eq.
(5.5), which the velocity must satisfy. But the quantity v(t0 ) canbe found unambiguously from Eq. (5.5):v(t0 ) = limt→t0r(t) − r(t0 ).t − t0(5.6)Therefore both the fundamental relation (5.5) and the relation (5.6) equivalent to itcan now be taken as the definition of the quantity v(t0 ), the instantaneous velocityof the body at time t0 .At this point we shall not allow ourselves to be distracted into a detailed discussion of the problem of the limit of a vector-valued function. Instead, we shall confineourselves to reducing it to the case of the limit of a real-valued function, which hasalready been discussed in complete detail.
Since the vector r(t) − r(t0 ) has coor0)0 ) y(t)−y(t0 )dinates (x(t) − x(t0 ), y(t) − y(t0 )), we have r(t)−r(t= ( x(t)−x(t, t−t0 ) andt−t0t−t0hence, if we regard vectors as being close together if their coordinates are closetogether, the limit in (5.6) should be interpreted as follows:r(t) − r(t0 )x(t) − x(t0 )y(t) − y(t0 )v(t0 ) = lim,= lim, limt→t0t→t0t→t0t − t0t − t0t − t0and the term o(t − t0 ) in (5.5) should be interpreted as a vector depending on t such0)that the vector o(t−tt−t0 tends (coordinatewise) to zero as t → t0 .|t − t0 | is the absolute value of the number t − t0 while |v| is the absolute value, or lengthof the vector v.4 Here5.1 Differentiable Functions175Finally, we remark that if v(t0 ) = 0, then the equationr − r(t0 ) = v(t0 ) · (t − t0 )(5.7)defines a line, which by the circumstances indicated above should be regarded asthe tangent to the trajectory at the point (x(t0 ), y(t0 )).Thus, the standard for defining the velocity of a motion is the velocity of uniformrectilinear motion defined by the linear relation (5.7).
The standard motion (5.7) isconnected with the motion being studied as shown by relation (5.5). The value v(t0 )at which (5.5) holds can be found by passing to the limit in (5.6) and is called thevelocity of motion at time t0 . The motions studied in classical mechanics, which aredescribed by the law (5.1), must admit comparison with this standard, that is, theymust admit of the linear approximation indicated in (5.5).If r(t) = (x(t), y(t)) is the radius-vector of a moving point m at time t, thenṙ(t) = (ẋ(t), ẏ(t)) = v(t) is the vector that gives the rate of change of r(t) at time t,and r̈(t) = (ẍ(t), ÿ(t)) = a(t) is the vector that gives the rate of change of v(t)(acceleration) at time t, then Eq. (5.1) can be written in the formm · r̈(t) = F(t),from which we obtain in coordinate form for motion in a gravitational field⎧x(t)⎨ ẍ(t) = −GM [x 2 (t)+y2 (t)]3/2 ,⎩ ÿ(t) = −GMy(t).[x 2 (t)+y 2 (t)]3/2(5.8)This is a precise mathematical expression of our original problem.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
