1, imagine two identical and synchronized turkeys going towards each other. Where they meet gives the answer.
2, for uniform temperature, obviously true. If not uniform, forming "properly" two closed curves passing through the point of the lowest temperature and the point of the highest temperature will do. "Properly" here means to keep the temperature along the curve either statically climbing or dropping, that is, never to go up and down the "hills" or "valleys" except for the two points mentioned. Which can be done because of 单值连续 .
3, apparently one can apply 2. for showing the existence of such a molecule ( I still need to think it through), but, imagine the extreme case of two molecules which just switch positions at boiling... which seems to contradict the supposed answer. Perhaps I miss something here.
Assuming the water body behave both as a continuum (therefore 单值连续 can apply ) and as a discreet set with each unit (fixed position in space) small enough to hold a water molecule.
Applying result from 2, there exist a position whose temperature history is identical to the temperature history of some particular molecule. Once can show this being true just by switching coordinates.
synchronized... This would could have different man's. E.g same speed, or they started at the same time from their starting points... The first, but the second, one is not necessary.
1。有个爱好登山的火鸡。从某山脚下A点上山到达山顶B点。再从B点沿原路返回到山脚A 点。火鸡在A点出发时用秒表开始计时。同样地，返回时，火鸡从B点开始计时。试证明，在这条路上肯定存在一点C, 火鸡在上山和下山时，到达这一点时所用的时间一样。
2。假定在任意一时刻，全球的温度在空间上的分布是单值连续的。那么在任何时刻，地球表面上都存在至少两条环绕地球的闭合曲线带，在该时刻这两条曲线上的温度是彼此一一对应的。
3。将一锅冷水烧成沸腾的开水，试证明，沸腾的开水锅中至少有一个“处惊不变”的水分子，其所处的位置与冷水时所处的位置一样。
关键是连续可微，因而路径上一定有一点，其瞬时速度等于路径的平均速度，这个对上下山都适用，另外上下山路径又是相同的，如果再假定所费时间也相同，那么上山时该点和下山时该点必须是同一点。
1, imagine two identical and synchronized turkeys going towards each other. Where they meet gives the answer.
2, for uniform temperature, obviously true. If not uniform, forming "properly" two closed curves passing through the point of the lowest temperature and the point of the highest temperature will do. "Properly" here means to keep the temperature along the curve either statically climbing or dropping, that is, never to go up and down the "hills" or "valleys" except for the two points mentioned. Which can be done because of 单值连续 .
3, apparently one can apply 2. for showing the existence of such a molecule ( I still need to think it through), but, imagine the extreme case of two molecules which just switch positions at boiling... which seems to contradict the supposed answer. Perhaps I miss something here.
Assuming the water body behave both as a continuum (therefore 单值连续 can apply ) and as a discreet set with each unit (fixed position in space) small enough to hold a water molecule.
Applying result from 2, there exist a position whose temperature history is identical to the temperature history of some particular molecule. Once can show this being true just by switching coordinates.
This particular molecule is the desired one.
上山下山时间不需要一样。当然不能停留在出发点不出发。
synchronized... This would could have different man's. E.g same speed, or they started at the same time from their starting points... The first, but the second, one is not necessary.
plot x 位置，y温度成一图，看不出来。它要运动。
f(A)= 0, g(B) = 0
f(B) > 0, g(A) > 0
Therefore h(A) < 0 and h(B) > 0
by Intermediate value theorem, there exists a point C between A and B such that h(C) = 0.
Brouwer's fixed point theorem --> there is an x0 such that g(x0) = x0, which implies that g(x0) is f(x0).
or, one can imagine the two trips of the same turkey being superimposed.
The solution I gave for the second problem I think is the general solution though there might be a specific one that is easier.
Still unsure of the third one.
如果明确最后结论 f(C)=g(C) at C 就更完美了。
上山与下山速度不同也没关系。也不需要匀速运动，走走歇歇都行。
Brouwer Theorem 其中寓意着很有意义的一个生活道理。值得了解一下。比如揉面，面团里肯定有一点，怎么揉都是固定不变的。你把一张包括您家所在地的一张地图摊开在你的餐桌上查地图的时候，地图上肯定有一点和你餐桌所在的位置正好重回。投资股票也有用，不过说清楚需要付费啦：））
可以考你孩子，老婆，或女朋友啊：））
我覺得現實中對一鍋水也是不成立的，除非對水分子的運動方式予以限制，使之接近連續體。如不能"跳''或"擠"到某個非鄰近位置，等等。
揉麵團在現實中估计也如此。
但地圖的例子是成立的，因為是連續體。
布劳威尔是个异类，认为数学不是对客观世界的描述和反应，而是只存在于人的头脑中（意识中）的产物。所以证明数学定理必须把它给构造出来，构造不出来就不算数。
他是荷兰人，和荷兰量子力学学派的哲学观点不谋而合。他和希尔伯特观点对立，当时受到打击。可能今后其思想会越来越证明是对的。
当然用化学观点看是另一回事，但各学科各有其道，不能混为一谈。。。
但得的結果是可以違反題意的。
在物理界，这些都是连续介质。。。
可作思考實驗用，實際生活中是不成立的。
My guess.
水分子之间空隙相對於分子非常大。
麵團的例子也不太可能成立。
把大圆在任意一点切开拉直作为横轴，看纵轴温度曲线。。。（两个端点温度？）
因全球温度"连续单值''，故每个圓 C 上必有一個點 p ，其 d =0.
連所有 p 得一閉合曲线 Q.
取分别從左右兩面任意鄰近每個 p 的两个温度相等的點 p1 和 p2 。
連所有 P1 得一閉合曲线 Q1 。連所有 p 2，得 Q2。
Q1 , Q2 即满足要求。