作者anoymouse (没有昵称)
看板Math
标题[微积] continuous derivative 问题
时间Thu Mar 21 16:37:52 2024
Thomas' calculus 关於积分弧长，有先做一个前提:
Suppose the curve whose length we want to find is the graph of the function
y = f(x) from x= a to x = b.
In order to derive an integral formula for the length of the curve, we assume
that ƒ has a continuous derivative at every point of [a, b]. Such a function
is called smooth,and its graph is a smooth curve because it does not have any breaks, corners,
or cusps.
最後说到:
(1 + [f'(x)]^2)^1/2 is continuous on [a,b]
想问f has a continuous derivative ... of [a,b]的意思是
(1) f'(x) is continuous on [a,b] 还是
(2) f在[a,b]每一点都有derivative
如果是(2) 举一个example
https://imgur.com/Wv2wDLl
g(x)每一点都有derivative 但是 g'(0)在x = 0 并不连续。
这样的话 "(1 + [f'(x)]^2)^1/2 is continuous " 就不一定会成立吧?
所以原文的意思是应该是(1)?
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1F：推 cuteSquirrel: 您po的原文叙述有强调是平滑曲线，排除断点 折角 03/21 18:46
2F：→ cuteSquirrel: "not have any breaks, corners or cusps." 03/21 18:47
3F：→ anoymouse : 所以你认为是（2）？ 03/21 20:33
4F：→ anoymouse : 我的举例g of x虽然靠近0会剧烈跳动 但应该每个点都 03/21 20:33
5F：→ anoymouse : 可以微分且g'在0不连续吧 这样不符合圆滑曲线吗 03/21 20:33
6F：推 cuylerLin : 这是英文问题不是数学问题，a continuous derivativ 03/21 23:06
7F：→ cuylerLin : e 的意思是 a derivative that is continuous 也就 03/21 23:06
8F：→ cuylerLin : 是导数是一个连续函数。 03/21 23:06
9F：推 arrenwu : 就 (1) 03/21 23:12
10F：→ cmrafsts : 我还以为是要问在端点微分是什麽意思XD 03/22 04:19
11F：→ anoymouse : 好的 谢谢各位 03/22 08:39
12F：推 alan23273850: 当然是 1，语意蛮明确的 03/22 11:05