3 ms·
Consider a disconnected domain (say, union of a few open balls in R^n), and f being constant in each connected component, but having different values in each ba
by flebron 5y ago
Consider a disconnected domain (say, union of a few open balls in R^n), and f being constant in each connected component, but having different values in each ball. The differential is indeed everywhere 0 in the entire domain.
- deleted 5y ago[deleted]
- dataflow 5y agoD'oh, I missed the possibility of an open domain. Thanks!
- bmitc 5y agoYea, the statement is missing "locally constant". It's a poor example given the assumptions are not clearly stated.
- pfortuny 5y agoOh, these kind of problems happen time and again in real-life maths: you use a lemma until someone points out that you are assuming something which may not take place (like connectedness of the domain, here).
- bmitc 5y agoYea, but like I said, it's a poor example as a "false belief" in this context, because as stated, it isn't false for some unwritten assumptions and is false for other unwritten assumptions. In mathematics, well and actually everything, the importance of assumptions is paramount. The problem is just ill-stated. This happens in engineering and software all the time. Time and time again, issues being resolved usually revolves around clarifying assumptions.
- sannee 5y agoAnd, even better, the dimension of the kernel of d just counts how many connected components the domain has. There is a similar thing in graph theory, where the kernel of the incidence matrix counts how many connected components a graph has.