Kevin Buzzard
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 Nov 15 awarded Popular Question Aug 13 awarded Yearling Apr 3 awarded Popular Question Aug 13 awarded Yearling Jul 2 awarded Curious May 20 comment Proving that sin(x)/x=(1-x^2/pi^2)(1-x^2/4pi^2)(…) @abel: my gut feeling is that any attempt to prove that the RHS satisfies the differential equation will get bogged down in trying to e.g. relate sum 1/n^2 to sum 1/n^4 (when one is checking coefficients of the power series match up) and this is not elementary. May 20 asked Proving that sin(x)/x=(1-x^2/pi^2)(1-x^2/4pi^2)(…) May 15 awarded Nice Question Mar 14 awarded Nice Question Feb 11 awarded Popular Question Dec 12 awarded Popular Question Oct 24 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? Sure -- the issue is not proving the assertion, the issue is deciding whether the argument I give in the original question is a valid proof or not. Aug 13 awarded Yearling Mar 22 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? @Rhys: Consider the equation $x^2+y^2=7$. Would you be happy to say that $2xdx+2ydy=0$? Would you be happy to say that $dy/dx=-x/y$? I am absolutely convinced that on some level that I can make completely rigorous, the assertion $dy/dx=-x/y$ is true -- I could talk about modules of Kaehler differentials in algebraic geometry, for example. However $y$ is not a function of $x$. Hence I've now decided that I don't buy your objection raised in the first comment. I am genuinely confused. Mar 22 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? I have now found two respectable mathematicians in my department, one of whom claims the solution deserves full marks and the other one claims that it does not. My conclusion is that I finally now realise why I struggled with applied mathematics -- unlike pure mathematics at this level it still seems to be the case that whether or not an answer is correct is a matter of opinion! Mar 22 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? @Rhys: I don't believe your last comment. $x$ is defined as the solution to a differential equation which mentions a function $V$ about which we know nothing other than the implicit assertion that it's continuously differentiable. Mar 21 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? What does it mean, on a formal level, for $\dot{x}$ to change sign? I am thinking of a continuous function which one might argue changes sign on the Cantor set (for some definition of "changes sign") and that is a long way from "$t_1,t_2,\ldots,t_k$". Does this worry you, or can you rule it out, or are your methods robust enough not to care? I'm pretty sure the UG will only recently have seen a rigorous definition of Riemann integral at this point in his education ;-) Mar 21 asked Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? Jan 10 accepted Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”? Jan 10 answered Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?