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November 29

Coordinate Changes

Could someone check these tangent vector coordinate changes for me, please. It's been a long time since I did any calculations like this, and had to do it all from scratch. I'm changing from a Cartesian basis {∂/∂x,∂/∂y} to a polar basis {∂/∂r,∂/∂θ}, and back again. I got that

{\frac {\partial }{\partial r}}={\frac {x}{\sqrt {x^{2}+y^{2}}}}\,{\frac {\partial }{\partial x}}+{\frac {y}{\sqrt {x^{2}+y^{2}}}}\,{\frac {\partial }{\partial y}},{\text{ and }}\ {\frac {\partial }{\partial \theta }}=x\,{\frac {\partial }{\partial y}}-y\,{\frac {\partial }{\partial x}}.

I believe this one is correct: ∂/∂r is a unit, radial vector and ∂/∂θ is tangent to a circle giving a rotation. I'm not too sure about the next one. It ought to be correct, but it doesn't feel right:

{\frac {\partial }{\partial x}}=\cos \theta \,{\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \theta \,{\frac {\partial }{\partial \theta }}\ {\text{ and }}\ {\frac {\partial }{\partial y}}=\sin \theta \,{\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \theta \,{\frac {\partial }{\partial \theta }}.

Hopefully they'll be okay. If now then could you supply the correct ones, please? — Fly by Night (talk) 15:07, 29 November 2010 (UTC)[reply]

What you have is correct. An easy way to check to write out the equations using matrices, the matrices should be inverses of each other.--RDBury (talk) 23:26, 29 November 2010 (UTC)[reply]

Thanks for the reply. That's how I got them. I used the Jacobian matrix and it's inverse. But I couldn't remember if I had to use the Jacobian, its inverse, its transpose, or its transpose's inverse. TheCoordinate transformation law section of the vector field article seems to tell me to use the transpose. — Fly by Night (talk) 01:18, 30 November 2010 (UTC)[reply]

Irrationality of logarithms with rational bases

Hi. I'm interested in studying irrationality measures, and transcendence, of numbers of the form $\log _{a}(b)={\frac {\ln b}{\ln a}}$ , where both a and b are rational numbers. I'm searching mathscinet and coming up with nothing. I don't know if this means there's nothing published about such numbers (seems unlikely), of if I'm just not searching cleverly enough (much more likely). Can anyone help me locate information about Diophantine approximation (or irrationality measure, or transcendence) of logarithms with rational bases? Thanks in advance. -GTBacchus^(talk) 17:33, 29 November 2010 (UTC)[reply]

I don't know about their irrationality measure, but the Gelfond–Schneider theorem says that for rational (or even algebraic) a and b, log_ab is transcendental whenever it is irrational.—Emil J. 17:39, 29 November 2010 (UTC)[reply]

Linear forms in logarithms may be relevant, too.—Emil J. 17:42, 29 November 2010 (UTC)[reply]

It does seem that in these woods, all roads lead to linear forms in logarithms. Better get studying those. Thanks. -GTBacchus^(talk) 02:15, 30 November 2010 (UTC)[reply]

If log₂ 3 is a rational number n/m where n, m are positive integers, then 2ⁿ = 3^m, so an even number equals an odd number. Similar results entailing non-unique factorization of integers follow from some other assumptions of rationality of such numbers. Michael Hardy (talk) 21:43, 30 November 2010 (UTC)[reply]

Yes, I realize that the number is trivially irrational. I also know that it's transcendental. I'm asking about its irrationality measure, which is an altogether stickier ball of wax. -GTBacchus^(talk) 00:31, 1 December 2010 (UTC)[reply]