Wikipedia:Reference desk/Archives/Mathematics/2012 October 11

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October 11

what is the full Bessel function solution for the cylindrical wave equation?

Everywhere I look I get the large R approximation solution, but I don't get what solution they are approximating from. The equation is (v^2/r) * d/dr [r dE/dr] = d^2E / dt^2, where d signifies a partial derivative and E is the value of the wavefunction. 128.143.1.242 (talk) 11:27, 11 October 2012 (UTC)

Wave equation:

$\nabla ^{2}E={\frac {1}{v^{2}}}{\frac {\partial ^{2}E}{\partial t^{2}}}$

in cylindrical coords the Laplacian operator is:

$\nabla ^{2}E={\frac {1}{r}}{\frac {\partial }{\partial r}}({\frac {\partial E}{\partial r}})+$theta and z derivatives...

From your question I think you only want solutions where $E=E(r,t)$, so we have:

${\frac {v^{2}}{r}}{\frac {\partial }{\partial r}}(r{\frac {\partial E}{\partial r}})={\frac {\partial ^{2}E}{\partial t^{2}}}$

Following Boas, Mathematical Methods in the Physical Sciences (2nd Ed, Ch13.5), we separate the variables assuming that $E(r,t)=R(r)T(t)$. Subs in and divide by RT:

${\frac {v^{2}}{rR}}{\frac {d}{dr}}(r{\frac {dR}{dr}})={\frac {1}{T}}{\frac {d^{2}T}{dt^{2}}}$

We argue that both sides of the equation must be constant and equal to -k^2:

${\frac {1}{T}}{\frac {d^{2}T}{dt^{2}}}=-k^{2}$ and ${\frac {v^{2}}{rR}}{\frac {d}{dr}}(r{\frac {dR}{dr}})=-k^{2}$.

Then for T(t) we have:

$T(t)=Ae^{-ikt}+Be^{ikt}$.

For R(r) we have:

${\frac {d}{dr}}(r{\frac {dR}{dr}})=-k^{2}Rr/v^{2}$,

expand and multiply by r:

$r^{2}{\frac {d^{2}R}{dr^{2}}}+r{\frac {dR}{dr}}+{\frac {k^{2}}{v^{2}}}r^{2}R=0$,

which is Bessel's equation with p=0, so a solution is

$R(r)=CJ_{0}({\frac {k}{v}}r)$.

I'm not sure about the N_0(...) solution, what do you think? 77.86.104.103 (talk) 09:25, 13 October 2012 (UTC)

What are your boundary conditions? It's your boundary conditions which tell us the values k can take. Also, are you sure we are valid in assuming that E = E(r,t) rather than the most general $E=E(r,\theta ,z,t)$ for three-dimensions?77.86.104.103 (talk) 11:12, 13 October 2012 (UTC)

calculus partial derivatives problem [y^−3/2arctan(x/y)] *urgent* help me..?

mathematics is not my strong suit..i tried the problem from a couple of different angles..i am not getting the correct answer

here is the problem:

f(x,y) = y^−3/2arctan(x/y)...find f_x(x,y) and f_y(x,y) [as in derivatives with respect to x and with respect to y].

here is what i have tried doing so far..i used the product rule obviously. we know that derivative of (1/a)arctan(x/a) gives 1/(a²+x²)

i took y^−3/2 as 'u' and arctan(x/y) as 'v' for the implementation of the product rule. so am i getting y/(y²+x²) as the derivative (since (1/y) is missing from the arctan term)?

for f_x(x,y) i get the answer y^−1/2/(y²+x²)..this answer matches with the book's answer (ch-13.3 prob no.25- calculus 9th ed by anton, bivens, davis)

however i am not sure if i got it correct only by chance since i used the same method for f_y(x,y) only to get an incorrect answer..my answer for f_y(x,y) came −(3/2)y^−5/2arctan(x/y) − (xy^−5/2)/(y²+x²)..

the correct answer is −(3/2)y^−3/2arctan(x/y) − (xy^−3/2)/(y²+x²).....[note: xy^−3/2...not xy^−5/2]

f_y(x,y) = −(3/2)(y^−5/2)arctan(x/y) + (y^−3/2)(y/(y²+x²))(−x/(y²))...which gives:−(3/2)y^−5/2arctan(x/y) − (xy^−5/2)/(y²+x²)..so what went wrong there..?

been stuck for hours its quite frustrating....so can anyone please show me the workings with the steps so that i know where i am getting it wrong?

thanks in advance! :) Krunchychicken (talk) 16:13, 11 October 2012 (UTC)

I don't see where you get (y^−3/2)(y/(y²+x²))(−x/(y²)). Shouldn't that be (y^−3/2)(y²/(y²+x²))(−x/(y²))? 86.176.213.216 (talk) 17:27, 11 October 2012 (UTC)

${\frac {d}{dw}}\arctan w={\frac {1}{1+w^{2}}}$

So ${\frac {d}{dy}}\arctan({\frac {x}{y}})={\frac {1}{1+({\frac {x}{y}})^{2}}}{\frac {d}{dy}}({\frac {x}{y}})={\frac {1}{1+({\frac {x}{y}})^{2}}}({\frac {-x}{y^{2}}})={\frac {-x}{y^{2}+x^{2}}}$

Simply apply the standard formulas and the chain rule  :${\frac {d}{dy}}(f(g(y))=f'(g(y))\cdot g'(y)$, don't get caught up in substitutions when you're not good at this stuff. In this case f is arctan() and g is x/y (with y being the variable, and x just a constant) .

I'm not sure how you used "(1/a)arctan(x/a) gives 1/(a2+x2)", but you cannot use this for calculating the derivative to y! Constants cannot be substituted by variables or the other way around. Such formulas are only correct when a really is a constant. The partial derivative in x was correct, because in that case y is a constant (definition of partial dervative). Ssscienccce (talk) 20:08, 11 October 2012 (UTC)

See here [1] and here [2]. Bo Jacoby (talk) 00:31, 12 October 2012 (UTC).

on non-euclidean geometry

Please can anyone name a book which contains non-euclidean geometry free from application of differential geometry with a historical note of the works of lobachevsky and janos bolyai about euclid's parallel postulate and having theoreoms on spherical,elliptic and hyperbolic triangles.

• I know the names of following books but are not sure which are understandable to a high school student.
• Geometry revisited -Coxeter
• Geometry: Euclid and Beyond - Robin Hartshorne
• Plane Trigonometry - S. L. Loney — Preceding unsigned comment added bySolomon7968 (talk • contribs) 19:23, 11 October 2012 (UTC)

There may not exist a book that covers all the aspects you mention and is "free from application of differential geometry".—Tamfang (talk) 20:35, 11 October 2012 (UTC)

I taught a course from the textbook "Modern geometries" by Michael Henle (http://www.amazon.com/Modern-Geometries-Non-Euclidean-Projective-Discrete/dp/0130323136). This course was aimed primarily at undergraduate majors in mathematics education at a major US university, and it was suitable for that task in my opinion. The textbook covers the essential aspects of non-euclidean geometry rather nicely. It contained a modest amount of history which could easily be supplemented by other sources (such as Kline's "Mathematical thought", another book that I heartily recommend). Most of the results in Henle are essentially copied from Ahlfors' classic work "Complex analysis", which I would also recommend for a more advanced audience, although it is much less leisurely and contains no discussion of the history of the subject (and, in addition, is about "analysis" rather than "geometry"). Sławomir Biały (talk) 23:54, 11 October 2012 (UTC)