From: Cliff Y. <sta...@us...> - 2004-09-25 21:40:13
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Update of /cvsroot/maxima/maximabook/examples In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv2367/examples Added Files: examples.tex Log Message: Import main tex files --- NEW FILE: examples.tex --- %-*-EMaxima-*- \subsection*{Establishing a Minimum for the Rayleigh Quotient} We begin by defining the Rayleigh Quotient in general. From basic Regular Sturm-Liouville Eigenvalue principles, we know that the Rayleigh Quotient is defined as \[\lambda =\frac{-p\phi \left. \frac{d\phi }{dx}\right| ^{b}_{a}+\int _{a}^{b}\, \left[p \left( \frac{d\phi }{dx}\right)^{2}-q\phi ^{2}\right] \, dx}{\int _{a}^{b}\, \phi ^{2}\sigma \, dx}\] given the Sturm-Liouville differential equation \[\frac{d}{dx}\left( p\left( x\right) \frac{d\phi }{dx}\right) +q\left( x\right) \phi +\lambda \sigma \left( x\right) \phi =0\] where $a<x<b$. \beginmaxima RQ:(-p*('ev('ev(u(x)*'diff(u(x),x)),x=a)-'ev('ev(u(x)*diff(u(x),x)),x=b))+ 'integrate(p*'diff(u(x),x)^2-q*u(x)^2,x,a,b))/'integrate(u(x)^2*sigma,x,a,b); \maximatexoutput \[ \frac{\int_{a}^{b}{p\*\left(\frac{d}{d\*x}\*u\left(x\right)\right)^{2}-q\*u^{2 }\left(x\right)\;dx}-p\*\left(\mathrm{EV}\left(\mathrm{EV}\left(u\left(x\right)\*\left(\frac{d}{d\*x}\*u\left(x\right)\right)\right),\linebreak[0]x=a\right)-\mathrm{EV}\left(\mathrm{EV}\left(u\left(x\right)\*\left(\frac{d}{d\*x}\*u\left(x\right)\right)\right),\linebreak[0]x=b\right)\right)}{\sigma\*\int_{a}^{b}{u^{2 }\left(x\right)\;dx}} \] \endmaxima Now we evaluate it. This must be done in stages, otherwise the ev command will not understand its arguements. \beginmaxima ev(RQ,p=1,q=0,sigma=1,u(x)=x-x^2,a=0,b=1); ev(%,diff,integrate); ev(%,ev); \maximatexoutput \[ \frac{\mathrm{EV}\left(\mathrm{EV}\left(\left(x-x^{2}\right)\*\left(\frac{d}{d\*x}\*\left(x-x^{2}\right)\right)\right),\linebreak[0]x=1\right)-\mathrm{EV}\left(\mathrm{EV}\left(\left(x-x^{2}\right)\*\left(\frac{d}{d\*x}\*\left(x-x^{2}\right)\right)\right),\linebreak[0]x=0\right)+\int_{0}^{1}{\left(\frac{d}{d\*x}\*\left(x-x^{2}\right)\right)^{2}\;dx}}{\int_{0}^{1}{\left(x-x^{2}\right)^{2}\;dx}} \] \[ 30\*\left(\mathrm{EV}\left(\mathrm{EV}\left(\left(1-2\*x\right)\*\left(x-x^{2}\right)\right),\linebreak[0]x=1\right)-\mathrm{EV}\left(\mathrm{EV}\left(\left(1-2\*x\right)\*\left(x-x^{2}\right)\right),\linebreak[0]x=0\right)+\frac{1}{3}\right) \] \[ 10 \] \endmaxima This can be checked by hand. Seeing that it is correct, we now can use it to search for the minimum eigenvalue on a more difficult problem: \beginmaxima ev(RQ,p=1,q:-(x^2),sigma=1,u(x)=x-1,a=0,b=1)$ ev(%,diff,integrate)$ EV(%,EV,NUMER); \maximatexoutput \[ 6.1 \] \endmaxima \beginmaxima ev(RQ,p=1,q:-(x^2),sigma=1,u(x)=-2*x^2+2,a=0,b=1)$ ev(%,diff,integrate)$ ev(%,ev,NUMER); \maximatexoutput \[ 2.6428571428571432 \] \endmaxima \beginmaxima ev(RQ,p=1,q:-(x^2),sigma=1,u(x)=x^3+x^2-2,a=0,b=1)$ ev(%,diff,integrate)$ ev(%,ev,NUMER); \maximatexoutput \[ 2.7760840108401084 \] \endmaxima The smallest eigenvalue must therefore be less than or equal to 2.642857... \subsection*{Laplacian in Different Coordinate Systems} This will probably go in the main documentation somewhere, but for now I'll stick it here. It is possible to express the Laplacian in different coordinate systems, provided you know how to define the coordinate system. We will use Spherical Coordinates for our first example: \beginmaxima load(vect)$ scalefactors([[rho*cos(theta)*sin(phi),rho*sin(theta)*sin(phi),rho*cos(phi)],rho,theta,phi]); depends(f,[rho,theta,phi]); express(laplacian(f)); ev(%,diff)$ ratexpand(%); \maximatexoutput \p ; In: LAMBDA (X ANS A3) ; #'(LAMBDA (X ANS A3) NIL (COND # #)) ; Note: Variable A3 defined but never used. ; ; Note: Variable A3 defined but never used. ; ; Note: Variable A3 defined but never used. ; ; Note: Variable A3 defined but never used. \\ \[ \mathrm{DONE} \] \[ \left[ f\left(\rho,\linebreak[0]\vartheta,\linebreak[0]\varphi\right) \right] \] \[ \frac{\frac{d}{d\*\rho}\*\left(\frac{d}{d\*\rho}\*f\*\left| \sin \varphi\right| \*\rho^{2}\right)+\frac{d}{d\*\vartheta}\*\frac{\frac{d}{d\*\vartheta}\*f\*\left| \sin \varphi\right| }{\sin ^{2 }\varphi}+\frac{d}{d\*\varphi}\*\left(\frac{d}{d\*\varphi}\*f\*\left| \sin \varphi\right| \right)}{\left| \sin \varphi\right| \*\rho^{2}} \] \[ \frac{2\*\left(\frac{d}{d\*\rho}\*f\right)}{\rho}+\frac{\frac{d}{d\*\varphi}\*f\*\cos \varphi}{\sin \varphi\*\rho^{2}}+\frac{\frac{d^{2}}{d\*\vartheta^{2}}\*f}{\sin ^{2 }\varphi\*\rho^{2}}+\frac{\frac{d^{2}}{d\*\varphi^{2}}\*f}{\rho^{2}}+\frac{d^{2}}{d\*\rho^{2}}\*f \] \endmaxima |