︠96144497-8862-439c-a642-8af78ea24fa8i︠ %html TowardSpikeInConformalGauge system:sage

Calculo transfor. que me llevan spikes Martin a Conformal gauge

Comprobemos que las soluciones de Martin no están en el gauge conforme

︡ddda9e28-2b29-4d7a-bda5-0fe4b6c6bc1b︡{"html": "TowardSpikeInConformalGauge\r\nsystem:sage\r\n\r\n

Calculo transfor. que me llevan spikes Martin a Conformal gauge

\r\n\r\n

Comprobemos que las soluciones de Martin no est\u00e1n en el gauge conforme

"}︡ ︠1cb9ab8e-6d29-45cf-94dd-232e301326de︠ #auto sig, sig1, sig2, tau, omg, a, b, tta, r0, r1, x1, x2, u, x0n, x_1n, x1n, x2n = var('sigma, sigma1, sigma2, tau, omega, a, b, theta, ro,r1, x1, x2, u, x0n, x_1n, x1n, x2n') ︡b7625f5f-539a-48a9-8f4b-330f8ac27772︡︡ ︠6abff4e3-0575-4ec8-bff8-a6a43948c0e4︠ #auto rho = function('rho', sig); rho=arccosh(sqrt(tan(sig+pi/2)^2*cosh(2*r0)^2/(tan(sig+pi/2)^2-sinh(2*r0)^2)))/2 theta=tau+sig; rho.show() ︡295fca2b-3949-4e97-bf88-922769dcf957︡{"stdout": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2"}︡ ︠b1f577da-5917-486b-9ef4-83bf8afc10a1i︠ %html right)
}}} ︡ee48e867-ec46-48f5-b152-e5ec31b997af︡{"html": "right)\n}}}"}︡ ︠9d1ab993-a2b9-4b79-a6e0-505d34ba76c6︠ #auto x0=cosh(rho)*cos(tau); x_1=cosh(rho)*sin(tau); x1=sinh(rho)*cos(sig+tau); x2=sinh(rho)*sin(sig+tau); x = vector([x0, x_1, x1, x2]); for i in range(0,4): x[i].show() ︡ad555d93-2743-4ba5-bbab-452cab34240e︡{"stdout": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\cos\\left(\\tau\\right) \\cosh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2"}︡ ︠66c3ed75-64ae-48c8-83d1-7c9cb75edf16i︠ %html right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\tau\right) \cosh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(\sigma + \tau\right) \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma + \tau\right) \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)
}}} ︡8d48b379-243c-4ad9-a42b-a55f4bc2da06︡{"html": "right)\\right)\n
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\tau\\right) \\cosh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}}\\right)\\right)
\n
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\cos\\left(\\sigma + \\tau\\right) \\sinh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}}\\right)\\right)
\n
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma + \\tau\\right) \\sinh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}}\\right)\\right)
\n}}}"}︡ ︠8daa14aa-f263-4974-9457-660561ea03d3︠ #auto dtx=diff(x,tau); dsx=diff(x,sig); ︡30a81ced-6d20-429f-b248-2d9e1bce234c︡︡ ︠eb608d7c-2354-476d-b6ec-a5c72a60824e︠ #auto dtx2= -dtx[0]*dtx[0]-dtx[1]*dtx[1]+dtx[2]*dtx[2]+dtx[3]*dtx[3]; sdtx2 =dtx2.simplify_full(); sdtx2.show() ︡45a586d1-7314-4beb-b4d3-e6c76bcf951a︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1
"}︡ ︠5a43ee38-5b11-4c55-a734-5f7f6ce436ea︠ #auto dsx2= -dsx[0]*dsx[0]-dsx[1]*dsx[1]+dsx[2]*dsx[2]+dsx[3]*dsx[3]; sdsx2 =dsx2.simplify_full(); sdsx2.show() ︡c6a40967-53a8-45e8-8a25-f96dd138dafa︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(4 \\, \\cosh\\left(\\mbox{ro}\\right)^{8} - 8 \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + 5 \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\sin\\left(\\sigma\\right)^{2} \\cosh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\frac{\\sqrt{-\\sin\\left(\\sigma\\right) + 1} \\sqrt{\\sin\\left(\\sigma\\right) + 1} {\\left(2 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} - 1\\right)}}{\\sqrt{-{\\left(2 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} - 1\\right)} \\sin\\left(\\sigma\\right) + 1} \\sqrt{{\\left(2 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} - 1\\right)} \\sin\\left(\\sigma\\right) + 1}}\\right)\\right)^{2} + {\\left({\\left(16 \\, \\cosh\\left(\\mbox{ro}\\right)^{8} - 32 \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + 24 \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 8 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\sin\\left(\\sigma\\right)^{4} - {\\left(4 \\, \\cosh\\left(\\mbox{ro}\\right)^{8} - 8 \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + 13 \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 9 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} + 2\\right)} \\sin\\left(\\sigma\\right)^{2} + 1\\right)} \\sinh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\frac{\\sqrt{-\\sin\\left(\\sigma\\right) + 1} \\sqrt{\\sin\\left(\\sigma\\right) + 1} {\\left(2 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} - 1\\right)}}{\\sqrt{-{\\left(2 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} - 1\\right)} \\sin\\left(\\sigma\\right) + 1} \\sqrt{{\\left(2 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} - 1\\right)} \\sin\\left(\\sigma\\right) + 1}}\\right)\\right)^{2}}{{\\left(16 \\, \\cosh\\left(\\mbox{ro}\\right)^{8} - 32 \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + 24 \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 8 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\sin\\left(\\sigma\\right)^{4} - 2 \\, {\\left(4 \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 4 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\sin\\left(\\sigma\\right)^{2} + 1}
"}︡ ︠49276af8-23e4-4294-8b94-9264fb778e0c︠ #auto dtxdsx= -dtx[0]*dsx[0]-dtx[1]*dsx[1]+dtx[2]*dsx[2]+dtx[3]*dsx[3]; sdtxdsx =dtxdsx.simplify_full(); sdtxdsx.show() ︡ade67ddb-bb6f-4a3a-8d91-5e8f517ef255︡{"stdout": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sinh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\frac{{\\left(\\sinh\\left(\\mbox{ro}\\right)^{2} + \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} {\\left| \\cos\\left(\\sigma\\right) \\right|}}{\\sqrt{-4 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\mbox{ro}\\right)^{2} + {\\left(4 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma\\right)^{2"}︡ ︠45555ba3-9a10-4883-a26d-81d546fa6ef2i︠ %html right)\right)^{2}
}}}

As a suppose, este anzats no esta en el gauge conforme

︡98896e04-25fc-444a-8cb4-eab6369344eb︡{"html": "right)\\right)^{2}\n}}}\n\n

As a suppose, este anzats no esta en el gauge conforme

"}︡ ︠ebb48c9a-e18b-49d0-85fb-3f7701204e6bi︠ %html

Induced metric on AdS3 (fast rotating string ω=1)

︡d00ef37b-7d93-4545-a256-0059da633e16︡{"html": "

Induced metric on AdS3 (fast rotating string ω=1)

"}︡ ︠c91274c9-e096-4274-86a4-e7f39d186331︠ #auto t=tau; sp = vector([t,rho, theta]); g = matrix(SR,[[-cosh(rho)^2,0,0],[0,1,0],[0,0,sinh(rho)^2]]); pretty_print(g) ︡4d41e923-426c-4157-b16a-6cb630c7dc0b︡{"stdout": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n-\\cosh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2"}︡ ︠f8758fa8-6249-483e-bcdb-ff235d870586i︠ %html right)\right)^{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)^{2} \end{array}\right) }}} ︡f181f24d-8c9d-4954-b9ad-f139acede375︡{"html": "right)\\right)^{2} & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & \\sinh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}}\\right)\\right)^{2}\n\\end{array}\\right)\n}}}"}︡ ︠47d8ccf8-d91a-4fa1-b6b4-cf6bcb86f7ca︠ #auto dspt = diff(sp,tau); dsps = diff(sp, sig); ︡13f02bae-e82e-4ea7-952e-fdf1fe9cd760︡︡ ︠6f7aef82-8824-45fa-89a2-5da644865380︠ #auto h11=(dspt*g*dspt).simplify_full(); h12 = dspt*g*dsps; h21 = dsps*g*dspt; h22 =dsps*g*dsps; ︡f8e2bf34-d5d4-4b35-b91a-739c51ed3bfd︡︡ ︠0b1e9227-14bd-4118-bb39-951a4667ce4d︠ #auto h=matrix(SR,[[h11,h12],[h21,h22]]); pretty_print(h) ︡268579f0-06de-491a-9ad4-318cd81363c7︡{"stdout": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n-1 & \\sinh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2"}︡ ︠b6e199c4-cb2d-4859-9330-48802b6e3dcfi︠ %html right)\right)^{2} \\ \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)^{2} & \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)^{2} + \frac{{\left(\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}\right)} {\left(\frac{{\left(\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} + 1\right)} \tan\left(\frac{1}{2} \, \pi + \sigma\right)^{3} \cosh\left(2 \, \mbox{ro}\right)^{2}}{{\left(\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}\right)}^{2}} + \frac{-{\left(\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} + 1\right)} \tan\left(\frac{1}{2} \, \pi + \sigma\right) \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}\right)}^{2}}{4 \, {\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}} - 1\right)} {\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}} + 1\right)} \tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}} \end{array}\right) }}}

∫ƒ(σ)dσ=σ'

︡7f44db9e-cb3b-4dbe-a36d-ae5ec9e4e4b8︡{"html": "right)\\right)^{2} \\\\\n\\sinh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}}\\right)\\right)^{2} & \\sinh\\left(\\frac{1}{2} \\, {\\rm arccosh}\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}}\\right)\\right)^{2} + \\frac{{\\left(\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}\\right)} {\\left(\\frac{{\\left(\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} + 1\\right)} \\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{3} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{{\\left(\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}\\right)}^{2}} + \\frac{-{\\left(\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} + 1\\right)} \\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right) \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}\\right)}^{2}}{4 \\, {\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}} - 1\\right)} {\\left(\\sqrt{\\frac{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} - \\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}} + 1\\right)} \\tan\\left(\\frac{1}{2} \\, \\pi + \\sigma\\right)^{2} \\cosh\\left(2 \\, \\mbox{ro}\\right)^{2}}\n\\end{array}\\right)\n}}}\n\n

∫ƒ(σ)dσ=σ'

"}︡ ︠2fe9adf6-deb9-4ea8-9026-ef1ebc3bd7d4︠ #Si la worldsheet es lorentziana ︡f69a5838-0c2d-47b6-9fdf-04c6a353b427︡︡ ︠bed6d41b-b7b8-4a0e-9ecc-f608d5e3af76︠ #auto I1= integral(tan(sig)^2*r1^2/(tan(sig)^2-r0^2), sig); I1.simplify_trig().show() ︡415ed620-c92f-4d18-b640-56dc47d45350︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r_{1}^{2} \\mbox{ro} \\log\\left(\\frac{-\\mbox{ro} \\cos\\left(\\sigma\\right) - \\sin\\left(\\sigma\\right)}{\\cos\\left(\\sigma\\right)}\\right) - r_{1}^{2} \\mbox{ro} \\log\\left(\\frac{\\mbox{ro} \\cos\\left(\\sigma\\right) + \\sin\\left(\\sigma\\right)}{\\cos\\left(\\sigma\\right)}\\right) + 2 \\, r_{1}^{2} \\arctan\\left(\\frac{\\sin\\left(\\sigma\\right)}{\\cos\\left(\\sigma\\right)}\\right)}{2 \\, {\\left(\\mbox{ro}^{2} + 1\\right)}}
"}︡ ︠fd95b426-093c-4fe0-a708-f5fb4f32d01b︠ diff(I1,sig).simplify_full().show() ︡719dde6d-97f7-4c7e-a82f-338c34558aff︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{-r_{1}^{2} \\sin\\left(\\sigma\\right)^{2}}{\\mbox{ro}^{2} \\cos\\left(\\sigma\\right)^{2} - \\sin\\left(\\sigma\\right)^{2}}
"}︡ ︠fe3d0711-690a-4773-aeed-c6da8905a919︠ #auto #I5= integral(sinh(2*r0)/(sinh(2*u)*sqrt(sinh(2*u)^2-sinh(2*r0)^2)), (u, r0, r1) ); I5.show() ︡c9febf6b-6df9-4cff-9e5e-17650b5e8ffa︡︡ ︠b9b3ebd8-04ab-4adb-9f04-047836da0f00i︠ %html

∫ƒ(σ)dτ=τ'

︡39ad9b8e-924a-460d-9d7f-0976ab2596d5︡{"html": "

∫ƒ(σ)dτ=τ'

"}︡ ︠738b7d65-940f-4821-aa96-3ef8ab67ca03︠ #auto assume(1-a^2<0) ︡22028155-066b-475c-9c12-ce8f5f5d2287︡︡ ︠d0757d5c-8e45-4251-a15e-1c44456bb041︠ #auto I2= integral(1/(a*cosh(2*sig)+1), sig); I2.show() ︡0f208fe8-a344-4e7b-84ee-4aa0d4342d5a︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\arctan\\left(\\frac{a e^{\\left(2 \\, \\sigma\\right)} + 1}{\\sqrt{a^{2} - 1}}\\right)}{\\sqrt{a^{2} - 1}}
"}︡ ︠06072315-7b37-469d-9299-8025aea6c9a4︠ #auto I2.diff(sig).show() ︡90ef9e5e-6076-4e80-b8dd-893645054f54︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, a e^{\\left(2 \\, \\sigma\\right)}}{{\\left(\\frac{{\\left(a e^{\\left(2 \\, \\sigma\\right)} + 1\\right)}^{2}}{a^{2} - 1} + 1\\right)} {\\left(a^{2} - 1\\right)}}
"}︡ ︠712e5bdb-8b61-4758-8096-fe9b3c0826f4i︠ %html

Metrica inducida en las nuevas coordenadas (moño)

︡d9109836-a228-4070-84b4-585268fd8ac2︡{"html": "

Metrica inducida en las nuevas coordenadas (mo\u00f1o)

"}︡ ︠ce97ff23-d7f3-4b76-8126-051bb20d625e︠ #auto tau1 =(sig1+arctan((cosh(2*r0)-1)*tanh(sig2)/sinh(2*r0))).simplify_full(); rho1 = (arccosh(sqrt((cosh(2*r0)*cosh(2*sig2)+1)/2))).simplify_full(); tta1 =sig1+arctan((cosh(2*r0)-1)*tanh(sig2)/sinh(2*r0))+arctan(-sinh(2*r0)/tanh(2*sig2)); tau1.show(); rho1.show(); tta1.show() ︡7fc643f3-6ebe-448f-8d61-64c20834b0f0︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sigma_{1} + \\arctan\\left(\\frac{\\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)}{\\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\rm arccosh}\\left(\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sigma_{1} + \\arctan\\left(\\frac{{\\left(\\cosh\\left(2 \\, \\mbox{ro}\\right) - 1\\right)} \\tanh\\left(\\sigma_{2}\\right)}{\\sinh\\left(2 \\, \\mbox{ro}\\right)}\\right) + \\arctan\\left(\\frac{-\\sinh\\left(2 \\, \\mbox{ro}\\right)}{\\tanh\\left(2 \\, \\sigma_{2}\\right)}\\right)
"}︡ ︠844f3eb7-fb82-49cf-9fe5-79d44e6651cd︠ #auto spn = vector([tau1,rho1, tta1]); gn = matrix(SR,[[-cosh(rho1)^2,0,0],[0,1,0],[0,0,sinh(rho1)^2]]); pretty_print(gn) ︡f819d4e7-ec8f-4275-a635-91bfd2befaa8︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n-{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} + \\sinh\\left(\\mbox{ro}\\right)^{2} & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & {\\left(\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} - 1\\right)} {\\left(\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} + 1\\right)}\n\\end{array}\\right)"}︡ ︠010b2d35-a1d5-4be9-b265-59d8c35bec92︠ #auto dspns1 = diff(spn,sig1); dspns2 = diff(spn,sig2); ︡7634cd3e-d175-48cf-8fa8-68108103c18e︡︡ ︠fd53fc49-3e56-4e48-8dd6-84fb11d2ce50︠ #auto for in range(0,5): ︡f40311d9-a439-43e9-8ea2-1d6a0807b1d6︡{"stderr": "Traceback (most recent call last):\n File \"\", line 1, in \n File \"_sage_input_22.py\", line 10, in \n exec compile(u'open(\"___code___.py\",\"w\").write(\"# -*- coding: utf-8 -*-\\\\n\" + _support_.preparse_worksheet_cell(base64.b64decode(\"Zm9yIGluIHJhbmdlKDAsNSk6\"),globals())+\"\\\\n\"); execfile(os.path.abspath(\"___code___.py\"))' + '\\n', '', 'single')\n File \"\", line 1, in \n \n File \"/tmp/tmpvVazdC/___code___.py\", line 3\n for in range(_sage_const_0 ,_sage_const_5 ):\n ^\nSyntaxError: invalid syntax"}︡ ︠90308033-5715-44b5-a487-5cb714a2f7bd︠ #auto h11n=(dspns1*gn*dspns1).simplify_full(); h12n = (dspns1*gn*dspns2).simplify_full(); h21n = (dspns2*gn*dspns1).simplify_full(); h22n =(dspns2*gn*dspns2).simplify_full(); ︡897948f0-aed4-444b-9f32-9a8638ccb778︡︡ ︠8ef172e4-03b2-4c63-b83a-44d321c79bb4︠ #auto hn=matrix(SR,[[h11n,h12n],[h21n,h22n]]); pretty_print(hn) ︡2573a0fa-6528-4b1b-a23a-8a2731caf12f︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n-1 & 0 \\\\\n0 & 1\n\\end{array}\\right)"}︡ ︠30aa53e6-f1da-4484-b894-723c1a500501i︠ %html

:)

︡798c9a08-d0a8-4696-99f5-b119cbff74eb︡{"html": "

:)

"}︡ ︠c8d94024-1019-455b-b464-ee5d2ab403c8︠ #auto signa=arctan(-sinh(2*r0)/tanh(2*sig2)); ︡bb1f1b4b-09bb-43bc-ab0c-0c24db35b86e︡︡ ︠f5dc7af2-0a81-487b-9032-5180fe384949︠ #auto xdxp=-diff(tau1, sig2)+diff(signa, sig2)*(cosh(2*r0)*cosh(2*sig2)-1)/2; xdxp.show() ︡d2ec6166-6493-4dea-b41a-b107ef9c68d2︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\frac{\\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)^{2}}{\\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)^{2}} + \\frac{-\\sinh\\left(\\mbox{ro}\\right)}{\\cosh\\left(\\mbox{ro}\\right)}}{\\frac{\\sinh\\left(\\mbox{ro}\\right)^{2} \\sinh\\left(\\sigma_{2}\\right)^{2}}{\\cosh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\sigma_{2}\\right)^{2}} + 1} + \\frac{-{\\left(\\tanh\\left(2 \\, \\sigma_{2}\\right)^{2} - 1\\right)} {\\left(\\cosh\\left(2 \\, \\mbox{ro}\\right) \\cosh\\left(2 \\, \\sigma_{2}\\right) - 1\\right)} \\sinh\\left(2 \\, \\mbox{ro}\\right)}{{\\left(\\frac{\\sinh\\left(2 \\, \\mbox{ro}\\right)^{2}}{\\tanh\\left(2 \\, \\sigma_{2}\\right)^{2}} + 1\\right)} \\tanh\\left(2 \\, \\sigma_{2}\\right)^{2}}
"}︡ ︠e39119cc-f8de-49ca-a001-e34d13f6d25a︠ #auto #xdxp.simplify_full().show() ︡39577682-29db-4aeb-a065-71aa6172a5e8︡︡ ︠ea501e0a-e222-4efc-ac98-d899f3b013edi︠ %html

calculemos las restricciones de virasoro en las nuevas coordenadas

︡7fd1d5a8-6527-402b-b6dc-24cf26e37641︡{"html": "

calculemos las restricciones de virasoro en las nuevas coordenadas

"}︡ ︠6cbca032-965a-4242-9336-771510b1d69e︠ #auto x0n=cosh(rho1)*cos(tau1); x_1n=cosh(rho1)*sin(tau1); x1n=sinh(rho1)*cos(tta1); x2n=sinh(rho1)*sin(tta1); xn = vector([x0n, x_1n, x1n, x2n]); for i in range(0,4): xn[i].show() ︡2e0cc298-ebe4-4f24-a55c-b3dda54d76af︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} \\cos\\left(\\sigma_{1} + \\arctan\\left(\\frac{\\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)}{\\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)}\\right)\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} \\sin\\left(\\sigma_{1} + \\arctan\\left(\\frac{\\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)}{\\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)}\\right)\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} - 1} \\sqrt{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} + 1} \\cos\\left(\\sigma_{1} + \\arctan\\left(\\frac{{\\left(\\cosh\\left(2 \\, \\mbox{ro}\\right) - 1\\right)} \\tanh\\left(\\sigma_{2}\\right)}{\\sinh\\left(2 \\, \\mbox{ro}\\right)}\\right) + \\arctan\\left(\\frac{-\\sinh\\left(2 \\, \\mbox{ro}\\right)}{\\tanh\\left(2 \\, \\sigma_{2}\\right)}\\right)\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} - 1} \\sqrt{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} + 1} \\sin\\left(\\sigma_{1} + \\arctan\\left(\\frac{{\\left(\\cosh\\left(2 \\, \\mbox{ro}\\right) - 1\\right)} \\tanh\\left(\\sigma_{2}\\right)}{\\sinh\\left(2 \\, \\mbox{ro}\\right)}\\right) + \\arctan\\left(\\frac{-\\sinh\\left(2 \\, \\mbox{ro}\\right)}{\\tanh\\left(2 \\, \\sigma_{2}\\right)}\\right)\\right)
"}︡ ︠1ec8400c-abff-4fde-a6aa-6ddc14dacb87︠ #auto cos(tau1).simplify_full().show(); sin(tau1).simplify_full().show(); ︡369a304e-9a4f-4af1-8fbd-5f4e198ee42f︡{"stdout": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{-\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right) - \\cos\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)}{\\sqrt{\\sinh\\left(\\mbox{ro}\\right)^{2} \\sinh\\left(\\sigma_{2}\\right)^{2} + \\cosh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\sigma_{2}\\right)^{2"}︡ ︠d21e6adb-99ee-4ecf-a266-d30af2097696i︠ %html /div>
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)}{\sqrt{\sinh\left(\mbox{ro}\right)^{2} \sinh\left(\sigma_{2}\right)^{2} + \cosh\left(\mbox{ro}\right)^{2} \cosh\left(\sigma_{2}\right)^{2}}}
}}} ︡7bda12e4-6b12-4eb4-b904-48f81fc636df︡{"html": "/div>\n
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sin\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)}{\\sqrt{\\sinh\\left(\\mbox{ro}\\right)^{2} \\sinh\\left(\\sigma_{2}\\right)^{2} + \\cosh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\sigma_{2}\\right)^{2}}}
\n}}}"}︡ ︠8310631e-0e11-4902-9b9c-03c54df590b6︠ #auto assume(sinh(r0)>0); (cos(tta1)).simplify_full().factor().simplify_full().show() ︡65440a59-8305-46e6-a8a0-5af44547265a︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} {\\left| \\sinh\\left(\\sigma_{2}\\right) \\right|}}{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2} - 1} \\sinh\\left(\\sigma_{2}\\right)}
"}︡ ︠67851af6-c9ac-47fc-9d79-f68ee6391616︠ #auto assume(sinh(r0)>0); (sin(tta1)).simplify_full().factor().simplify_full().show() ︡fe51e01f-eb55-46ef-bcdb-45bee24b21a5︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right) - \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)\\right)} {\\left| \\sinh\\left(\\sigma_{2}\\right) \\right|}}{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2} - 1} \\sinh\\left(\\sigma_{2}\\right)}
"}︡ ︠54d58289-fa33-4d8f-b0a0-b4f16cfc23a1i︠ %html

coordenadas en el gauge conforme simplificadas

︡1728fbc7-6b58-4c94-ac8f-07bd245e521b︡{"html": "

coordenadas en el gauge conforme simplificadas

"}︡ ︠5a79f232-31b2-468a-aac7-0cd43a5068f0︠ #auto x0s=-(sin(sigma1)*sinh(ro)*sinh(sigma2) -cos(sigma1)*cosh(ro)*cosh(sigma2)); x_1s=(sin(sigma1)*cosh(ro)*cosh(sigma2) +cos(sigma1)*sinh(ro)*sinh(sigma2)); x1s=(sin(sigma1)*sinh(ro)*cosh(sigma2) +cos(sigma1)*sinh(sigma2)*cosh(ro)); x2s=(sin(sigma1)*sinh(sigma2)*cosh(ro) -cos(sigma1)*sinh(ro)*cosh(sigma2)); xs = vector([x0s, x_1s, x1s, x2s]); for i in range(0,4): xs[i].show() ︡733bc94d-3ef2-4c95-94a0-3061155849b7︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right) - \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)
"}︡ ︠b558d734-5224-48f7-bdd4-e9291167edbf︠ #auto dtxn=diff(xs,sig1); dsxn=diff(xs,sig2); dtxn; ︡97a2a429-345d-4a23-809c-41f6ac018cab︡{"stdout": "(-sin(sigma1)*cosh(ro)*cosh(sigma2) - cos(sigma1)*sinh(ro)*sinh(sigma2), -sin(sigma1)*sinh(ro)*sinh(sigma2) + cos(sigma1)*cosh(ro)*cosh(sigma2), -sin(sigma1)*sinh(sigma2)*cosh(ro) + cos(sigma1)*sinh(ro)*cosh(sigma2), sin(sigma1)*sinh(ro)*cosh(sigma2) + cos(sigma1)*sinh(sigma2)*cosh(ro))"}︡ ︠a19ee568-4795-472d-9fbe-0d63cd31887b︠ #auto dsxn[1].simplify().show(); ︡b8eadd14-a287-460d-b4b3-110a2c3c9549︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)
"}︡ ︠20406e36-dcde-4b5d-a307-6e55b6afceb3︠ #auto dtxndsxn= -dtxn[0]*dsxn[0]-dtxn[1]*dsxn[1]+dtxn[2]*dsxn[2]+dtxn[3]*dsxn[3]; dtxndsxn.show() ︡3d96ce34-8845-4fe7-9517-d2ef6c137ad1︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(\\sin\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) - \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)\\right)} {\\left(\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} - {\\left(\\sin\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)\\right)} {\\left(\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) - \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} - {\\left(\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right) - \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)\\right)} {\\left(\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)\\right)} + {\\left(\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)\\right)} {\\left(\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right) - \\cos\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)\\right)}
"}︡ ︠51ef4fce-932e-4ef2-9316-2d9d7a4823f8︠ #auto sdtxndsxn =dtxndsxn.simplify_full(); sdtxndsxn.show() ︡e060945a-ece1-4da0-a458-8ba4a1061fbf︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}0
"}︡ ︠40c77040-f1d5-4314-aca2-7b56ea77a6fa︠ #auto dtxn2= -dtxn[0]*dtxn[0]-dtxn[1]*dtxn[1]+dtxn[2]*dtxn[2]+dtxn[3]*dtxn[3]; sdtxn2 =dtxn2.simplify_full(); sdtxn2.show() ︡d8836dc7-4c2a-455e-a7e8-41e9ab2ec52c︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1
"}︡ ︠bca3089f-0010-44f6-b63e-6a63106b3b5e︠ #auto dsxn2= -dsxn[0]*dsxn[0]-dsxn[1]*dsxn[1]+dsxn[2]*dsxn[2]+dsxn[3]*dsxn[3]; sdsxn2 =dsxn2.simplify_full(); sdsxn2.show() ︡32f4a419-d847-4a61-9b97-371906ac316c︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}1
"}︡ ︠1c629fe1-a62b-48a2-959b-cbd04327b2c9︠ sdtxn2+sdsxn2 ︡07c722ba-37fa-4036-8bda-4e96f3b986fb︡{"stdout": "0"}︡ ︠fa625476-a13c-4b14-b28c-3bc265347ba7︠ N(ln(4)); N(ln(1/4)) ︡6438c3c5-9510-477a-8f20-aa0218ca8d45︡{"stderr": "Traceback (most recent call last):\n File \"\", line 1, in \n File \"_sage_input_84.py\", line 10, in \n exec compile(u'open(\"___code___.py\",\"w\").write(\"# -*- coding: utf-8 -*-\\\\n\" + _support_.preparse_worksheet_cell(base64.b64decode(\"Tihsbig0KSk7Ck4obG4oMS80KSk=\"),globals())+\"\\\\n\"); execfile(os.path.abspath(\"___code___.py\"))' + '\\n', '', 'single')\n File \"\", line 1, in \n \n File \"/tmp/tmpaq3CE6/___code___.py\", line 3, in \n N(ln(_sage_const_4 ));\n File \"free_module_element.pyx\", line 2555, in sage.modules.free_module_element.FreeModuleElement_generic_dense.__call__ (sage/modules/free_module_element.c:17127)\n File \"expression.pyx\", line 3426, in sage.symbolic.expression.Expression.__call__ (sage/symbolic/expression.cpp:15476)\n File \"ring.pyx\", line 638, in sage.symbolic.ring.SymbolicRing._call_element_ (sage/symbolic/ring.cpp:6460)\nValueError: the number of arguments must be less than or equal to 0"}︡ ︠32d6f42d-0fcd-4f0f-97ff-f2cbe1b52503︠ #parametric_plot((cos(u),sin(u)^3),(u,0,2*pi),rgbcolor=hue(0.6)) ︡5059e86a-4a7b-4c94-96c3-def38f399fc7︡︡ ︠5bbc61d2-b5c3-4ba5-8579-000b1cd90426︠ plot(sinh(1)/tanh(2*u), (u,1,3)) ︡1aa1c4f9-226d-4602-bfbd-df792b8738f0︡{"stderr": "Traceback (most recent call last):\n File \"\", line 1, in \n File \"_sage_input_86.py\", line 10, in \n exec compile(u'open(\"___code___.py\",\"w\").write(\"# -*- coding: utf-8 -*-\\\\n\" + _support_.preparse_worksheet_cell(base64.b64decode(\"cGxvdChzaW5oKDEpL3RhbmgoMip1KSwgKHUsMSwzKSk=\"),globals())+\"\\\\n\"); execfile(os.path.abspath(\"___code___.py\"))' + '\\n', '', 'single')\n File \"\", line 1, in \n \n File \"/tmp/tmp1WAgB2/___code___.py\", line 3, in \n exec compile(u'plot(sinh(_sage_const_1 )/tanh(_sage_const_2 *u), (u,_sage_const_1 ,_sage_const_3 ))' + '\\n', '', 'single')\n File \"\", line 1, in \n \nNameError: name 'u' is not defined"}︡ ︠5737d2ad-afad-4d94-aeb2-5721ee147076︠ plot(arctan(sinh(1)/tanh(2*u)), (u,-2,2)) ︡dac93237-7d08-4bd4-97d2-4b4495230fb1︡{"stderr": "Traceback (most recent call last):\n File \"\", line 1, in \n File \"_sage_input_87.py\", line 10, in \n exec compile(u'open(\"___code___.py\",\"w\").write(\"# -*- coding: utf-8 -*-\\\\n\" + _support_.preparse_worksheet_cell(base64.b64decode(\"cGxvdChhcmN0YW4oc2luaCgxKS90YW5oKDIqdSkpLCAodSwtMiwyKSk=\"),globals())+\"\\\\n\"); execfile(os.path.abspath(\"___code___.py\"))' + '\\n', '', 'single')\n File \"\", line 1, in \n \n File \"/tmp/tmpk6NsNd/___code___.py\", line 3, in \n exec compile(u'plot(arctan(sinh(_sage_const_1 )/tanh(_sage_const_2 *u)), (u,-_sage_const_2 ,_sage_const_2 ))' + '\\n', '', 'single')\n File \"\", line 1, in \n \nNameError: name 'u' is not defined"}︡ ︠6927bc13-4b45-40f9-97a9-9be757b32b8a︠ plot(arctan(u),(-100,100)) ︡1e6be8b4-900f-4ae5-8333-db854e3a320b︡{"stderr": "Traceback (most recent call last):\n File \"\", line 1, in \n File \"_sage_input_88.py\", line 10, in \n exec compile(u'open(\"___code___.py\",\"w\").write(\"# -*- coding: utf-8 -*-\\\\n\" + _support_.preparse_worksheet_cell(base64.b64decode(\"cGxvdChhcmN0YW4odSksKC0xMDAsMTAwKSk=\"),globals())+\"\\\\n\"); execfile(os.path.abspath(\"___code___.py\"))' + '\\n', '', 'single')\n File \"\", line 1, in \n \n File \"/tmp/tmpMHyhP0/___code___.py\", line 3, in \n exec compile(u'plot(arctan(u),(-_sage_const_100 ,_sage_const_100 ))' + '\\n', '', 'single')\n File \"\", line 1, in \n \nNameError: name 'u' is not defined"}︡ ︠e6e03b7a-e65b-41ee-a005-6d36533734c3︠

Intentemos respetir las cuentas en el caso coordenadas worldsheet z (metrica inducida euclidea)

︡807d1e42-126a-4152-9fab-7fb7a9267dbd︡{"stderr": "Traceback (most recent call last):\n File \"\", line 1, in \n File \"_sage_input_90.py\", line 10, in \n exec compile(u'open(\"___code___.py\",\"w\").write(\"# -*- coding: utf-8 -*-\\\\n\" + _support_.preparse_worksheet_cell(base64.b64decode(\"PGgzIHN0eWxlPSJjb2xvcjogIzgwMDA4MDsiPkludGVudGVtb3MgcmVzcGV0aXIgbGFzIGN1ZW50YXMgZW4gZWwgY2FzbyBjb29yZGVuYWRhcyB3b3JsZHNoZWV0IHogKG1ldHJpY2EgaW5kdWNpZGEgZXVjbGlkZWEpIDwvaDM+\"),globals())+\"\\\\n\"); execfile(os.path.abspath(\"___code___.py\"))' + '\\n', '', 'single')\n File \"\", line 1, in \n \n File \"/tmp/tmpcmku9g/___code___.py\", line 2\n

Intentemos respetir las cuentas en el caso coordenadas worldsheet z (metrica inducida euclidea)

\n ^\nSyntaxError: invalid syntax"}︡ ︠679fc6db-1a29-42a5-ae09-007f3756a07f︠ #integral tau(moño_sigma) ︡a52a4df3-6fcb-40c4-ba13-293be4b4dd15︡︡ ︠a1b1d3d7-0f42-4857-88bc-6a6e7d49e1a8︠ #auto Ie1= integral(1/(a*cos(2*sig)+1), sig); Ie1 ︡d460c496-d71f-4e6f-9561-606c0c9791d5︡{"stderr": "Traceback (most recent call last):\n File \"\", line 1, in \n File \"_sage_input_92.py\", line 10, in \n exec compile(u'open(\"___code___.py\",\"w\").write(\"# -*- coding: utf-8 -*-\\\\n\" + _support_.preparse_worksheet_cell(base64.b64decode(\"SWUxPSBpbnRlZ3JhbCgxLyhhKmNvcygyKnNpZykrMSksIHNpZyk7IEllMQ==\"),globals())+\"\\\\n\"); execfile(os.path.abspath(\"___code___.py\"))' + '\\n', '', 'single')\n File \"\", line 1, in \n \n File \"/tmp/tmpQfWS49/___code___.py\", line 3, in \n exec compile(u'Ie1= integral(_sage_const_1 /(a*cos(_sage_const_2 *sig)+_sage_const_1 ), sig); Ie1' + '\\n', '', 'single')\n File \"\", line 1, in \n \nNameError: name 'a' is not defined"}︡ ︠a141b40f-2908-499f-a5a2-757d365cd0d1︠ (log(((a - 1)*sin(2*sigma)/(cos(2*sigma) + 1) - sqrt(a^2 - 1))/((a - 1)*sin(2*sigma)/(cos(2*sigma) + 1) + sqrt(a^2 - 1)))).simplify_full().show() ︡52f660a4-57fe-4830-8189-ecc3a77a7b4a︡{"stderr": "Traceback (most recent call last):\n File \"\", line 1, in \n File \"_sage_input_93.py\", line 10, in \n exec compile(u'open(\"___code___.py\",\"w\").write(\"# -*- coding: utf-8 -*-\\\\n\" + _support_.preparse_worksheet_cell(base64.b64decode(\"KGxvZygoKGEgLSAxKSpzaW4oMipzaWdtYSkvKGNvcygyKnNpZ21hKSArIDEpIC0gc3FydChhXjIgLSAxKSkvKChhIC0KMSkqc2luKDIqc2lnbWEpLyhjb3MoMipzaWdtYSkgKyAxKSArIHNxcnQoYV4yIC0gMSkpKSkuc2ltcGxpZnlfZnVsbCgpLnNob3coKQ==\"),globals())+\"\\\\n\"); execfile(os.path.abspath(\"___code___.py\"))' + '\\n', '', 'single')\n File \"\", line 1, in \n \n File \"/tmp/tmpxT2arr/___code___.py\", line 3, in \n (log(((a - _sage_const_1 )*sin(_sage_const_2 *sigma)/(cos(_sage_const_2 *sigma) + _sage_const_1 ) - sqrt(a**_sage_const_2 - _sage_const_1 ))/((a -\nNameError: name 'a' is not defined"}︡ ︠316702d2-48e1-4a18-a90d-c2d32a49c657︠ #auto tae =(sig1-(1/2)*ln((-sinh(r0)*sin(sig2)+cosh(r0)*cos(sig2))/(sinh(r0)*sin(sig2)+cosh(r0)*cos(sig2)))).simplify_full(); rhoe = (arccosh(sqrt((cosh(2*r0)*cos(2*sig2)+1)/2))).simplify_full(); ttae =(sig1-(1/2)*ln((-sinh(r0)*sin(sig2)+cosh(r0)*cos(sig2))/(sinh(r0)*sin(sig2)+cosh(r0)*cos(sig2))))+arctan(-I*sinh(2*r0)/tan(2*sig2)); tae.show(); rhoe.show(); ttae.show() ︡53ab4ec7-8f07-439f-9bcf-fd248b79814e︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sigma_{1} - \\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right) + \\frac{1}{2} \\, \\log\\left(\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\rm arccosh}\\left(\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sigma_{1} - \\frac{1}{2} \\, \\log\\left(\\frac{-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) - \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)}{\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)}\\right) + \\arctan\\left(\\frac{-i \\, \\sinh\\left(2 \\, \\mbox{ro}\\right)}{\\tan\\left(2 \\, \\sigma_{2}\\right)}\\right)
"}︡ ︠e249d876-d0c1-47f9-b09c-9f8db175f5a2︠ #auto spe = vector([tae,rhoe, ttae]); ge = matrix(SR,[[-cosh(rho1)^2,0,0],[0,1,0],[0,0,sinh(rho1)^2]]); pretty_print(ge) ︡762a335c-0a34-4124-9fdc-c5a3f0901f67︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n-{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} + \\sinh\\left(\\mbox{ro}\\right)^{2} & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & {\\left(\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} - 1\\right)} {\\left(\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} + 1\\right)}\n\\end{array}\\right)"}︡ ︠bb44bad7-bc7e-4fe9-9b2d-e02233880957︠ #auto dspes1 = diff(spe,sig1); dspes2 = diff(spe,sig2); ︡46b0168c-4a44-434b-a598-855af2047041︡︡ ︠08565387-0fc1-47e0-8d00-ed9ed379cf3d︠ #auto h11e=(dspes1*ge*dspes1).simplify_full(); h12e = (dspes1*ge*dspes2).simplify_full(); h21e =(dspes2*ge*dspes1).simplify_full(); h22e =(dspes2*ge*dspes2).simplify_full().factor().simplify_full(); ︡ef7fd35e-c071-4e4a-97cb-9179fed6458c︡︡ ︠58620724-125e-432e-9317-6ded868f5ea7︠ #auto he=matrix(SR,[[h11e,h12e],[h21e,h22e]]); pretty_print(he) ︡fab912b5-9538-4ac9-9e8f-6445ddd925a2︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n-1 & \\frac{\\left(i + 1\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + {\\left(4 \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + 4 \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) + \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sin\\left(\\sigma_{2}\\right)^{4} + \\left(i + 1\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) - {\\left(\\left(2 i + 4\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + \\left(i + 4\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) + \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sin\\left(\\sigma_{2}\\right)^{2} - {\\left(\\left(-2 i\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + \\left(-3 i\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) + {\\left(\\left(4 i\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + \\left(4 i\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) + i \\, \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sin\\left(\\sigma_{2}\\right)^{2} - i \\, \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sinh\\left(\\sigma_{2}\\right)^{2}}{\\sin\\left(\\sigma_{2}\\right)^{6} \\sinh\\left(\\mbox{ro}\\right)^{4} \\cosh\\left(\\mbox{ro}\\right)^{2} - \\cos\\left(\\sigma_{2}\\right)^{6} \\sinh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\mbox{ro}\\right)^{4} - {\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{4} \\cosh\\left(\\mbox{ro}\\right)^{2} + {\\left(\\cosh\\left(\\mbox{ro}\\right)^{4} + 1\\right)} \\sinh\\left(\\mbox{ro}\\right)^{2}\\right)} \\sin\\left(\\sigma_{2}\\right)^{4} \\cos\\left(\\sigma_{2}\\right)^{2} + {\\left(\\sinh\\left(\\mbox{ro}\\right)^{4} \\cosh\\left(\\mbox{ro}\\right)^{2} + 2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\mbox{ro}\\right)^{4} + \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\sin\\left(\\sigma_{2}\\right)^{2} \\cos\\left(\\sigma_{2}\\right)^{4}} \\\\\n\\frac{\\left(i + 1\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + {\\left(4 \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + 4 \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) + \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sin\\left(\\sigma_{2}\\right)^{4} + \\left(i + 1\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) - {\\left(\\left(2 i + 4\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + \\left(i + 4\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) + \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sin\\left(\\sigma_{2}\\right)^{2} - {\\left(\\left(-2 i\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + \\left(-3 i\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) + {\\left(\\left(4 i\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{5} \\cosh\\left(\\mbox{ro}\\right) + \\left(4 i\\right) \\, \\sinh\\left(\\mbox{ro}\\right)^{3} \\cosh\\left(\\mbox{ro}\\right) + i \\, \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sin\\left(\\sigma_{2}\\right)^{2} - i \\, \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sinh\\left(\\sigma_{2}\\right)^{2}}{\\sin\\left(\\sigma_{2}\\right)^{6} \\sinh\\left(\\mbox{ro}\\right)^{4} \\cosh\\left(\\mbox{ro}\\right)^{2} - \\cos\\left(\\sigma_{2}\\right)^{6} \\sinh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\mbox{ro}\\right)^{4} - {\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{4} \\cosh\\left(\\mbox{ro}\\right)^{2} + {\\left(\\cosh\\left(\\mbox{ro}\\right)^{4} + 1\\right)} \\sinh\\left(\\mbox{ro}\\right)^{2}\\right)} \\sin\\left(\\sigma_{2}\\right)^{4} \\cos\\left(\\sigma_{2}\\right)^{2} + {\\left(\\sinh\\left(\\mbox{ro}\\right)^{4} \\cosh\\left(\\mbox{ro}\\right)^{2} + 2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} \\cosh\\left(\\mbox{ro}\\right)^{4} + \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\sin\\left(\\sigma_{2}\\right)^{2} \\cos\\left(\\sigma_{2}\\right)^{4}} & \\frac{-{\\left(256 \\, \\cosh\\left(\\mbox{ro}\\right)^{16} - 1024 \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + 1792 \\, \\cosh\\left(\\mbox{ro}\\right)^{12} - 1792 \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + 1120 \\, \\cosh\\left(\\mbox{ro}\\right)^{8} - 448 \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + 112 \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 16 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{16} + \\left(2 i + 1\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} - {\\left(1024 \\, \\cosh\\left(\\mbox{ro}\\right)^{16} - 4224 \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + 7616 \\, \\cosh\\left(\\mbox{ro}\\right)^{12} - 7840 \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + 5040 \\, \\cosh\\left(\\mbox{ro}\\right)^{8} - 2072 \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + 532 \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 78 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} + 5\\right)} \\cos\\left(\\sigma_{2}\\right)^{14} + \\left(-8 i - 5\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + {\\left(1792 \\, \\cosh\\left(\\mbox{ro}\\right)^{16} - 7616 \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + 14080 \\, \\cosh\\left(\\mbox{ro}\\right)^{12} - 14800 \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + 9680 \\, \\cosh\\left(\\mbox{ro}\\right)^{8} - 4036 \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + 1048 \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 155 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} + 10\\right)} \\cos\\left(\\sigma_{2}\\right)^{12} + \\left(12 i + 10\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + {\\left(\\left(-64 i - 1792\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(224 i + 7808\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(-320 i - 14688\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(240 i + 15600\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(-100 i - 10240\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(22 i + 4256\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(-2 i - 1094\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} + 159 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} - 10\\right)} \\cos\\left(\\sigma_{2}\\right)^{10} + \\left(-8 i - 10\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + {\\left(\\left(160 i + 1120\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(-576 i - 4976\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(848 i + 9440\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(-656 i - 9992\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(282 i + 6454\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(-64 i - 2603\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(6 i + 639\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 87 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} + 5\\right)} \\cos\\left(\\sigma_{2}\\right)^{8} + \\left(2 i + 5\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + {\\left(\\left(-160 i - 448\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(592 i + 2024\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(-888 i - 3844\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(692 i + 3994\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(-296 i - 2471\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(66 i + 925\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(-6 i - 202\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} + 23 \\, \\cosh\\left(\\mbox{ro}\\right)^{2} - 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{6} - \\cosh\\left(\\mbox{ro}\\right)^{6} + {\\left(\\left(80 i + 112\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(-304 i - 516\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(460 i + 980\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(-352 i - 989\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(142 i + 569\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(-28 i - 185\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(2 i + 31\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - 2 \\, \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\cos\\left(\\sigma_{2}\\right)^{4} + {\\left(\\left(-20 i - 16\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(78 i + 76\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(-118 i - 146\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(86 i + 144\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(-30 i - 76\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(4 i + 20\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} - 2 \\, \\cosh\\left(\\mbox{ro}\\right)^{4}\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} + {\\left(\\left(-4 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(18 i + 2\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(-32 i - 9\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + {\\left(\\left(128 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(-512 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(864 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(-800 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(440 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(-144 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(26 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} + \\left(-2 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\cos\\left(\\sigma_{2}\\right)^{10} + \\left(28 i + 16\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + {\\left(\\left(-320 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(1312 i + 32\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(-2272 i - 112\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(2160 i + 160\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(-1220 i - 120\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(410 i + 50\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(-76 i - 11\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} + \\left(6 i + 1\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\cos\\left(\\sigma_{2}\\right)^{8} + \\left(-12 i - 14\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + {\\left(\\left(320 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(-1344 i - 64\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(2368 i + 240\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(-2272 i - 368\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(1284 i + 296\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(-428 i - 132\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(78 i + 31\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} + \\left(-6 i - 3\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\cos\\left(\\sigma_{2}\\right)^{6} + \\left(2 i + 6\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + {\\left(\\left(-160 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(688 i + 48\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(-1224 i - 192\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(1164 i + 312\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(-636 i - 264\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(198 i + 123\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(-32 i - 30\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} + \\left(2 i + 3\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\cos\\left(\\sigma_{2}\\right)^{4} - \\cosh\\left(\\mbox{ro}\\right)^{4} + {\\left(\\left(40 i\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{16} + \\left(-176 i - 16\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{14} + \\left(314 i + 68\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{12} + \\left(-290 i - 116\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{10} + \\left(146 i + 101\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{8} + \\left(-38 i - 47\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{6} + \\left(4 i + 11\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{4} - \\cosh\\left(\\mbox{ro}\\right)^{2}\\right)} \\cos\\left(\\sigma_{2}\\right)^{2}\\right)} \\cosh\\left(\\sigma_{2}\\right)^{2}}{{\\left(256 \\, \\sinh\\left(\\mbox{ro}\\right)^{16} + 1024 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + 1792 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} + 1792 \\, \\sinh\\left(\\mbox{ro}\\right)^{10} + 1120 \\, \\sinh\\left(\\mbox{ro}\\right)^{8} + 448 \\, \\sinh\\left(\\mbox{ro}\\right)^{6} + 112 \\, \\sinh\\left(\\mbox{ro}\\right)^{4} + 16 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{16} + \\sinh\\left(\\mbox{ro}\\right)^{16} - {\\left(1024 \\, \\sinh\\left(\\mbox{ro}\\right)^{16} + 3968 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + 6720 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} + 6496 \\, \\sinh\\left(\\mbox{ro}\\right)^{10} + 3920 \\, \\sinh\\left(\\mbox{ro}\\right)^{8} + 1512 \\, \\sinh\\left(\\mbox{ro}\\right)^{6} + 364 \\, \\sinh\\left(\\mbox{ro}\\right)^{4} + 50 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 3\\right)} \\cos\\left(\\sigma_{2}\\right)^{14} + 3 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + {\\left(1792 \\, \\sinh\\left(\\mbox{ro}\\right)^{16} + 6720 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + 10944 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} + 10096 \\, \\sinh\\left(\\mbox{ro}\\right)^{10} + 5760 \\, \\sinh\\left(\\mbox{ro}\\right)^{8} + 2076 \\, \\sinh\\left(\\mbox{ro}\\right)^{6} + 460 \\, \\sinh\\left(\\mbox{ro}\\right)^{4} + 57 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 3\\right)} \\cos\\left(\\sigma_{2}\\right)^{12} + 3 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} - {\\left(1792 \\, \\sinh\\left(\\mbox{ro}\\right)^{16} + 6496 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + 10096 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} + 8752 \\, \\sinh\\left(\\mbox{ro}\\right)^{10} + 4600 \\, \\sinh\\left(\\mbox{ro}\\right)^{8} + 1486 \\, \\sinh\\left(\\mbox{ro}\\right)^{6} + 283 \\, \\sinh\\left(\\mbox{ro}\\right)^{4} + 28 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{10} + \\sinh\\left(\\mbox{ro}\\right)^{10} + 5 \\, {\\left(224 \\, \\sinh\\left(\\mbox{ro}\\right)^{16} + 784 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + 1152 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} + 920 \\, \\sinh\\left(\\mbox{ro}\\right)^{10} + 430 \\, \\sinh\\left(\\mbox{ro}\\right)^{8} + 117 \\, \\sinh\\left(\\mbox{ro}\\right)^{6} + 17 \\, \\sinh\\left(\\mbox{ro}\\right)^{4} + \\sinh\\left(\\mbox{ro}\\right)^{2}\\right)} \\cos\\left(\\sigma_{2}\\right)^{8} - {\\left(448 \\, \\sinh\\left(\\mbox{ro}\\right)^{16} + 1512 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + 2076 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} + 1486 \\, \\sinh\\left(\\mbox{ro}\\right)^{10} + 585 \\, \\sinh\\left(\\mbox{ro}\\right)^{8} + 120 \\, \\sinh\\left(\\mbox{ro}\\right)^{6} + 10 \\, \\sinh\\left(\\mbox{ro}\\right)^{4}\\right)} \\cos\\left(\\sigma_{2}\\right)^{6} + {\\left(112 \\, \\sinh\\left(\\mbox{ro}\\right)^{16} + 364 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + 460 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} + 283 \\, \\sinh\\left(\\mbox{ro}\\right)^{10} + 85 \\, \\sinh\\left(\\mbox{ro}\\right)^{8} + 10 \\, \\sinh\\left(\\mbox{ro}\\right)^{6}\\right)} \\cos\\left(\\sigma_{2}\\right)^{4} - {\\left(16 \\, \\sinh\\left(\\mbox{ro}\\right)^{16} + 50 \\, \\sinh\\left(\\mbox{ro}\\right)^{14} + 57 \\, \\sinh\\left(\\mbox{ro}\\right)^{12} + 28 \\, \\sinh\\left(\\mbox{ro}\\right)^{10} + 5 \\, \\sinh\\left(\\mbox{ro}\\right)^{8}\\right)} \\cos\\left(\\sigma_{2}\\right)^{2}}\n\\end{array}\\right)"}︡ ︠f58bdb35-4756-4b73-bbd1-94c16c8fb990︠ #calculemos rescricciones, en las nuevas coordenadas ︡178eda07-512d-4f29-9ea0-b2d06d038d45︡︡ ︠b65bc232-69a7-486f-b3d9-3c26ec1ab251︠ #auto x0e=cosh(rhoe)*cos(tae); x_1e=cosh(rhoe)*sin(tae); x1e=sinh(rhoe)*cos(ttae); x2e=sinh(rhoe)*sin(ttae); xe = vector([x0e, x_1e, x1e, x2e]); for i in range(0,4): xe[i].show() ︡38911ff7-5a29-43e8-bf2b-64e66d144d9e︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} \\cos\\left(\\sigma_{1} - \\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right) + \\frac{1}{2} \\, \\log\\left(\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} \\sin\\left(\\sigma_{1} - \\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right) + \\frac{1}{2} \\, \\log\\left(\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} - 1} \\sqrt{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} + 1} \\cos\\left(\\sigma_{1} - \\frac{1}{2} \\, \\log\\left(\\frac{-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) - \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)}{\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)}\\right) + \\arctan\\left(\\frac{-i \\, \\sinh\\left(2 \\, \\mbox{ro}\\right)}{\\tan\\left(2 \\, \\sigma_{2}\\right)}\\right)\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} - 1} \\sqrt{\\sqrt{{\\left(2 \\, \\sinh\\left(\\mbox{ro}\\right)^{2} + 1\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} - \\sinh\\left(\\mbox{ro}\\right)^{2}} + 1} \\sin\\left(\\sigma_{1} - \\frac{1}{2} \\, \\log\\left(\\frac{-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) - \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)}{\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)}\\right) + \\arctan\\left(\\frac{-i \\, \\sinh\\left(2 \\, \\mbox{ro}\\right)}{\\tan\\left(2 \\, \\sigma_{2}\\right)}\\right)\\right)
"}︡ ︠92562900-cb21-40a0-bb7a-46aa54518ed0︠ #auto cos(tae).show(); sin(tae).show(); ︡7eaa9644-c120-48da-af75-cc037f0736d7︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\cos\\left(\\sigma_{1} - \\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right) + \\frac{1}{2} \\, \\log\\left(\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1} - \\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right) + \\frac{1}{2} \\, \\log\\left(\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right)
"}︡ ︠e7966cc9-516d-49bc-8e00-bdc7e614ee2d︠ #auto #assume(sinh(r0)>0); (cos(ttae)).simplify_full().factor().simplify_full().show() ︡35390801-f618-49ff-8cbc-c5b335468d62︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left(\\left(2 i\\right) \\, \\cos\\left(\\sigma_{1}\\right) \\cos\\left(\\sigma_{2}\\right)^{2} \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right) - \\sin\\left(\\sigma_{1}\\right) \\sin\\left(\\sigma_{2}\\right) \\cos\\left(\\sigma_{2}\\right) - i \\, \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\cos\\left(\\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right) + {\\left(\\left(2 i\\right) \\, \\sin\\left(\\sigma_{1}\\right) \\cos\\left(\\sigma_{2}\\right)^{2} \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right) - i \\, \\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\sin\\left(\\sigma_{2}\\right) \\cos\\left(\\sigma_{1}\\right) \\cos\\left(\\sigma_{2}\\right)\\right)} \\sin\\left(\\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right)\\right)} \\sin\\left(\\frac{1}{2} \\, \\log\\left(\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right) + {\\left({\\left(\\left(-2 i\\right) \\, \\cos\\left(\\sigma_{1}\\right) \\cos\\left(\\sigma_{2}\\right)^{2} \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\sin\\left(\\sigma_{1}\\right) \\sin\\left(\\sigma_{2}\\right) \\cos\\left(\\sigma_{2}\\right) + i \\, \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)} \\sin\\left(\\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right) + {\\left(\\left(2 i\\right) \\, \\sin\\left(\\sigma_{1}\\right) \\cos\\left(\\sigma_{2}\\right)^{2} \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right) - i \\, \\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\sin\\left(\\sigma_{2}\\right) \\cos\\left(\\sigma_{1}\\right) \\cos\\left(\\sigma_{2}\\right)\\right)} \\cos\\left(\\frac{1}{2} \\, \\log\\left(-\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right)\\right)} \\cos\\left(\\frac{1}{2} \\, \\log\\left(\\sin\\left(\\sigma_{2}\\right) \\sinh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)\\right)\\right)}{\\sqrt{-2 \\, \\cos\\left(\\sigma_{2}\\right)^{2} \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\sin\\left(\\sigma_{2}\\right) \\cos\\left(\\sigma_{2}\\right) + \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)} \\sqrt{2 \\, \\cos\\left(\\sigma_{2}\\right)^{2} \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\sin\\left(\\sigma_{2}\\right) \\cos\\left(\\sigma_{2}\\right) - \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\mbox{ro}\\right)}}
"}︡ ︠f23c7f56-5461-4a58-b1ed-598fa56a333b︠ #auto dtxe=diff(xe,sig1); dsxe=diff(xe,sig2); dtxe; ︡a7c3d5e6-73b1-41cf-a826-430b433ebcb5︡{"stdout": "(-sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2)*sin(sigma1 - 1/2*log(-sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro)) + 1/2*log(sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro))), sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2)*cos(sigma1 - 1/2*log(-sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro)) + 1/2*log(sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro))), -sqrt(sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2) - 1)*sqrt(sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2) + 1)*sin(sigma1 - 1/2*log(-(sin(sigma2)*sinh(ro) - cos(sigma2)*cosh(ro))/(sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro))) + arctan(-I*sinh(2*ro)/tan(2*sigma2))), sqrt(sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2) - 1)*sqrt(sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2) + 1)*cos(sigma1 - 1/2*log(-(sin(sigma2)*sinh(ro) - cos(sigma2)*cosh(ro))/(sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro))) + arctan(-I*sinh(2*ro)/tan(2*sigma2))))"}︡ ︠a2ed4215-92ca-4cb4-b9cb-8b6af359545b︠ #auto sum(dtxe[i]*dsxe[i]*adsv[i] for i in range(0,4)).simplify_full() ︡dfeecbbd-02a1-4d00-9d6e-6d2ea189f723︡{"stdout": "(((4*I + 4)*cosh(ro)^5 + (-4*I - 4)*cosh(ro)^3 + (I + 1)*cosh(ro))*cos(sigma2)^4*sinh(ro) + ((-4*I - 4)*cosh(ro)^5 + (4*I + 4)*cosh(ro)^3 + (-I - 1)*cosh(ro))*cos(sigma2)^2*sinh(ro) + ((I + 1)*cosh(ro)^5 + (-I - 1)*cosh(ro)^3)*sinh(ro))/(sin(sigma2)^6*sinh(ro)^4*cosh(ro)^2 - cos(sigma2)^6*sinh(ro)^2*cosh(ro)^4 - (2*sinh(ro)^4*cosh(ro)^2 + (cosh(ro)^4 + 1)*sinh(ro)^2)*sin(sigma2)^4*cos(sigma2)^2 + (sinh(ro)^4*cosh(ro)^2 + 2*sinh(ro)^2*cosh(ro)^4 + cosh(ro)^2)*sin(sigma2)^2*cos(sigma2)^4)"}︡ ︠698b7bbd-bd89-48e7-ae72-4a2508d58972︠ ((((4*I + 4)*cosh(ro)^5 - (4*I - 4)*cosh(ro)^3 + (I + 1)*cosh(ro))*cos(sigma2)^4*sinh(ro) + (-(4*I - 4)*cosh(ro)^5 + (4*I + 4)*cosh(ro)^3 - (I - 1)*cosh(ro))*cos(sigma2)^2*sinh(ro) + ((I + 1)*cosh(ro)^5 - (I - 1)*cosh(ro)^3)*sinh(ro))).factor().simplify_full().show() ︡adae7baa-2375-4306-a9ae-77e3c31ef92b︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(\\left(4 i + 4\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{5} + \\left(-4 i + 4\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{3} + \\left(i + 1\\right) \\, \\cosh\\left(\\mbox{ro}\\right)\\right)} \\cos\\left(\\sigma_{2}\\right)^{4} \\sinh\\left(\\mbox{ro}\\right) + {\\left(\\left(-4 i + 4\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{5} + \\left(4 i + 4\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{3} + \\left(-i + 1\\right) \\, \\cosh\\left(\\mbox{ro}\\right)\\right)} \\cos\\left(\\sigma_{2}\\right)^{2} \\sinh\\left(\\mbox{ro}\\right) + {\\left(\\left(i + 1\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{5} + \\left(-i + 1\\right) \\, \\cosh\\left(\\mbox{ro}\\right)^{3}\\right)} \\sinh\\left(\\mbox{ro}\\right)
"}︡ ︠7c38650a-b425-4ef1-8d3e-c87c927a7294i︠ %html

calculemos α y P(σ\pm) correspondiente al caso lorentziano

︡0e4877f4-4741-4e9d-8bd9-0db893a9c59a︡{"html": "

calculemos α y P(σ\\pm) correspondiente al caso lorentziano

"}︡ ︠ad061ee6-3e81-4773-b696-a94fcf5aba4e︠ #auto i, j, k, l, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4 = var('i, j, k, l, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4') ︡3ac078c2-96d2-48b7-a26a-2dccf9885b1a︡︡ ︠d41a259c-6af0-4f63-adb3-204f822f7e35︠ #auto eps=((j-i)*(k-i)*(l-i)*(k-j)*(l-j)*(l-k))/12 ︡f8bcb039-d0d3-4e23-a1dc-46d37bb44065︡︡ ︠82b9a5f2-953b-4606-9777-8502fb3021ff︠ eps(2,4,1,3) ︡13acafca-e076-4b54-aa51-e3364a27fb50︡{"stdout": "__main__:3: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)\n-1"}︡ ︠f54e2bdb-aeb0-42a2-aed9-1e32a2a64280︠ #auto dsp = dsxn+dtxn; dsm = dsxn-dtxn; dsp[1].show() ︡e2eeea58-e0c7-4ba5-8d4f-8909829952a1︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right) + \\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)
"}︡ ︠bae6d854-159e-47ec-87a1-da969e76b8d1︠ adsv=vector([-1,-1,1,1]); ︡76c79d8e-5381-4d20-aa7a-1cd92598cc97︡︡ ︠11b28cfa-67fa-4d82-aff8-bea619f6a65e︠ #auto N = vector(SR,[1,2,3,4]); for i in range(0,4): N[i]=((1/2)*sum(sum(sum(eps(i,j,k,l)*xs[j]*adsv[j]*dsp[k]*adsv[k]*dsm[l]*adsv[l] for j in range(0,4)) for k in range(0,4)) for l in range(0,4))).simplify_full() ︡e89115be-224f-43fa-b458-ed396c776aa7︡︡ ︠0d8134bb-3b77-4e83-acda-7a03698581fa︠ #auto for i in range(0,4): N[i].show() ︡3d9cd20b-5042-4d14-afbb-9cab5f35f234︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) - \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) - \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right) - \\cos\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)
"}︡ ︠59899698-1419-4775-9bfb-252f156e8e8b︠ #auto for i in range(0,4): xs[i].show() ︡cdeeb8cc-093b-48b6-9b52-68aff7b9e2da︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\cosh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\sinh\\left(\\sigma_{2}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right) - \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right)
"}︡ ︠f6491c15-5ea1-4133-9151-0b6d2ab2c879︠ #auto prue=vector(SR,[a1, a2, a3, a4]); prue1=vector(SR,[b1, b2, b3, b4]); prue2=vector(SR,[c1, c2, c3, c4]); adsv=vector([-1,-1,1,1]); ︡46107bba-bd0b-4001-aed0-31558ac0cacb︡︡ ︠9d889328-3a76-4cea-bff8-a81beb41273f︠ #auto sum(dtxn[i]*dsxn[i]*adsv[i] for i in range(0,4)).simplify_full() ︡461c1714-4c0a-4129-a35e-2a621febfb8d︡{"stdout": "0"}︡ ︠fc6e2e5c-97b0-431c-9313-f7362cfb7bea︠ #auto sum(dsp[i]*dsp[i]*adsv[i] for i in range(0,4)).simplify_full() ︡26600503-df81-4ee6-90b7-5471fbfec773︡{"stdout": "0"}︡ ︠029ee62b-cdc6-4469-bc21-84697651cb83︠ #auto sum(dsm[i]*dsm[i]*adsv[i] for i in range(0,4)).simplify_full() ︡2657533c-ced8-4d4d-b73a-9d218e47ad70︡{"stdout": "0"}︡ ︠d64af336-fec4-471a-b083-0e6b7e958b85︠ #auto sum(sum(sum(eps(0,j,k,l)*prue[j]*prue1[k]*prue2[l] for j in range(0,4)) for k in range(0,4)) for l in range(0,4)) ︡4e1f56d8-0b6d-4163-b985-def1eb5c2366︡{"stdout": "a2*b3*c4 - a2*b4*c3 - a3*b2*c4 + a3*b4*c2 + a4*b2*c3 - a4*b3*c2"}︡ ︠aff8605f-da6f-474b-9da3-8a47d524653ei︠ %html

Observe que Y es ortogonal a ∂Y \bar∂Y

︡3f707dfc-c318-42d8-ae2c-e2a700d7c1d2︡{"html": "

Observe que Y es ortogonal a ∂Y \\bar∂Y

"}︡ ︠9e6c56b4-a31d-4591-92eb-99f59b39b9e5︠ #auto sum(dsp[i]*xs[i]*adsv[i] for i in range(0,4)).simplify_full() ︡9c7af06d-35f1-4530-b972-51a21f35cc91︡{"stdout": "0"}︡ ︠f0e710df-5d78-41ed-8e12-706190b671ad︠ #auto sum(dsm[i]*xs[i]*adsv[i] for i in range(0,4)).simplify_full() ︡c17c6a5f-6613-429c-98be-5109843cc14a︡{"stdout": "0"}︡ ︠9e0163d2-ce31-4bc3-b9c5-59aa3b749b31i︠ %html

Observe que N es ortogonal a ∂Y \bar∂Y

︡e0a408aa-19bf-4767-92ea-0c5023186e1d︡{"html": "

Observe que N es ortogonal a ∂Y \\bar∂Y

"}︡ ︠d4061a1b-d272-41f1-8ebe-561f25255945︠ #auto sum(N[i]*xs[i]*adsv[i] for i in range(0,4)).simplify_full() ︡e8a616af-b7e5-4adb-a683-b1a6ce127c22︡{"stdout": "0"}︡ ︠0dd7023a-c32e-4067-a872-7b40b6a02767︠ #auto sum(N[i]*dsm[i]*adsv[i] for i in range(0,4)).simplify_full() ︡e77186ae-7995-4fc6-8ab0-6b5aaa62f034︡{"stdout": "0"}︡ ︠b1ef20e2-f821-467e-8b0f-105c2284826d︠ #auto sum(N[i]*dsp[i]*adsv[i] for i in range(0,4)).simplify_full() ︡1e439472-66bb-4476-b65c-fb46f5e05c8e︡{"stdout": "0"}︡ ︠785c28e8-59e5-497c-b616-56981b0eeb97︠ #auto sum(N[i]*N[i]*adsv[i] for i in range(0,4)).simplify_full() ︡7777424e-8040-4302-80ab-844d7d6ae9be︡{"stdout": "1"}︡ ︠43ec3e2c-309c-4dc4-b66c-ab1e839f5061i︠ %html

Ahora calculamos p(σ\pm)

︡6efc7dfd-133b-4133-bd6f-e2a8c7bbc6cc︡{"html": "

Ahora calculamos p(σ\\pm)

"}︡ ︠667ab966-3d33-4d3c-b556-5b4335bf8401︠ #auto ddsp =(1/4)*(diff(dsp,sig1)+diff(dsp,sig2)); ddsm =(1/4)*(diff(dsm,sig2)-diff(dsm,sig1)); ddsp[1].show(); ddsm[1].show(); ︡69c5e618-5670-4786-8239-2b4f4e2f83aa︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, \\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) + \\frac{1}{2} \\, \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)
"}︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sin\\left(\\sigma_{1}\\right) \\sinh\\left(\\mbox{ro}\\right) \\cosh\\left(\\sigma_{2}\\right) - \\frac{1}{2} \\, \\cos\\left(\\sigma_{1}\\right) \\sinh\\left(\\sigma_{2}\\right) \\cosh\\left(\\mbox{ro}\\right)
"}︡ ︠13587d7e-5c2d-4ffa-9cbb-539d37c0d304︠ #auto p=sum((1/2)*ddsp[i]*N[i]*adsv[i] for i in range(0,4)).simplify_full(); p.show() ︡9d3469ef-211e-4100-b76b-40002194786e︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4}
"}︡ ︠1994761f-e931-4b9d-af71-133aa47dd3b4︠ #auto pb=sum((1/2)*ddsm[i]*N[i]*adsv[i] for i in range(0,4)).simplify_full(); pb.show() ︡535c3259-c772-48f2-a47e-8325ad7a380c︡{"html": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{4}
"}︡