Η Εξίσωση Της Αγάπης

Κυριακή 13 Φεβρουαρίου 2011

Η Εξίσωση Της Αγάπης

Input:

$x^2+(y-x^2^(1\/3))^2 = 1$

Implicit plot:

Alternate forms:

$(x^2^(1\/3)-y)^2+x^2 = 1$

$-2 x^2^(1\/3) y+x^2+(x^2)^(2\/3)+y^2 = 1$

$x^2+(y-x^(2\/3) e^((2 i pi (left floor)1\/2-(arg(x))\/pi(right floor))\/3))^2 = 1$

RawBoxes[RowBox[{arg, RowBox[{(, z, )}]}]] is the complex argument

RawBoxes[TemplateBox[{x}, Floor]] is the floor function

Alternate form assuming x and y are positive:

$(x^(2\/3)-y)^2+x^2 = 1$

Integer solutions:

Solutions for the variable y:

$y = x^2^(1\/3)-sqrt(1-x^2)$

$y = sqrt(1-x^2)+x^2^(1\/3)$

Implicit derivatives:

$((partial d)x(y))\/((partial d)y) = -(3 (y-8 x^2 y+26 x^4 y-46 x^6 y+51 x^8 y-38 x^10 y+19 x^12 y-6 x^14 y+x^16 y-4 x^2 y^2+24 x^4 y^2-88 x^6 y^2+164 x^8 y^2-148 x^10 y^2+52 x^12 y^2-4 x^14 y^2-8 y^3+56 x^2 y^3-158 x^4 y^3+240 x^6 y^3-222 x^8 y^3+130 x^10 y^3-46 x^12 y^3+8 x^14 y^3+40 x^2 y^4-200 x^4 y^4+540 x^6 y^4-680 x^8 y^4+340 x^10 y^4-40 x^12 y^4+28 y^5-168 x^2 y^5+336 x^4 y^5-308 x^6 y^5+233 x^8 y^5-84 x^10 y^5+28 x^12 y^5-140 x^2 y^6+560 x^4 y^6-1036 x^6 y^6+756 x^8 y^6-140 x^10 y^6-56 y^7+280 x^2 y^7-284 x^4 y^7+8 x^6 y^7-4 x^8 y^7+56 x^10 y^7+240 x^2 y^8-720 x^4 y^8+548 x^6 y^8-240 x^8 y^8+70 y^9-280 x^2 y^9+90 x^4 y^9+50 x^6 y^9+70 x^8 y^9-220 x^2 y^10+440 x^4 y^10-220 x^6 y^10-56 y^11+168 x^2 y^11-38 x^4 y^11+56 x^6 y^11+104 x^2 y^12-104 x^4 y^12+28 y^13-56 x^2 y^13+28 x^4 y^13-20 x^2 y^14-8 y^15+8 x^2 y^15+y^17))\/(x (3-22 x^2+66 x^4-112 x^6+125 x^8-94 x^10+49 x^12-16 x^14+3 x^16-24 y^2+156 x^2 y^2-414 x^4 y^2+612 x^6 y^2-570 x^8 y^2+342 x^10 y^2-126 x^12 y^2+24 x^14 y^2-8 y^3+96 x^2 y^3-528 x^4 y^3+1312 x^6 y^3-1480 x^8 y^3+624 x^10 y^3-56 x^12 y^3+84 y^4-474 x^2 y^4+1080 x^4 y^4-1332 x^6 y^4+975 x^8 y^4-414 x^10 y^4+84 x^12 y^4+48 y^5-480 x^2 y^5+1944 x^4 y^5-3264 x^6 y^5+2040 x^8 y^5-288 x^10 y^5-168 y^6+672 x^2 y^6-924 x^4 y^6+932 x^6 y^6-420 x^8 y^6+168 x^10 y^6-120 y^7+960 x^2 y^7-2664 x^4 y^7+2592 x^6 y^7-600 x^8 y^7+210 y^8-426 x^2 y^8+18 x^4 y^8-12 x^6 y^8+210 x^8 y^8+160 y^9-960 x^2 y^9+1096 x^4 y^9-640 x^6 y^9-168 y^10+108 x^2 y^10+90 x^4 y^10+168 x^6 y^10-120 y^11+480 x^2 y^11-360 x^4 y^11+84 y^12-38 x^2 y^12+84 x^4 y^12+48 y^13-96 x^2 y^13-24 y^14+24 x^2 y^14-8 y^15+3 y^16))$

$((partial d)y(x))\/((partial d)x) = (-3 x+22 x^3-66 x^5+112 x^7-125 x^9+94 x^11-49 x^13+16 x^15-3 x^17+24 x y^2-156 x^3 y^2+414 x^5 y^2-612 x^7 y^2+570 x^9 y^2-342 x^11 y^2+126 x^13 y^2-24 x^15 y^2+8 x y^3-96 x^3 y^3+528 x^5 y^3-1312 x^7 y^3+1480 x^9 y^3-624 x^11 y^3+56 x^13 y^3-84 x y^4+474 x^3 y^4-1080 x^5 y^4+1332 x^7 y^4-975 x^9 y^4+414 x^11 y^4-84 x^13 y^4-48 x y^5+480 x^3 y^5-1944 x^5 y^5+3264 x^7 y^5-2040 x^9 y^5+288 x^11 y^5+168 x y^6-672 x^3 y^6+924 x^5 y^6-932 x^7 y^6+420 x^9 y^6-168 x^11 y^6+120 x y^7-960 x^3 y^7+2664 x^5 y^7-2592 x^7 y^7+600 x^9 y^7-210 x y^8+426 x^3 y^8-18 x^5 y^8+12 x^7 y^8-210 x^9 y^8-160 x y^9+960 x^3 y^9-1096 x^5 y^9+640 x^7 y^9+168 x y^10-108 x^3 y^10-90 x^5 y^10-168 x^7 y^10+120 x y^11-480 x^3 y^11+360 x^5 y^11-84 x y^12+38 x^3 y^12-84 x^5 y^12-48 x y^13+96 x^3 y^13+24 x y^14-24 x^3 y^14+8 x y^15-3 x y^16)\/(3 y (1-8 x^2+26 x^4-46 x^6+51 x^8-38 x^10+19 x^12-6 x^14+x^16-4 x^2 y+24 x^4 y-88 x^6 y+164 x^8 y-148 x^10 y+52 x^12 y-4 x^14 y-8 y^2+56 x^2 y^2-158 x^4 y^2+240 x^6 y^2-222 x^8 y^2+130 x^10 y^2-46 x^12 y^2+8 x^14 y^2+40 x^2 y^3-200 x^4 y^3+540 x^6 y^3-680 x^8 y^3+340 x^10 y^3-40 x^12 y^3+28 y^4-168 x^2 y^4+336 x^4 y^4-308 x^6 y^4+233 x^8 y^4-84 x^10 y^4+28 x^12 y^4-140 x^2 y^5+560 x^4 y^5-1036 x^6 y^5+756 x^8 y^5-140 x^10 y^5-56 y^6+280 x^2 y^6-284 x^4 y^6+8 x^6 y^6-4 x^8 y^6+56 x^10 y^6+240 x^2 y^7-720 x^4 y^7+548 x^6 y^7-240 x^8 y^7+70 y^8-280 x^2 y^8+90 x^4 y^8+50 x^6 y^8+70 x^8 y^8-220 x^2 y^9+440 x^4 y^9-220 x^6 y^9-56 y^10+168 x^2 y^10-38 x^4 y^10+56 x^6 y^10+104 x^2 y^11-104 x^4 y^11+28 y^12-56 x^2 y^12+28 x^4 y^12-20 x^2 y^13-8 y^14+8 x^2 y^14+y^16))$

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[via]Computed by Wolfram Mathematicahttp://www.wolframalpha.com

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Κυριακή 13 Φεβρουαρίου 2011