円の数学

初版：2008.11.11
Update:2011.11.24

中心角と円周角の関係^{１），２）}　

円群「(x-a)²+y²=a²」に直交する曲線群

円群 (x-a)2+y2=a2(a>0) 2ydy+2(x-a)dx=0 y'=-(x-a)/x 直交曲線群の微分方程式 y'=-x/(x-a)

(%i1) f:(x-a)^2+y^2-a^2=0;        diff(f); (%o1) y^2+(x-a)^2-a^2=0 (%o2) 2*y*del(y)+2*(x-a)*del(x)+(-2*(x-a)-2*a)*del(a)=0 (%i3) f:(x-a)^2+y^2-a^2=0;        solve(diff(f),del(y)); (%o3) y^2+(x-a)^2-a^2=0 (%o4) [del(y)=-((x-a)*del(x)-x*del(a))/y] (%i5) yd:-(x-a)/y;expand(f); (%o5) (a-x)/y (%o6) y^2+x^2-2*a*x=0 (%i7) yd:-(x-a)/y;solve(expand(f),a); (%o7) (a-x)/y (%o8) [a=(y^2+x^2)/(2*x)] (%i9) factor(ev(yd,a=(y^2+x^2)/(2*x))); (%o9) ((y-x)*(y+x))/(2*x*y) (%i10) yd1:factor(ev(yd,a=(y^2+x^2)/(2*x))); (%o10) ((y-x)*(y+x))/(2*x*y) (%i11) ydd:-1/yd1; (%o11) -(2*x*y)/((y-x)*(y+x)) (%i12) dy/dx=ydd; (%o12) dy/dx=-(2*x*y)/((y-x)*(y+x)) (%i13) diff(z*y); (%o13) y*del(z)+z*del(y) (%i14) diff(z*x); (%o14) x*del(z)+z*del(x) (%i15) z+x*diff(z,x)=(2*z)/(1-z^2); (%o15) z=(2*z)/(1-z^2) (%i16) x*dz/dx=factor((2*z)/(1-z^2)-z); (%o16) (dz*x)/dx=-(z*(z^2+1))/((z-1)*(z+1)) (%i17) dz*partfrac((1-z^2)/(z*(1+z^2)))=-dx/x; (%o17) (dz*(1-z^2))/(z*(z^2+1))=-dx/x (%i18) dz*partfrac((1-z^2)/(z*(1+z^2)),z)=-dx/x; (%o18) dz*(1/z-(2*z)/(z^2+1))=-dx/x (%i19) log(z)-log(1+z^2)=-log(x)+c1; (%o19) log(z)-log(z^2+1)=c1-log(x) (%i20) factor(ev(z/(1+z^2)=c/x,z=y/x)); (%o20) (x*y)/(y^2+x^2)=c/x (%i21) x^2+y^2-x^2*y/c=0; (%o21) y^2-(x^2*y)/c+x^2=0 (%i22) factor(ev(abs(z)/(1+z^2)=c/abs(x),z=y/x)); (%o22) (x^2*abs(y))/(abs(x)*(y^2+x^2))=c/abs(x) (%i23) x^2*abs(y)=c*(x^2+y^2); (%o23) x^2*abs(y)=c*(y^2+x^2)

正方形と円弧で分割された領域の面積

領域（一辺10の正方形に対する緑の部分	面積S
	S1=(π*102/4)*2-102   =50(π-2)
	S2=100+100π/3-100√3
	S3=25π/3 +50√3 -100
	S4=25(√3 -π/3)
	S5=100 -50π/3 -25√3
	S6=100-25π
solve([S1=50*(%pi-2),S1=S2+2*S3, S4*2+S3+S2=25*%pi,S5+S3=S4, S4+%pi*100/6-25*sqrt(3)=%pi*100/12], [S1,S2,S3,S4,S5]);
	S7=25(5π/6 -√3)
	S8=(125π/4)-50-(75/2)sin-1(3/5) ≒24.04347884493286
	S9=(100π/3)-25√3 -75sin-1(3/5) ≒13.15590177094152
	S10=S8-S9  =25√3 +(75/2)sin-1(3/5) -50-(25π/12) ≒10.88757707399136
	S11=(25π/2)-S8  =50+(75/2)sin-1(3/5) -(75π/4) ≒15.22642932493955
	S12=25π/2
	S13=
	S14=
	S15=25(π-2)/2
	S16=75sin-1(3/5)-(25π/2) ≒8.992674989623914
	S17=S15-S16  =25(π-1)-75sin-1(3/5)
	S18=S8-S15 =S8-25(π-2)/2
	S19=25π/4-S15  =25(4-π)/4
	S20=S16=25(π-2)/2
	S21=
	S22=100-25π+S16  =75-25π/2
	S23=
	S24=
	S25=
	S26=
	S27=

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初版：2008.11.11
Update:2011.11.19/21/23/24