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Memorandum of Mathematical Physics

Cat: SCI
Pub: 2011
#2010

Takumi (of Yobinori)

20625u
Title

Memorandum of Mathematical Physics

数理物理メモ

Index
  1. Preface:
  2. Equation of motion (EOM)
  3. Pendulum isochronism:
  4. Conservation law:
  5. Damping oscillation:
  6. Conservation law of angular momentum:
  7. Inertial frame:
  8. Coriolis force:
  9. Archimedean spiral
  10. Hyperbolic function:
  11. Inverse trigonometric function:
  12. Two-particle system motion:
  13. Rigid body dynamics:
  14. Inertia moment of rigid body:
  15. Mechanical energy of rigid body:
  16. Matrix exponential:
  17. Chaucy's functional equation:
  18. Taylor expansion:
  19. Fourier expansion:
  20. Linear algebra:
  21. Delta function:
  22. Gaussian function:
  23. Vector analysis:
  24. Integration method:
  25. Multiple integral:
  1. 序文:
  2. 運動方程式
  3. 振り子の等時性:
  4. 保存則:
  5. 減衰振動:
  6. 角運動量保存則:
  7. 慣性系:
  8. コリオリの力:
  9. アルキメデスの螺旋:
  10. 双曲線関数:
  11. 逆三角関数:
  12. 二粒子系の運動:
  13. 剛体の力学:
  14. 剛体の慣性モーメント:
  15. 剛体の力学エネルギー:
  16. 行列指数関数:
  17. コーシーの関数方程式:
  18. テイラー展開:
  19. フーリエ展開:
  20. 線形代数:
  21. デルタ関数:
  22. ガウス関数:
  23. ベクトル解析:
  24. 積分法:
  25. 重積分:
Tag
; Angular momentum; Azmuthal direction; Center of gravity; Coriolis force; Commutative law; Conservation law; Constant area velocity; Critical damping; Damped oscillation; Diagonalization; Eigenvalue; EOM; Equivalence principle; Forced oscillation; Fourier expansion; Inertial frame; Integral method; Integration by parts; Inverse matrix ; Jacobian; Multiple integral; Orthogonal axes theorem; Parallel axes theorem; Pendulum isochronism; Rank of matrix; Reduced mass; Relative vector; Rigid body dynamics; Rotation; Simple harmonic motion; Solid angle; Spherical coordinates; Taylor expansion; Torque; Transposed matrix; Uniform linear motion; Vector product; ;
Why
  • Physical phenomena could have been more easily understood with differential equations; the world is written in DE.
  • Dinosaurs recognized all movings are eatable. They know movement by the acceleration.
Original resume
Remarks

>Top 0. Preface:

  • LaTex is convenient in writing of mathematical expression, though it need some kind of patience.

0. 序文:

>Top 1. Equation of motion (EOM):

  • ¶1: Free-fall motion:
    • md2rdt2=F [2nd-order LDE]
    • my=mg [m=mass]
      y=gt+Cy=gt2+Ct+C
      y(0)=0C=0;y(0)=y0=C [initial condition]
    • y(t)=12gt2+y0

  • ¶2: >Top Parabolic motion:
    • {mx=0my=mg
    • mx=0x=0x=C=v0cosθ
    • x(0)=0C=0
      • x=Ct+C=v0cosθt [uniform linear motion]
    • my=mgy=gy=gt+C=gt+v0sinθ
      • y=12gt2+v0sinθt+C
    • y(0)=0=Cy(t)=12gt2+v0sinθt [uniform acceleration motion]■

  • ¶3: Motion with air resistance:
    • mx=kx
    • let: x=vmv=kvv=kmv
    • v(0)=v0=Cv(t)=v0ekmt
    • x=vv0ekmt
      x=kmv0ekmt+C
      • x(0)=0→\;C'=\frac{m}{k}v_0
    • x(t)=mkv0(1ekmt)

  • ¶4: >Top Simple harmonic motion:
    • mx=kx...(*) [k: sprint constant; x: displacement from equilibrium position]
      x=kmx=ω2x;$here,\omega=\sqrt{\frac{m}{k}}$
    • assume GS of (*): x=C1sint+C2cost=Csinωt1+C2cosωt [linear combination]
      • x=Csinωt1+C2cosωt
    • initial condition: {x(0)=0x(0)=v0
      {C2=0C1ω=V0
      • C1=v0ω,C2=0
    • x(t)=v0ωsinωt

  • ¶5: Uniform circular motion:
    • [r0=raidus, ω0=angular velocity]
    • v=rer+rϕeϕ=0er+r0ω0eϕ=r0ω0eϕ
    • a=(rrϕ2)er+(rϕ+2rϕ)eϕ=r0ω20er=v20r02er
    • [using EOM]
      F=ma=mr0ω20er [centripetal]

1. 運動方程式:

  • radial direction: 動径方向
  • azmuthal direction: 方位角方向 <Ar. as-sumut
  • pendulum isochronism: 振り子の等時性

 

  • r(x): trace of position:
  • to know r(t) by integration
  • ¶3.
  • attenuation
  • Harmonic Motion:
  • harmonicmotion
  • Polar coordinate system:
    {er=ϕereϕ=ϕer
  • <Rec. cord.>
    {r=xex+yeyv=xex+yeya=xex+yey
  • <Por. cord.>
    {r=rerv=rer+rϕeϕa=(rrϕ2)er+(rϕ+2rϕ)eϕ
  • Taylor expansion:
    $\sin x=x-\frac{x^3}{3!}
    +\frac{x^5}{5!}-\dots

>Top 2. Pendulum isochronism:

  • Single pendulum: [|ϕ|<<1]
    • [azmuth direction of EOM]
    • m(rϕ+2rϕ)=mgsinϕ=ml [m=mass; l=length of pendulum]
      mlϕ=mgsinϕ...(*)
      ϕ=glsinϕ [sinϕϕ]
      ϕ=glϕ
      let: ω=gl:
      ϕ=ω2ϕ [=uniform circular motion]
    • let: ϕ(t)=C1sinωt+C2cosωt
      here: ϕ(0)=ϕ0,ϕ=o [initial condition]
      C2=ϕ0,C1ω=0C1=0,C2=ϕ0 [ω0]
      ϕ(t)=ϕ0cosωt
    • here: T=2πω=2πlg[T=period]

  • Exact solution of single pendulum:
    • multiply ϕ:ϕϕ+glϕsinϕ=0
      ddt(12ϕ2glcosϕ)=0
      12ϕ2glcosϕ=E [E=constant; mechanical energy]
    • when: θ=θ0,θ=0
      E=glcosθ0
      12ϕ2gl(cosϕcosϕ0)
      ϕ2=2gl(cosϕcosϕ0)
      here: ϕ=2gl(cosϕcosϕ0)[0ϕϕ0]
    • T=4T40dt=40ϕ012gl(cosϕcosϕ0)dϕ
      =22lgϕ001cosϕcosϕ0dϕ
    • change of variables: θ such that sinϕ2=sinϕ02sinθ
      T=4lgπ20dθ1sin2ϕ02sin2θ
    • substitute to: sin(ϕ02)=a
      T=4ωπ20dθ1a2sin2θ=4ωK(a)[K:complete elliptic integral 1st]1

2. 振り子の等時性:

  • complete elliptic integral of the 1st kind: 第1種完全楕円積分
  • 1第1種完全楕円積分:
  • F(x,k)=x0dt(1t2)(1k2t2)
  • F(x,0)=x011t2dt=sin1x
  • F(x,1)=x011t2dt=tanh1x

= >Top 3. Conservation law:

  • Momentum conservation law:
    • md2rdt2=F
    • P=mdrdt\; [P=momentum]
      dpdt=FP(t2)P(t1)=t2t1F(t)dt
      P(t2)P(t1)=I [I=impulse=change of momentum]
    • when I=0:→P(t2)=P(t1) [conservation of momentum]

  • Energy conservation law:
    • W=Fcosθs=FS[Inner vector: W=work is constant]
    • dW=Fdr [W is changeable]
      W=cFdr
    • conservative force: W=rr0Fdr
    • potential energy: U(r)=rr0Fdr

  • Kinetic energy:
    • k=12m|v|2=12m(v2x+v2y+v2z)
      dkdt=mvdvdt
    • v=drdt,mdvdt=Fdkdt=Fdrdt
      t2t1dkdtdt=t2t1Fdrdtdt
      k(t2)k(t1)=cFdr
    • Δk=W
    • energy conservation law: k(t2)k(t1)=cFdr
    • [F=conservative force]:
      cFdr=r2r1Fdr=r2r0Fdrr1r0Fdr=U(r2)+U(r1)
    • mechanical energy:
      k(t2)k(t1)=U(r2)+U(r1)
      k(t2)+U(r2)=k(t1)+U(r1)
    • If F is conservative force, the potential energy is defined as:
      U(r)=rr0Fr...(*)
      then, U(r)U(r)=rr0Fdr(rr0Fdr=(r0rFdr+rr0Fdr)=rrFdr
    • thus, conservative force: W=ΔU

    • ¶1: Potential energy & Force [1 dimension]:
      • U(x)=xx0F(x)dx
      • U(x+Δx)U(x)=x+ΔxxF(x)dxF(x)Δx
        F(x)=limΔx0U(x+Δx)U(x)Δx=U(x)dx

    • ¶2: Potential energy & Force [3D]
      • U(r)=rr0F(r)dr
      • U(x+Δx,y+Δy,z+Δz)U(x,y,z)(FxΔx+FyΔy+FzΔz)
      • when: Δy=Δz=0
        U(x;Δx,y,z)U(x,y,z)FxΔx
        Fx=limΔx0U(x;Δx,y,z)U(x,y,z)Δx
      • thus: {Fx=U(r)xFy=U(r)yFz=U(r)z
      • F(r)=(U(r)x,U(r)y,U(r)z)=(x,y,z)U(r)=U(r)

    • ¶3: Harmonic oscillation (2D):
      • U(x,y)=12k(x2+y2)
        Fx=(x,y))x=kx
        Fy=(x,y))y=ky
        F(r)=k(x,y)=kr
      • =(x,y,z)×F=0

3. 保存則:

  • momentum: 運動量
  • impulse: 力積
  • inner product/vector product: 内積/外積
  • conservative force: 保存力
  • gradient: 勾配
  • harmonic oscillator: 調和振動子 =spring
  • integration by substitution: 置換積分
  • Harmonic oscillator:
  • harmonic oscillator
  • >Top Vector product:
    a×b=(aybzazby,azbxaxbz,axbyaybx)
  • vectorproduct

>Top 4. Damped oscillation:

  • Damping oscillation:
    • mx=kxbx [k=spring constant; b=damping coefficient]
      x=ω20x2γx[ω0=km,;γ=b2m]...(*)
    • assume:x=eλt=ω20eλt
      λ+2γλ+ω20=0
      λ±=ω±γ2ω20[λ is -real number]
    • 1) when: γ>ω0: [over-damped]
      assume GS of (*) is x=C1eλ+t+C2eλt
      here: x(0)=a,x(0)=0 [exponentially damped]
    • 2) when: γ<ω0:
      λ±=γ±γ2ω20=γ±iω20γ2=γ±iω
      assume GS of (*) is x=C1eγ+iw)t+C2eγiw)t
      =eγt(C1eiωt+C2eiωt)
      =eγt((C1+C2)cosωt+i(C1C2)sinωt)
      =eγt(Acosωt+iBsinωt)=Ceωtcos(ωt+ϕ);
      [C=A2+B2,tanϕ=BA]
    • 3) when: γ=ω0
      λ+=λ=λ
    • >Top assume GS of (*) is x=C(t)eγt
      then, x=CeγtC(γeγt)
      x=Ceγt2C(γeγt)+C(γ2eγt)
      substitute to (*):
      Ceγt2γCeγt+γ2Ceγt=ω20Ceγt+2γ2eγt2γCeγt
      C+C(ω2γ2)=0[eγt0]
      C=0[ω20γ2=0]
      C=At+B x=(At+B)eγt[critical damping]

    • Energetic consideration:
      mx=kxbx
      mxx=akxxbx2
      ddt(12mv2+12kx2)=bv2<0 [mechanical energy]
      bv2bvdxdt [work efficient]

    • >Top Forced oscillation:
      mx=kx+Fcosωt
      x=w20x+fcosωt...(*) [w0=km,f=Fm]
    • assume SS of (*): x=acosωt
    • substitute to (*): aω2cosωt=aω20cosωt+fcosωt
      a=fω2ω20
      SS of (*) is x=fω2ω20cosωt [resonance]
    • GS of (*)=GS of homo.eq + SS of (*)
      assume GS of (*): x=Asinω0t+Bcosω0tfω2ω20cosωt
    • consider: ωω0;initial condition: x(0)=0,x(0)=0
      a=0,B=fω2ω20
      x=fω2ω20(cosω0tcosωt)
      x=2f(ω+ω0)(ωω0)
      sinω0+ω2tsinω0ω2t
    • here: ω0ω=Δω
      x=2fω0+ωsinω0+ω2tt2sinΔω2tΔω2t
      Δω0x=ft2ω0sinω0t [increasing ]

4. 減衰振動:

  • damped oscillation/vibration: 減衰振動
  • damping coefficient:
  • over-damped: 過減衰
  • critical damping: 臨界減衰
  • restoring force: 復元力
  • external force: 外力
  • forced oscillation: 強制振動
  • resonance: 共振・共鳴
  • Euler's formula:
    eiθ=cosθ+isinθ
  • Damping oscillation:
  • dampingoscillation
  • Critical damping:
  • criticaldamping
  • limx0sinxx=1
  • Forced oscillation:
  • forcedoscillation

 

>Top 5. Conservation law of angular momentum:

  • moment of force (torque):
    • N=r×F[r=distance from the origin]
    • conservation of angular momentum:
      L=r×p[L=angular momentum; p=momentum]...(*)
    • differentiate (*): dLdt=drdt×p+r×dpdt=v×mv+r×F here $[\frac{dr}{dt}=v,\;\frac{p}{dt}=F]
    • dLdt=N[N=torque]
      thus, when Fr,N=0
      dLdt=0L=C [conservation of angular momentum]

  • Law of constant area velocity:
    • ΔS=12rvΔtsinϕ
      ΔSΔt12rvsinϕ...(*)
    • while: angular momentum of planetary movement:
      L=rmvsinϕ[L=angular momentum]...(**)
      L=2mΔSΔt;[ΔSΔt=area velocity]
      →\;[constant area velocity is one of law of constant angular momentum]

5. 角運動量保存則:

  • angular momentum: 角運動量
  • moment of force: 力のモーメント
  • 外積の微分:
  • ddt(a×b)=dadt×b+a×dbdt
  • Constant area velocity: ΔS=vΔtsinϕ
  • areavelocity

>Top 6. Inertial Frame:

  • Inertial/Noninertial frame:
    • r=R+r0
      substitute to mr=F:
      m(R+r0)=Fmr0=FmR...(*)
    • when, O0 is R=Vt [uniform linear motion]
      R=V,R=0
      substitute to (*): mr0=FS0 is inertial frame.
    • right side of (*) FmR looks like a kind of force: [inertial force]

  • >Top Equivalence principle:
    • Fg=mgg[mg=gravitational mass]
    • while: mia=F[mi=inertial mass]
    • in only gravitational field: mia=mgga=mgmig
      regardless of any object: a=gmg=mi [equivalence principle]

6. 慣性系:

  • noninertial frame: 非慣性系
  • uniform linear motion: 等速直線運動
  • equivalence principle: 等価原理
  • Inertial frame:
  • coliorisforce

>Top 7. Coriolis Force:

  • Rotating coordinate system:
    • r=xoeox+yoeoy=xex+yey...(*)
      here: {ex=cosωteox+sinωteoyey=sinωteox+cosωteoy
      {ex=ωsinωteox+ωcosωteoy=ωeyey=ωcosωteoxωsinωteoy=ωex
      {ex=ω2exey=ω2ey
    • here EOM of (*): F=mr
      r=xex+yey+2(xex+yey)+xex+yey
      =xex+yey+2ω(xeyyex)ω2(xex+yey)
    • substitute to (*)mr=F+mω2r+2mω(yexxey)
    • here: mω(yexxey)=2mωv×ez [Coriolis force]

7. コリオリの力:

  • Coriolis force: コリオリの力
  • apparent force: 見かけの力
  • centrifugal force: 遠心力
  • {v=(x,y,z)ez=(0,0,1)
    v×ez=(yx)
  • inertialframe

>Top 8. Archimedean spiral:

  • Length in polar coordinates:
    • dL=(rdθ)2+(dr)2...(*1)
      =r2+(drdθ)2dθ[r=f(θ)]
  • from (*1):
    • L=π0r2+(drdθ)2dθ
      =π0θ2+1dθ
    • let θ=sinhtdθ=coshtdt
      L=T0sinh2t+1coshtdt
      =T0cosh2tdt=T01+cosh2t2dt
      =12[t+12sinh2t]T0
      =12T+14sinh2T
      =12T+12sinhTcoshT
      =12log(π+π2+1)+12ππ2+16.1099
    • Re:
      • θ=ete12(et)22θet1=0
        et=θ+θ2+1t=log(θ+θ2+1)
      • θ0πt0log(π+π2+1)=T
      • sinhT=π[θ sinht]
      • coshT=sinh2T+1=π2+1

8. アルキメデスの螺旋:

  • polardistance

>Top 9. Hyperbolic function:

  • Hyperbolic function:
    • sinhx=exe12
    • coshx=ex+e12
    • tanhx=
    • (sinhx)=coshx
    • cosh2xsinh2x=1
    • cosh2x=1+cosh2x2
    • sinh2x=2sinhxcoshx
  • comparison:
    • cosh2xsinh2x=1cos2x+sin2x=1tanhx=sinhxcoshxtanx=sinxcosx1tanh2x=1cosh2x1+tan2x=1cos2x
    • (sinhx)=coshx(sinx)=cosx(coshx)=sinhx(cosx)=sinx(tanhx)=1cosh2x(tanx)=1cos2x
    • sinh(x+y)=sinhxcoshy+coshxsinhysin(x+y)=sinxcosy+cosxsiny
  • parametric:
    • x=coshθ,y=sinhθx2y2=1 [hyperbola]
    • cf: x=cosθ,y=sinθx2+y2=1 [circle]
  • Euler's formula:
    • eiθ=cosθ+isinθ
    • cosθ=eiθ+eiθ2
      sinθ=eiθeiθ2i

9. 双曲線関数:

  • hyperbolic function: 双曲線関数
  • catenary: 懸垂線
  • hyperbolicsin

>Top 10. Inverse trigonometric function:

  • y=arcsinx[1x1,π2yπ2]
    • x=sinydxdycosy=1sin2y
      dydx=11sin2y=11x2
  • y=arccosx[1x1,0yπ]
    • x=cosydxdy=siny=1cos2y
      dydx=11cos2y=11x2
  • y=arctanx[x,π2yπ2]
    • x=tanydxdy1cos2y=1+tan2y
      dydx=11+tan2y=11+x2

10: 逆三角関数:

  • arcsin.gif

>Top 11. Two-particle system motion:

  • EOM:
    • m1r1=F1+F12...(1)
    • m2r2=F2+F21...(2)
    • (1)+(2): m1r1+m2r2=F1+F2
      here: rG=m1r1+m2r2m1+m2=m1r1+m2r2M
      [G=center of gravity; M=total mass]
      MrG=F1+F2 [external force only]

  • >Top relative vector: r=r2r1 [viewpoint from r1]
    • r=r2r1=F2+F21m2F1+F12m1=(1m1+1m2)F21+F2m2F1m1
      =\frac{1}{\mu}+\frac{F_2}{m_2}+\frac{F_1}{m_1}
      [\mu:\; reduced mass] 
    • →\;\mu r''=F_{21}+\frac{\mu}{m_2}F_2-\frac{\mu}{m_1}F_1
      →\;\mu r''=F_{21}\; [unless external force]

  • Motion of connected objects: (>Fig.)
    • motion of center of gravity: Mx_G''=0\; [constant linear motion]
    • relative motion: \mu x''=-k(x_2-x_1-l) [\mu: reduced mass]
      here: \omega=\sqrt{\frac{k}{\mu}}\; [simple oscillation]

  • Total momentum:
    • P=p_1+p_2=m_1r_1'+m_2r_2'
    • P'=m_1r_1''+m_2r_2''=F_1+F_2
      →\;P'=0→\;P=C\; [unless external force]

  • Total kinetic energy:
    • K=k_1+k_2=\frac{1}{2}m_1r_1'^2+\frac{1}{2}m_2r_2'^2...(*)
    • here: \cases{r_G=\frac{m_1r_1+m_2r_2}{M}\\r=r_2-r_1}
      →\;\cases{r_1=r_G-\frac{m_2}{M}r\\r_2=r_G+\frac{m_1}{M}r}
    • substitute to (*):
      K=\frac{1}{2}m_1(r_g-\frac{m_2}{M}r)^2+\frac{1}{2}m_2(r_g+\frac{m_1}{M})^2
      →\;=\frac{1}{2}(m_1+m_2)r_G'^2 +\frac{1}{2}\frac{m_1m_2(m_1+m_2)}{M}r'^2
      →\;=\frac{1}{2}Mr_G'^2+\frac{1}{2}\mu Mr'^2 [momentum of COG & reduced mass]

  • Total angular momentum:
    • L=L_1+L_2=r_1\times p_1+r_2\times p_2=r_1\times m_1r_1' +r_2\times m_2r_2'...(*)
    • →\;L'=r_1'\times m_1r_1'+r_1\times m_1r_1''+r_2'\times m_2r_2'+r_2\times m_2r_2''
      =r_1\times (F1+F_{12})+r_2\times (F_2+F_{21})
      =(r_2-r_1)\times F_{21}+r_1\times F_1+r_2\times F_2 =r_1\times F_1+r_2\times F_2\; [each torque]
    • consider COG and substitute to (*):
      L=(r_G-\frac{m_2}{M}r)\times m_1(r_G'-\frac{m_2}{M}r') +(r_G+\frac{m_1}{M}r)\times m_2(r_G'+\frac{m_2}{M}r')
      =m_1r_G\times r_G'-\frac{m_1m_2}{M}r_G\times r' -\frac{m_1m_2}{M}r\times r_G'+\frac{m_1m_2^2}{M^2}r\times r'
      +m_2r_G\times r_G'+\frac{m_1m_2}{M}r_G\times r' +\frac{m_1m_2}{M}r\times r_G'+\frac{m_1^2m_2}{M^2}r\times r'
      =r_G\times (m_1+m_2)r_G'+r\times\frac{m_1m_2(m_1+m_2)}{M^2}r'
    • \therefore\;L=r_G\times Mr_G'+r\times\mu r'\; [torque of COG and torque of relative vector]

11. 二粒子系の運動:

  • relative vector: 相対位置ベクトル
  • reduced mass: 換算質量
  • total momentum: 全運動量
  • center of gravity: COG
  • twoparticle
  • relativevector
  • reducedmass
  • >Top Commutative law of vector product:
    \mathbf{a}\times (\mathbf{b}+\mathbf{c}) =\mathbf{a}\times \mathbf{b} +\mathbf{a}\times \mathbf{c}
  • <Summary>:
  • EOM of COG:
    • Mr_G''=F_1+F_2
  • Relative EOM:
    • \mu r''=F_{21} [no ex-force]
  • Total momentum:
    • P=p_1+p_2
      P=constant [without ex-force]
  • Total kinetic energy:
    • K=k_1+k_2
      \frac{1}{2}Mr_G'^2 +\frac{1}{2}\mu r'^2
  • Total angular momentum:
    • L=L_1+L_2
      =r\times Mr_G'+r\times\mu r'
    • L'=N_1+N_2 [torque]

>Top 12. Rigid body dynamics:

  • Rigid body: deformation is zero or negligible; the distance between any two point remains constant regardless of external forces exerted; continuous distribution of mass.
  • Rigid body of EOM:
    • P_G'=F [EOM of COG]
    • L'=N [rotation around the axis passing COG]
  • EOM of COG:
    • m_ir_i''=f_i+\displaystyle\sum_jFf_{ij}\; [ex-power+in-power;\;i≠j]
    • here; Mr_G''=\frac{\sum_im_ir_i}{\sum_im_i}\; [COG]
      • feature: 1) degree of freedom: 3n→6; 2) offset of internal-power
    • Mr_G''=\sum_if_i
      • here: P_G=Mr_G',\; F=\sum_if_i
        P_G'=F
  • Rotary motion:
    • P_i'=f_i+\sum_jf_{ij}\; [momentum=exforce + inforce] →\;r_i\times P_i'=r_i\times (f_i+\sum_jf_{ij})\;
      r_i\times p_i'=\frac{d}{dt}(r_i\times p_i)...(*1)
      • \because\; \frac{d}{dt}(r_i\times p_i)=r_i'\times p_i+r_i\times p_i'...(*2)
        here: r_i'\times p_i=0 [parallel vector]
      • here: \sum_i\sum_j(r_i\times f_{ij})=\sum_{i>j}(r_i-r_j)\times f_{ij}=0
      • \because\;if\; n=3→\;\sum_i\sum_j(r_i\times f_{ij})
        =r_1\times f_{12}+r_1\times f_{13}+r_1\times f_{21} +r_1\times f_{23}+r_1\times f_{31}+r_1\times f_{32} =0
    • from (*2): \frac{d}{dt}\sum_i(r_i\times p_i)=\sum_i(r_i\times f_i)...(*3)
      from (*3): L'=\sum_i(r_i\times p_i);\;N=\sum_i(r_i\times f_i)=\; [L= angular momentum; N=moment of force]
      \therefore\; \boxed{L'=N} [around the origin]

  • Relative position from COG:
    • here;r_G=\frac{\sum_im_ir_i}{M};\;r_G'=\frac{\sum_im_ir_i}{M} =frac{\sum_ip_i}{M}
    • substitute to (*3): r_i=r_o+r_G
      →\;\frac{d}{dt}\sum_i((r_{oi}+r_G)\times p_i)=\sum_i((r_{oi}+r_G)\times f_i)
      →\;\frac{d}{dt}\sum_i(r_{oi}\times p_i)+\frac{d}{dt}\sum_i(r_G\times p_i) =\sum_i(r_{oi}\times f_i)+\sum_i(r_G\times f_i)...(*4)
      • here: \frac{d}{dt}\sum_i(r_G\times p_i)=\sum_i(r_G'\times p_i+r_G\times p_i') =\sum_i(r_G\times p_i')
      • here: p_i'=f_i+\sum_jf_{ij}=f_i
      • (*4) is: \frac{d}{dt}\sum_i(r_{oi}\times p_i)=\sum_i(r_{oi}\times f_i)
    • →\;p_{oi}'=p_i-m_ir_G'
      →\;\frac{d}{dt}\sum_i(r_i'\times (p_i'+m_ir_G'))=\sum_i(r_{oi}\times f_i)
      →\;\frac{d}{dt}\sum_i(r_{oi}\times p_{oi}') +\frac{d}{dt}((\sum_im_ir_{oi})\times r_G')=\sum_i(r_{oi}\times f_i)
      • here: \sum_im_ir_{oi}=\sum_i(r_i-r_G)=\sum_im_ir_i-r_G\sum_im_i=0
      • here: L_o=\sum_ir_{oi}\times p_{oi};\; N_o=\sum_i(r_{oi}\times f_i)
        \therefore\; \boxed{L_o'=N_o} [rotary motion around COG]

12. 剛体の力学:

  • rigid body: 剛体
  • point mass: 質点
  • point of application: 作用点
  • line of action: 作用線
  • degree of freedom: 自由度
  • manifold: 多様体
  • deformation: 変形
  • nonlocal action; action at a distance: 遠隔作用
  • roll/yaw/pitch: position of a rigid body
  • Rotation around the axis passing COG:
  • \frac{d}{dt}(r_i\times p_i) =r_i'\times p_i +r_i\times p_i'
    here; r_i'\times p_i=0
  • Relative position:
  • relativeposition
  • 原点:
    L'=N
  • 重心:
    L_o'=N_o

>Top 13. Inertia moment of rigid body:

  • Inertia moment:
    • Motion with fixed axis: →degree of freedom is 1.
      • L_z'=N_z
      • here: L_z=\sum_i(r_i\times p_i)_z=\sum_i(x_iy_i'-y_ix_i') =\int \rho (r)(xy'-yx')d^3r ..(*)
      • here: x=r\cos\phi,\; y=r\sin\phi
        x'=-r\phi'\sin\phi,\; y'=r\phi'\cos\phi
        →\;xy'-yx'=r^2\phi'\cos^2\phi+r^2\phi'\sin^2\phi=r^2\phi'
    • substiture to (*): L_z=(\int r^2\rho (r)d^3r)\phi'
      =I_z\phi'\;[inertia moment around z-axis]
      \therefore\;\boxed{I_z\phi''=N_z}\; [I_z: difficulty of rotation]

  • Inertia moment of COG:
    • I=\int r^2\rho(r)d^3r...(*2)
  • >Top Parallel axes theorem:
    I=I_G+Mh^2\; [M=total mass; h=distance between axes]
    • here: r_o=r-r_G
    • substitute to (*2): I=\int r^2\rho(r)d^3r
      =\int (x^2+y^2)\rho(r)d^3r
      =((x_o+x_g)^2+(y_o+y_G)^2)\rho(r)d^3r
      =(x_g^2+y_g^2)\int \rho(r)d^3r+2x_g\int x_o\rho(r)d^3r
      ++2y_g\int y_o\rho(r)d^3r+\int (x_o^2+y_o^2)\rho(r)d^3r
      • here: \int x_o\rho(r)d^3r=\int(x-x_G)\rho(r)d^3r
        =\int x\rho(r)d^3r-x_g\int\rho(r)d^3r =Mx_g-x_gM=0
    • \therefore\; \boxed{I=I_g+Mh^2}
  • >Top Orthogonal axes theorem:
    • I_z=\in r^2\sigma(r)d^3r=\int (x^2+y^2)\sigma(r)d^2r
      =\int x^2\sigma(r)d^2r+\int y^2\sigma(r)d^2r=I_y+I_x
    • \therefore\; \boxed{I_z=I_x+I_y}

13. 剛体の慣性モーメント:

  • inertia moment: 慣性モーメント
  • orthogonal axis: 直交軸

  • Parallel axis theorem:
  • parallelaxes
  • Orthogonal axis:
    parallelplane

>Top 14. Mechanical energy of rigid body:

  • Kinetic energy of rigid body:
    • \sum_i\frac{1}{2}m_i|r_i'|^2
      =\sum_i\frac{1}{2}m_i|r_{oi}'+r_G'|^2
      =\frac{1}{2}\sum_im_i|r_G'|^2+\sum_im_ir_{oi}'r_G' +\frac{1}{2}\sum_im_i|r_{oi}|^2
      =\frac{1}{2}|r_g'|^2+r_g(\sum_im_ir_{oi}')+\frac{1}{2}\sum_im_i|r_{oi}|^2
      =translational motion + kinetic energy around COG [\because \sum_im_ir_{oi}=0]
    • here [motion around fixed axis passing COG]
      \frac{1}{2}\sum_im_i|r_{oi}'|^2=\frac{1}{2}\sum_im_i(r_i\omega)^2
      =\frac{1}{2}\omega^2\sum_im_ir_i^2=\frac{1}{2}\omega^2\int r^2\rho(r)d^3r
      =\frac{1}{2}I\omega^2
    • \therefore\; K=\frac{1}{2}Mv_G^2+\frac{1}{2}I\omega^2 [translational.e+rotational.e]
  • Potential energy of rigid body:
    • \sum_im_igz_i=g\sum_im_iz_i=Mgz_G
      \therefore\; \boxed{U=Mgh} [h=height of COG from the origin]

14. 剛体の力学的エネルギー:

  • translational motion: 並進運動

>Top 15. Matrix exponential:

  • Proof:
    • e^0=E+0+\frac{1}{2}0^2+\cdots=E
    • \frac{d}{dt}e^{At}=\frac{d}{dt}(E+At+\frac{1}{2}(At)^2+\cdots)
      =0+A+A^2t+\frac{1}{2!}A^3t^2+cdots
      =A(E+At+\frac{1}{2!}A^2+cdots)=Ae^{At}
  • \frac{d}{dt}\mathbf{y}(t)=A\mathbf{y}(t)
    • →\;y(t)=e^{At}y_0\; [y_0=y(0)]

15. 行列指数関数:

  • matrix exponential: 行列乗
  • commutative: 可換
  • Maclaurin expansion:
    e^x=1+x+\frac{1}{2!}x^2 +\frac{1}{3!}x^3+\cdots
    =\displaystyle\sum_{k=0}^{\infty} \frac{1}{k!}x^k
  • e^A=\displaystyle\sum_{k=0}^{\infty} \frac{1}{k!}A^k\; [A=square matrix]
    1. e^0=E
    2. \frac{d}{dt}e^{At}=Ae^{At}
    3. if AB=BA →e^Ae^B=e^{A+B}

>Top 16. Chauchy's functional equation:

  • definition: f(x+y)=f(x)+f(y)\;...(*) [real continuous function]
    • is f(x)=ax\; the only function to satisfy (*)?]
  • proof:
    1. when x=y=0→\;f(0)=f(0)+f(0)→\;f(0)=0\; [cross the origin]
    2. when y=-x→\;f(0)=f(x)+f(-x)→\;f(x)=-f(x)\; [odd function]...(*2)
    3. when n is natural number, and x is real number, then f(nx)=nf(x)
      • when n=1 is established.
      • if n=k is established, then replace x→kx and y→x
        then, f(kx+x)=f(kx)+f(x)=kf(x)+f(x)
      • \therefore\;f((k+1)x)=(k+1)f(x) is established.
      • then, any n of natural number: f(nx)=nf(x)
    4. from (*2), substitute: x→nx, then f(-nx)=-f(nx)=-nf(x)\;[-n: negative number]
      →\;f(mx)=mf(x)\; [m: integer]
    5. when rational number r=\frac{m}{n}:
      →nf(rx)=f(nrx)=f(mx)=mf(x)
      →f(rx)=\frac{m}{n}f(x)=rf(x)...(*3)
      substitute to (*3): x=1→\;f(r)=rf(1)
      let f(1)=a then: f(r)=ar\;[x:rational number]
    6. ^\exists \{q_n\} such that ^\forall x=\displaystyle\lim_{n\to\infty}q_n=x\; [\{q_n\}: rational series] and f(x) is continuous function.
      →f(x)=\displaystyle\lim_{n\to\infty}f(q_n) =\displaystyle\lim_{x\to\infty}aq_n=ax
      \therefore\;f(x)=ax
  • when: f(x) is continuous function, then consider 'rational number' only: [density]
    • when x is real number, which is expressed as follows:
      x=m+\displaystyle\sum_{i=1}^{\infty}C_i10^{-i}\; [m, C_i: integer, 0≤C_i≤9]
    • rational series: q_n=m+\displaystyle\sum_{i=1}^nC_i10^{-i}→x (n→\infty)

16. コーシーの関数方程式:

  • real function: 実関数
  • mathematical induction: 数学的帰納法
  • density: 稠密性

 

  • 数学的帰納法, mathematical induction:
    1. base case: show f(0) is clearly true.
    2. inductive step: show for any k≥0, if f(k) holds, then f(k+1) also holds.

>Top 17. Taylor expansion:

  • f(x)=f(a)+f'(a)(x-a)+\frac{1}{2!}f''(a)(x-a)^2+\frac{1}{3!}f'''(a)(x-a)^3+\cdots...(*)
    • where: h=x-a, then (*) is:
      f(a+h)=f(a)+f'(a)h+\frac{1}{2!}f''(a)h^2+\frac{1}{3!}f'''(a)h^3+\cdots\;
      [Taylor expansion around a]
    • where: a=0=x→\;f(x)=f(0)+f'(0)x+\frac{1}{2!}f''(0)x^2+\frac{1}{3!}f'''(0)x^3+\cdots
      =\displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n
      [Maclaurin expansion]

  • ¶1: f(x)=e^x\; x=0...(*)
    • to find linear function in the vicinity of (*):
      f(0)=1,\; f'(0)=1→\;g(x)=1+x
    • to find quadratic function in the vicinity of (*):
      $f(0)=1,\; f'(0)=1,\;f''(0)=1→\; g(x)=1+x+\frac{1}{2}x^2
    • e^x=e^0+e^0x+\frac{1}{2!}e^0x^2+\frac{1}{3!}e^0x^3+\frac{1}{4!}e^0x^4+\cdots
      =1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\frac{1}{4!}x^4+\frac{1}{5!}x^5+\cdots
      \therefore\;\boxed{e^x=\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}x^n}
    • \log (1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+\cdots =\displaystyle\sum_{n=0}^{infty}\frac{(-1)^n}{n+1}x^{n+1}
      • f(x)=\log(1+x)→\;f^{(n)}(x)=
      • \log 2=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{2}+\cdots =\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}=\log 2\;[Mercator series]
    • (1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots
    • \sin x=x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-\frac{1}{7!}x7+\cdots
      f(0)=\sin 0=0,\;f'(0)=\cos 0=1,\;f''=-\sin 0=0,\;f'''=-\cos 0=-1, \cdots
      \sin x=0+1x+\frac{0}{2!}x^2+\frac{1}{3!}x^3+\frac{1}{4!}x^4+\cdots
      =0+1x+\frac{0}{2!}x^3+\frac{-1}{3!}x^5-\frac{0}{4!}x^6+\cdots
      =x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-\frac{1}{7!}x^7+\cdots
      \therefore\;\boxed{\sin x=\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}}
    • \cos x=1-\frac{1}{2!}x^2+\frac{1}{4!}x^4-\frac{1}{6!}x^6+\cdots
      \therefore\;\boxed{\cos x=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}}
    • \tan x=x+\frac{1}{3}x^3+\frac{2}{15}x^5+\cdots

  • Euler's formula:
    • i\sin x=ix-i\frac{1}{3!}x^3+i\frac{1}{5!}x^5-i\frac{1}{7!}x^7+\cdots
      e^{ix}=1+ix-\frac{1}{2!}x^2-i\frac{1}{3!}x^3+\frac{1}{4!}x^4+i\frac{1}{5!}x^5-\cdots
    • →e^{ix}=\cos x+i\sin x\; [Euler' formula]
      →\cos x=\frac{e^{ix}+e^{-ix}}{2},\;\sin x=\frac{e^{ix}-e^{-ix}}{2}
    • e^{i(a+b)}=e^{ia}e^{ib}
      →\;\cos (a+b)+i\sin (a+b)=(\cos a+i\sin a)(\cos b+i\sin b)
      =(\cos a\cos b-\sin a\sin b)+i(\cos a\sin b+\sin a\cos b)\; [addition theorem]
    • e^{in\theta}=(e^{i\theta})^n
      \cos(2\theta)+i\sin(2\theta)=(\cos\theta+i\sin\theta)^2
      (\cos^2\theta-\sin^2\theta)+i2\cos\theta\sin\theta\; [double-angle formula]
    • \cos(3\theta)+i\sin(3\theta)=(\cos\theta+i\sin\theta)^3
      =(\cos^3\theta-3\cos\theta\sin^2\theta)+i(3\cos^2\theta\sin\theta-\sin^3\theta)
      =(4\cos^3\theta-3\cos\theta)+i(3\sin\theta-4\sin^3\theta)\; [tripple-angle formula]
    • \sin^2(\frac{x}{2})=\bigl(\frac{e^{i\frac{x}{2}}-e^{-i\frac{x}{2}}}{2i}\bigr)^2=\frac{e^{ix}+e^{-ix}-2}{-4}=\frac{1-\cos x}{2\;} [half-angle formula]
    • \cos^2(\frac{x}{2})=\bigl(\frac{e^{i\frac{x}{2}}+e^{-i\frac{x}{2}}}{2}\bigr)^2=\frac{e^{ix}+e^{-ix}+2}{4}=\frac{1+\cos x}{2\;}
    • \cos a\sin b=\frac{e^{ia}+e^{-ia}}{2}・\frac{e^{ib}-e^{-ib}}{2i}
      =\frac{e^{i(a+b)}-e^{-i(a+b)}-e^{i(a-b)}+e^{-i(a-b)}}{4i}
      =\frac{1}{2}\bigl(\frac{e^{i(a+b)}-e^{-i(a+b)}}{2i}-\frac{e^{i(a-b)}-e^{-i(a-b)}}{2i}\bigr)=\frac{1}{2}(\sin (a+b)-\sin (a-b))\;[sum of products]
      • \cos a\cos b=\frac{1}{2}(\cos (a+b)+\cos (a-b))
      • \sin a\sin b=-\frac{1}{2}(\cos (a+b)-\cos (a-b))
      • \sin a\cos b=\frac{1}{2}(\sin (a+b)+\sin (a-b))
    • e^{ia}+e^{ib}=e^{\frac{i(a+b)}{2}}e^{\frac{i(a-b)}{2}} +e^{\frac{i(a+b)}{2}}e^{\frac{-i(a-b)}{2}}
      =e^{\frac{i(a+b}{2}}\left(e^{\frac{i(a-b)}{2}}+e^{\frac{-i(a-b)}{2}}\right)
      =\left(\cos (\frac{a+b}{2})+i\sin (\frac{a+b}{2})\right) ・\left(\cos (\frac{a-b}{2})+i\sin (\frac{a-b}{2})+\cos (\frac{a-b}{2})-i\sin (\frac{a-b}{2})\right)
      =2\cos (\frac{a-b}{2})\left(\cos (\frac{a+b}{2})+i\sin (\frac{a+b}{2})\right)
      =2\cos (\frac{a-b}{2})\cos (\frac{a+b}{2})+i2 \cos (\frac{a-b}{2})\sin (\frac{a+b}{2})
    • while e^{ia}+e^{ib}=(\cos a+i\sin a)+(cos b+\sin b)
      =(\cos a+\cos b)+i(\sin a+\sin b)→\;
    • \cos a+\cos b=2\cos (\frac{a+b}{2})\cos (\frac{a-b}{2})\; [product of sum]
    • \cos a-\cos b=-2\sin (\frac{a+b}{2})\sin (\frac{a-b}{2})
    • \sin a+\sin b=2\sin (\frac{a+b}{2})\cos (\frac{a-b}{2})
    • \sin a-\sin b=2\cos (\frac{a+b}{2})\sin (\frac{a-b}{2})

17. テイラー展開:

  • radius of convergence: 収束半径
  • : ダランベールの収束判定法
  • d'Alembert's ratio test:
    \displaystyle\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|=r
  • r<1:\; convergence
  • r>1:\; divergence

>Top 18. Fourier expansion:

  • continuous f(x) defined in 0≤x≤2\pi can be expressed by the combination of trigonometric funtions, i.e.:
    • f(x)=c+\displaystyle\sum_{n=1}^{\infty}\bigl(a_n\cos (nx)+b_n\sin (nx)\bigr)...(*1)
    • where c is the average of (*1): c=\frac{1}{2\pi}\displaystyle\int_0^{2\pi}f(x)dx...(*2)
    • coefficient a_n, \;b_n can be:
      \cases{a_n=\frac{1}{\pi}\displaystyle\int_0^{2\pi}f(x)\cos(nx)dx \\b_n=\frac{1}{\pi}\displaystyle\int_0^{2\pi}f(x)\sin(nx)dx}
      • where, n=0 of (*2): $→\;c=\frac{1}{2}a_0
    • so, \boxed{f(x)=\frac{a_0}{2}+\displaystyle\sum_{n=1}^{\infty} \bigl(a_n\cos (nx)+b_n\sin (nx)\bigr)}...(*3) [Fourier series]
    • here: \displaystyle\int_0^{2\pi}\cos(mx)\cos(nx)dx
      =\frac{1}{2}\displaystyle\int_0^{2\pi}\bigl(\cos(m+n)x+\cos(m-n)x\bigr)dx
      =\frac{1}{2}\left[\frac{1}{m+n}\sin (m+n)x+\frac{1}{m-n}\sin (m-n)x\right]_0^{2\pi}=\pi \delta_{mn}\;
      [Kronecker delta: \delta_{mn}=1\;(m=n),\;0\;(m\ne n)]
      • similarly: \displaystyle\int_0^{2\pi}\sin (mx)\sin (nx)dx=\pi\delta_{mn}
      • \displaystyle\int_0^{2\pi}\sin (mx)\cos (nx)dx=0

18. フーリエ展開:

>Top 19. Linear algebra:

  • Gaussian elimination method:
    • multiply 1st row: -\frac{d}{a} then add to 2nd row:
      \pmatrix{a&b&c\\d&e&f\\g&h&i}\;
    • multiply 1st row: -\frac{g}{a} then add to 3rd row:
      \pmatrix{a&b&c\\0&e'&f'\\g&h&i}\;
    • multiply 1st row: -\frac{g}{a} then add to 3rd row:
      \pmatrix{a&b&c\\0&e'&f'\\0&h'&i'}\;
    • multiply 2nd row: -\frac{h'}{e'} then add to 3rd row:
      \pmatrix{a&b&c\\0&e'&f'\\0&0&i''}\;
  • Cofactor expansion:
    • \pmatrix{a&b&c&d&\\0&e&f&g\\0&0&h&-i&\\0&0&0&j}=a・e・h・j
    • |A|=a_{i1}\tilde{a}_{i1}+a_{i2}\tilde{a}_{i2}+\cdots+a_{in}\tilde{a}_{in} =\displaystyle\sum_{j=1}^na_{ij}\tilde{a}_{ij}
      =a_{1j}\tilde{a}_{1j}+a_{2j}\tilde{a}_{2j}+\cdots+a_{nj}\tilde{a}_{nj} =\displaystyle\sum_{i=1}^na_{ij}\tilde{a}_{ij}
  • Properties of matrix (elementary row operations):
    • Row switching:
      • R_i \leftrightarrow R_j
    • Scalar times:
      • row multiplication: kR_i→R_i, \;\mathrm{where}\;k≠0
      • row additon: R_i+kR_j\leftrightarrow R_i, \;\mathrm{where}\; i≠j
      • invere of matrix: T_{ij}^{-1}=T_{ij}
      • same two rows makes the matrix is 0.
      • k times of a row and then add it to other row makes no change of the matrix.
    • Commutative law:
      • AB\ne BA
      • (A+B)+C=A+(B+C)
    • Associative law: A(BC)=(AB)C
    • Indenty matrix:
      • E=\pmatrix{1&0&0\\0&1&0\\0&0&1}
      • AE=A,\; EA=A
    • >Top Inverse matrix:
      • BA=E→\;A^{-1}A=E→\;AA^{-1}=E\; [regular matrix]
      • from Cramer's rule:
        • \pmatrix{a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}} \pmatrix{x_{11}&x_{12}&x_{13}\\x_{21}&x_{22}&x_{23}\\ x_{31}&x_{32}&x_{33}} =\pmatrix{1&0&0\\0&1&0\\0&0&1}
        • x_{11}=\frac{\tilde{a}_{11}}{|A|},\; x_{21}=\frac{\tilde{a}_{12}}{|A|},\; x_{31}=\frac{\tilde{a}_{13}}{|A|},\;
        • X=\frac{1}{|A|}\pmatrix{\tilde{a}_{11}&\tilde{a}_{21}&\tilde{a}_{31}\\ \tilde{a}_{12}&\tilde{a}_{22}&\tilde{a}_{32}\\ \tilde{a}_{13}&\tilde{a}_{23}&\tilde{a}_{33}} =\frac{1}{|A|}^tA^{ij}
      • A=\pmatrix{a&b\\c&d},\; A^{-1}=\frac{1}{ad-bc}\pmatrix{d&-b\\-c&a}
    • Matrix multiplication :
      • |AB|=|A||B|
      • |AA^{-1}|=|A||A^{-1}|=1→\;|A^{-1}|=\frac{1}{|A|}
    • >Top Linearly dependent:
      • if there exist scalars a_1,a_2,\cdots,a_n , not all zero, such that:
        • a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+a_n\mathbf{v}_n=\mathbf{0}...(*)
        • \mathbf{v}_1=\frac{-a_2}{a_1}\mathbf{v}_2+\cdots+\frac{-a_n}{a_1}\mathbf{v}_n\; [if a_1≠0]
          thus \mathbf{v}_1 is shown to be a linear combination of the remaining vectors.
      • a sequence of vectors (\mathbf{v}_1,\mathbf{v}_2,\cdots,\mathbf{v}_n) is linearly independent if (*) can only be satisfied by a_i=0 for i=1,\cdots,n. this implies that no vector in the sequence can be represented as a linear combination of the remaining vectors.
        • \displaystyle\sum_{i=1}^na_i\mathbf{v}_i=\mathbf{0} \Rightarrow a_1=\cdots=a_n=0
        • a certain vector \mathbf{x} can be uniquely expressed by linear combination of linearly independent vectors (\mathbf{v}_1,\mathbf{v}_2,\cdots,\mathbf{v}_n).
    • >Top Rank of a matrix A:
      • is the maximum number of linearly independent row of vectors of the matrix: rank A=n
      • is the dimension of the image of the linear map represented by A; which is converted to number of the rank dimension.
      • by simplification of a square matrix, if ^\exists EA, then ^\exists A^{-1}
      • eg.: B=\pmatrix{1&0&1\\-2&-3&1\\3&3&0}=\pmatrix{1&0&1\\0&-3&3\\0&0&0}:
        • the first two columns linearly independent, but the third is a linear combination of the first two; thus the rank is 2.
        • when the upper triangle matrix B has a column (or row) of all 0 elements (=rank down); the rank is n-1, here 3.
    • Trace: sum of its diagonal entries.
      • tr(AB)=\displaystyle\sum_{i=1}^m\displaystyle\sum_{j=1}^na_{ij}b_{ij} =\mathrm{tr}(BA)
      • tr(A)=\mathrm{tr}(^tA)
    • >Top Transposed matrix:
      • A=\pmatrix{a_{11}&\cdots&a_{1n}\\ \vdots&&\vdots\\a_{m1}&\cdots&a_{mn}}
      • ^tA=\pmatrix{a_{11}&\cdots&a_{m1}\\ \vdots&&\vdots\\a_{1n}&\cdots&a_{mn}}
      • ^{tt}A=A,\;^t(A+B)=^tA+^tB,\;^t(kA)=k^tA,\; ^t(kA+lB)=k^tA+l^tB,\;^t(AB)=^tB^tA
      • (square matrix): ^t(A^{-1})=(^tA)^{-1},\; \mathrm{tr}A=\mathrm{tr}^tA
      • \mathrm{det} A=\mathrm{det}^tA
      • \langle Ax,y\rangle=\langle x,^tAy\rangle
  • Coordinates transformation:
    • \pmatrix{p_x&q_x&r_x\\p_y&q_y&r_y\\p_z&q_z&r_z}\pmatrix{x'\\y'\\z'} =\pmatrix{x\\y\\z}
      →\;\pmatrix{x'\\y'\\z'} =\pmatrix{p_x&q_x&r_x\\p_y&q_y&r_y\\p_z&q_z&r_z}^{-1}\pmatrix{x\\y\\z}
    • \pmatrix{p_x&q_x&r_x\\p_y&q_y&r_y\\p_z&q_z&r_z}\pmatrix{x'\\y'\\z'} =\pmatrix{s_x&t_x&u_x\\s_y&t_y&u_y\\s_z&t_z&u_z}\pmatrix{x\\y\\z}
      →\;\pmatrix{x'\\y'\\z'} =\pmatrix{p_x&q_x&r_x\\p_y&q_y&r_y\\p_z&q_z&r_z}^{-1} \pmatrix{s_x&t_x&u_x\\s_y&t_y&u_y\\s_z&t_z&u_z}\pmatrix{x\\y\\z}
  • >Top Eigenvalue and eigenvector:
    • \bf{Ax}=\lambda \bf{x}\;[x: eigenvector]
      \bf{Ax}-\lambda \bf{Ex}=0→\;(\bf{A}-\lambda \bf{E})\bf{x}=0
      (\bf{A}-\lambda \bf{E})=0\; [x\ne 0; Eigen equation]
      \lambda=\lambda_i,\;(i=1,2,\cdots,n)\; [eigenvalue]
      solve: (A-\lambda_iE=0)x=0\; [n-unknown simul.linear.eq]
      →\;x=x_\;(i=1,2,\cdots,n)\; [eigenvector]

    • ¶1:A=\pmatrix{2&3\\4&1}
      \pmatrix{2-\lambda&3\\4&1-\lambda}=(2-\lambda)(1-\lambda)-12
      →\;=(\lambda-5)(\lambda+2)=0\;\therefore\;\lambda=5, -2
      • i) when: \lambda=5:→\;\pmatrix{-3&3\\4&-4}\pmatrix{x\\y} =\pmatrix{0\\0} \Leftrightarrow \\cases{-3x+3y=0\\4x-4y=0} \Leftrightarrow x-y=0
        x=s_1:, y=s_1→\;x_1=s_1\pmatrix{1\\1} \; (s_1≠0)
      • ii) when: \lambda=-2:→\;\pmatrix{4&3\\4&3}\pmatrix{x\\y} =\pmatrix{0\\0} \Leftrightarrow 4x+3y=0
        x=-3s_2:, y=4s_2→\;x_2=s_2\pmatrix{-3\\4} \; (s_2≠0)

    • ¶2:A=\pmatrix{2&1$1\\1&2$1\\1&1&2}
      \pmatrix{2-\lambda&1$1\\1&2-\lambda&1\\1&1&2-\lambda} =-(\lambda-1)^2(\lambda-4)
      →\therefore\;\lambda=1, 4
      • i) when: \lambda=1:→\;\pmatrix{1&1&1\\1&1&1\\1&1&1}\pmatrix{x\\y\\z} =\pmatrix{0\\0\\0} \Leftrightarrow x+y+z=0
        x=s_1:, y=t_1→\;z=-s_1-t_1
        →\;x_1=\pmatrix{s_1\\t_1\\-s_1-t_1} \; (s_1, t_1≠0)
        =s_1\pmatrix{1\\0\\-1}+t_1\pmatrix{0\\1\\-1}
      • ii) when: \lambda=4:→\;\pmatrix{-2&1&1\\1&-2&1\\1&1&-2}\pmatrix{x\\y\\z} =\pmatrix{0\\0\\0}
        →\left(\begin{array}{ccc|c}1&0&-1&0\\0&1&-1&0\\0&0&0&0\\\end{array}\right)
        \cases{x-z=0\\y-z=0}→\;x=y=z=s_2→\;s_2\pmatrix{1\\1\\1}

  • >Top Diagonalization-I (no-double solution):
    • diagonalizable square matrix A:
      • Diagonal matrix of A exists in the form of:
        P^{-1}AP=\pmatrix{\lambda_1&&&0\\&\lambda_2&&\\ &&\ddots&\\0&&&\lambda_n\\}\; [P\; is transformation matrix; \lambda_n\; eigenvalue]
      • (P^{-1}AP)^n=\underbrace{(P^{-1}AP)(P^{-1}AP)\cdots(P^{-1}AP)}_{n}=P^{-1}A^nP\therefore\;A^n=P(P^{-1}A^nP)P^{-1}
      • Formula:
        where, linearly independent eigenvector (x_1,x_2,\cdots,x_n) of a square matrix A:
        let P=(x_1,x_2,\cdots,x_n) then,
        P^{-1}AP=\pmatrix{\lambda_1&&&0\\&\lambda_2&&\\ &&\ddots&\\0&&&\lambda_n\\}\; [P\; is transformation matrix; \lambda_n
        • Proof: P has inverse matrix:
          C_1x_1+C_2x_2+\cdots+C_nx_n=(x_1,x_2,\cdots,x_n) \pmatrix{C_1\\C_2\\\vdots\\C_n}=0\;
          [n-dimensional simultaneous linear eq.] ...(*)
          if rank(P<0), then (*) has the solution other than trivial one (C_1=C_2=\cdots=C_n=0), which contracts linearly independence of x_1,x_2,\cdots,x_n; thus rank (P)=n→\;P is regular matrix.
        • Proof-2: being diagonal matrix:
          P^{-1}AP^=P^{-1}A(x_1,x_2,\cdots,x_n)=P^{-1}(Ax_1,Ax_2,\cdots,Ax_n)
          =P^{-1}(\lambda_1x_1,\lambda_2x_2,\cdots,\lambda_nx_n)
          =P^{-1}(x_1,x_2,\cdots,x_n)\pmatrix{\lambda_1&&&0\\&\lambda_2&&\\ &&\ddots&\\0&&&\lambda_n\\}\; [P=(x_1,x_2,\cdots,x_n)]
          • here cf.: (\lambda_1x_1,\lambda_2x_2) =\pmatrix{\lambda_1x_1&\lambda_2x_2\\\lambda_1y_1&\lambda_2y_2} =\pmatrix{x_1&x_2\\y_1&y_2}\pmatrix{\lambda_1&0\\0&\lambda_2}
      • Formula:
        where, eigenvector x_1,x_2,\cdots,x_k corresponding different eigenvalue of \lambda_1,\lambda_2,\cdots,\lambda_k of n-sized square matrix A is linearly independent. (1≤k≤n)...(*0)
      • Proof:
        when k=1, (*0) is obvious. [c_1x_1=0,\; x_1≠\mathbf{0}→\;c_1=0]
        assume k=m (*) is true, and consider: C_1x_1+C_2x_2+\cdots+C_mx_m+C_{m+1}x_{m+1}=0...(*1)
        multiply A to (*1) from left:
        →\;C_1\lambda_1x_1+C_2\lambda_2x_2+\cdots+ C_m\lambda_mx_m+C_{m+1}\lambda_{m+1}x_{m+1}=0...(a)
        multiply \lambda_{m+1} to (*1) from left:
        →\;C_1\lambda_{m+1}x_1+C_2\lambda_{m+1}x_2+\cdots+ C_m\lambda_{m+1}x_m+C_{m+1}\lambda_{m+1}x_{m+1}=0...(b)
        (a)-(b):C_1(\lambda_1-\lambda_{m+1})x_1 +C_2(\lambda_2-\lambda_{m+1})x_2+\cdots +C_m(\lambda_m-\lambda_{m+1})x_m=0
        here: x_1,x_2,\cdots,x_m is linearly independent, then:
        C_i(\lambda_i-\lambda_{m+1})=0,\;(i=1,2,\cdots,m)
        here: \lambda_i-\lambda_{m+1}≠0,\; (i=1,2,\cdots,m)
        →\;C_1=C_2=\cdots=C_m=0
        substitute this to (*1): →\;C_{m+1}x_{m+1}=0→\;C_{m+1}=0
        thus, x_1,x_2,\cdots,x_{m+1} is linearly independent.

    • ¶1: diagonalize: A=\pmatrix{-2&1\\5&2}
      • \pmatrix{-2-\lambda&1\\5&2-\lambda}=(-2-\lambda)(2-\lambda)-5 =(\lambda-3)(\lambda+3)=0→\;\lambda03, -3
      • i) when \lambda=3:
        \pmatrix{-5&1\\5&-1}\pmatrix{x\\y}=\pmatrix{0\\0} \Leftrightarrow -5x+y=0
        →\;x=x_1,\;y=5s_1→\;x_1=s_1\pmatrix{1\\5}
      • ii) when \lambda=-3
        \pmatrix{1&1\\5&5}\pmatrix{x\\y}=\pmatrix{0\\0}
        \Leftrightarrow x+y=0→\;x=s_2, \;y=-s_2→\;x_1=s_2\pmatrix{1\\-1}
        thus: P=\pmatrix{1&1\\5&-1} then P^{-1}AP=\pmatrix{3&0\\0&-3}
      • check: P^{-1}=\frac{1}{-6}\pmatrix{-1&-1\\-5&1} =\frac{1}{6}\pmatrix{1&1\\5&-1}
        →\;P^{-1}AP=\frac{1}{6}\pmatrix{1&1\\5&-1} \pmatrix{-2&1\\5&2}\pmatrix{1&1\\5&-1}
        • here: \frac{1}{6}\pmatrix{1&1\\5&-1}\pmatrix{-2&1\\5&2}\bigl(\bigr)
          =\frac{1}{6}\pmatrix{3&3\\-15&3}\pmatrix{1&1\\5&-1} =\frac{1}{6}\pmatrix{18&0\\0&18}=\pmatrix{3&0\\0&-3}

  • Diagonalization-II (double solution):
    • diagonalize square matrix A=\pmatrix{-2&2&4\\-2&3&2\\-2&1&4}
      \pmatrix{-2-\lambda&2&4\\-2&3-\lambda&2\\12&1&4-\lambda}
      =-(\lambda-1)(\lambda-2)^2=0→\;\lambda=1,\;2(double)$
    • i) \lambda=1:
      \pmatrix{-3&2&4\\-2&2&2\\-2&1&3}\pmatrix{x\\y\\z} =\pmatrix{0\\0\\0}→\;x_1=s_1\pmatrix{2\\1\\1}
    • ii) \lambda=2
      \pmatrix{-4&2&4\\-2&1&2\\-2&1&2}\pmatrix{x\\y\\z} =\pmatrix{0\\0\\0}\;\Leftrightarrow -2x+y+2z=0→\;x=s_2,\;z=t_2, y=2s_2-2t_2
      →\;x_2=s_2\pmatrix{1\\2\\0}+t_2\pmatrix{0\\-2\\1}
      • here: \pmatrix{2\\1\\1},\;\pmatrix{1\\2\\0},\;\pmatrix{0\\-2\\1} are linealy independent, and eigenvector.
      • p^{-1}AP=\pmatrix{1&0&0\\0&2&0\\0&0&2}
  • Diagonalization-III (no diagnalizable case)
    • A=\pmatrix{-3&-1\\1&-1}
      \pmatrix{-3-\lambda&-1\\1&-1-\lambda}=(-3-\lambda)(-1-\lambda)+1 =\lambda^2+4\lambda+4=(\lambda+2)^2=0→\;\lambda=-2\; (double)
      \Leftrightarrow x+y=0→\;x=s_1,\;y=-s_1 →\; x_1=s_1\pmatrix{1\\-1}
      there is no other independent eigenvector than this: thus 'no diagonal'

19. 線形代数:

  • Gaussian elimination: ガウスの消去法, 掃き出し法
  • identity (unit) matrix: 単位行列
  • upper triangle matrix: 上三角行列
  • cofactor expansion: 余因子展開
  • rank: 階数
  • trace: 跡, 対角線成分
  • linear combination: 線形結合
  • transposed matrix: 転置行列
  • coordinate transformation: 座標変換
  • eigenvalue: 固有値
  • eigenvector: 固有ベクトル
  • diagonalization: 対角化
  • diagonalizable: 対角化可能
  • regular matrix =reversible matrix: 正則行列 (=可逆行列)
  • transformation (=diagonalizable) matrix: 変換(=対角化)行列
  • 掃き出し法:

  • 余因子展開:

  • 特徴:
    • スカラー倍
    • 可換法則
    • 結合法則
    • 単位行列
    • 乗算
    • 線形独立(=一次独立)
    • ランク, 階数
    • 転置行列

  • 逆行列, Inverse matrix:
    1. make ^tA
    2. make \tilde{A}
    3. consider sgn \tilde{A}
  • 座標変換:

  • 固有値と固有ベクトル:

  • 対角化:

>Top 20. Delta function:

  • Paul Dirac's delta function:
    • definition-1: \delta(x-a)=\cases{\infty,\;(x=a)\\0,\;(x≠0)}
    • definition-2: \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a)
      • when f(x)=1→\;\int_{-\infty}^{\infty}\delta(x-a)dx=1 [size of infinity]...(*)
      • point mass of a mass m on x=a:\;\int_{-\infty}^{\infty}m\delta(x-a)dx=m
      • but, mathematically (*) normal function will be zero; which is called distribution (by Schwarz) or hyperfunction (by Sato).

20. デルタ関数:

  • point mass: 質点

>Top 21. Gaussian integration:

  • Definition: \boxed{\displaystyle\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}}
    • proof: I=\displaystyle\int_{-\infty}^{\infty}e^{-x^2}dx
    • I^2=(\displaystyle\int_{-\infty}^{\infty}e^{-x^2}dx) (\displaystyle\int_{-\infty}^{\infty}e^{-y^2}dy)\;[x↔y]
      =\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty}e^{x^2+y^2}dxdy
    • let: x=r\cos\theta,\;y=r\sin\theta, then
      I^2=\displaystyle\int_0^{2\pi}\displaystyle\int_0^{\infty}e^{-r^2}rdrd\theta
      • surface integral: ds=\pi(r+dr)^2\frac{d\theta}{2\pi}-\pi r^2\frac{d\theta}{2\pi} =rdrd\theta+\frac{1}{2}(dr)^2d\theta \simeq rdrd\theta
    • I^2=\displaystyle\int_0^{2\pi}d\theta\displaystyle\int_0^{\infty}re^{-r^2}rd
      =2\pi\left[-\frac{1}{2}e^{-r^2}\right]_0^{\infty}=\pi\;→I=\sqrt{\pi},\;(I>0)
    • Net outflow:=(\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y} +\frac{\partial V_z}{\partial z} )\Delta x\Delta y\Delta z
    • thus outflow per volume: =div \mathbf{V}

21. ガウス積分:

  • gaussintegral.gif
  • surfaceintegral.gif

>Top 22. Vector analysis:

  • Divergence: outflow-inflow
    • V_x(x+\frac{\Delta x}{2},y,z)\Delta y\Delta z-V_x(x-\frac{\Delta x}{2},y,z)\Delta y\Delta z
      =\bigl(V_x(x+\frac{\Delta x}{2},y,z)-V_x(x-\frac{\Delta x}{2},y,z)\bigr)\Delta y\Delta z...(*)
      • Taylor expansion: f(a+h)=f(a)+f'(a)h+\frac{1}{2!}f''(a)h^2+\cdots
      • Vx(x±\frac{\Delta x}{2},y,z)\simeq Vx(x,y,z) ±\frac{\partial V_x}{\partial x}\frac{\Delta x}{2}
    • (*)\simeq\bigl(V_x(x,y,z)+\frac{\partial V_x}{\partial x}\frac{\Delta x}{2} -V_x(x,y,z) +\frac{\partial V_x}{\partial x}\frac{\Delta x}{2}\bigr)\Delta y\Delta z =\frac{\partial V_x}{\partial x} \Delta x\Delta y\Delta z

  • >Top Rotation:
    • Definition: rot \mathbf{V} =\bigl(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z}, \frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x}, \frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}\bigr)
      =\nabla \times \mathbf{V}\; [\nabla=(\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z})]
    • =\bigl(V_y(x+\frac{\Delta x}{2},y)-V_y(x-\frac{\Delta x}{2},y)\bigr)\Delta y +\bigl(V_x(x,y-\frac{\Delta y}{2})-V_x(x,y+\frac{\Delta x}{2})\bigr) \Delta x
      \simeq \bigl(V_y(x,y)+\frac{\partial V_y}{\partial x}\frac{\Delta x}{2} -V_y(x,y)+\frac{\partial V_y}{\partial x}\frac{\Delta x}{2}\bigr)\Delta y +\bigl(V_x(x,y)-\frac{\partial V_x}{\partial y}\frac{\Delta y}{2} -V_x(x,y)-\frac{\partial V_x}{\partial y}\frac{\Delta y}{2}\bigr)\Delta x
      =(\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y})\Delta x\Delta y
    • per unit space: =\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y} \; [z-component of rot\;\mathbf{V}]

  • Gradient:
    • Definition: grad f=(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})\;[grad f means mostly† increasing direction]
    • Proof (†):\Delta f=f(r+\Delta r)-f(r)
      =f(x+\Delta x, y+\Delta y, z+\Delta z)-f(x,y,z)
      \simeq f(x,y,z)+\frac{\partial f}{\partial x}\Delta x +\frac{\partial f}{\partial y}\Delta y+\frac{\partial f}{\partial z}\Delta z
      =(\mathrm{grad}\;f)・\Delta r=|\mathrm{grad}\;f||\Delta r|\cos\theta ≤|\mathrm{grad}\;f||\Delta \mathbf{r}|\; [\theta=0\;maximum]

  • >Top Solid angle: (unit: Steradian d\Omega)
    • dS=r^2\sin\theta d\theta d\phi
      d\Omega\equiv \frac{dS}{r^2}=\sin\theta d\theta d\phi

    • ¶1: [global surface]
      =\displaystyle\int_0^{2\pi}\displaystyle\int_0^{\pi}\sin\theta d\theta d\phi=4\pi

    • ¶2: [\theta_1→\theta_2,\;\phi_1→\phi_2]
      =(\phi_2-\phi_1)(\cos\theta_1-\cos\theta_2)

    • ¶3: [half-apex angle=\alpha]
      =\displaystyle\int_0^{2\pi}\displaystyle\int_0^{\alpha}\sin\theta d\theta d\phi =2\pi(1-\cos\alpha)

  • >Top Spherical coordinates:
    • \cases{x=r\sin\theta\cos\phi\\y=r\sin\theta\sin\phi\\z=r\cos\theta}
      \cases{r=\sqrt{x^2+y^2+z^2}\;(0≤r)\\\theta=\cos^{-1}(\frac{z}{\sqrt{s^2+y^2+z^2}}) \;(0≤\theta ≤\pi) \\\phi=\mathrm{sgn}(y)\cos^{-1}(\frac{x}{\sqrt{x^2+y^2}})\; (0≤\phi<2\pi) }
      • here: sgn(y)=\cases{1\;(0≤y)\\-1\;(y<0)}

22. ベクトル解析:

  • vector field: ベクトル場
  • position vector: 位置ベクトル
  • solid angle: 立体角
  • steradian: 立体角の単位 (sr)
  • divergence.gif
  • rotation.gif
  • gradient.gif
  • solidangle.gif
  • sphericalcoordinates.gif

>Top 23. Integration method:

  • Some Calculus formula:
    • y=fg\\ \log y=\log(fg)=\log f+\log g\\ \frac{y'}{y}=\frac{f'}{f} +\frac{g'}{g}\\ y'=f'g+fg'
    • y=(\frac{f}{g})
      \log y=\log\frac{f}{g} =\log f-\log g\\ \frac{y'}{y}=\frac{f'}{f} -\frac{g'}{g}\\ y'=\frac{f'}{g}-\frac{fg'}{g^2} =\frac{f'g-fg'}{g^2}
    • (e^x)'=e^x
    • (a^x)'=a^x\log a
    • y=(x^x)'
      \log y=x\log x
      \frac{y'}{y}=\log x+1\\ y'=y(\log x+1)=x^x(\log x+1)
    • (\log x)'=\frac{1}{x}
    • (\log y)'=\frac{y'}{y}
    • (\log_ax)'\\ =(\frac{\log x}{\log a})'\\ =\frac{1}{\log a}(\log x)'\\ =\frac{1}{x\log a}

  • Some Integral formula:
    • \int\frac{1}{x}=\log|x|+C
    • \int\frac{1}{x^2}=-\frac{1}{x}+C
    • \int \frac{f'}{f}=\log|f|+C
    • \int e^xdx=e^x+C
    • \int a^x=\frac{a^x}{\log a}+C
    • \int a^x\log adx=a^x+C
    • \int\log xdx=\int x'\log xdx=x\log x-x(\log x)'dx=x\log x-\int dx
      =x\log x-x+C\; [\log = 1・\log]
    • \int \frac{1}{x\log a}dx=\log_ax
    • \int \sin xdx=-\cos x+C
    • \int \cos xdx=\sin x+C
    • \int \tan xdx=\int \frac{\sin x}{\cos x}dx=-\int\frac{(\cos x)'}{\cos x}dx
      =-log|\cos x|+C
    • \int \sec^2xdx=\tan x+C
    • \int \csc^2xdx=-cotx+C
    • \int \frac{1}{\cos^2x}dx=\tan x+C
    • \int \frac{1}{\sin^2x}dx=-\frac{1}{\tan x}+C
    • \int\frac{1}{x^2+a^2}dx=\frac{1}{a}\tan^{-1}\frac{x}{a}+C,\;(a≠0)
    • \int\frac{1}{\sqrt{x^2-a^2}}dx=\sin^{-1}\frac{x}{a}+C,\;(a>0)
    • \int\frac{1}{\sqrt{x^2+a}}dx=\log |x+\sqrt{x^2+a}|+C,\;(a≠0)

  • Integration by parts:
  • \int f(x)g'(x)dx=f(x)g(x)-\int f'(x)g(x)dx
    • \int\log|x|dx=\int x'\log|x|dx=x\log|x|-\int x(\log|x|)'dx
      =x\log|x|-\int x\frac{1}{x}dx=x\log|x|-x+C
  • \int f(g(x))g'(x)dx= [differential contact type]
    • let:g(x)=t
  • \int \frac{f'(x)}{f(x)}dx=\log|f(x)|+C
  • \int(f(x))^af'(x)dx=\frac{(f(x)^{a+1})}{a+1}+C
  • \int x^ne^xdx=x^ne^x-\int (x^n)'e^x
    • \int \frac{1}{\cos^3xdx}
      =\int \frac{1}{\cos^2x}\frac{1}{\cos x}dx =\tan x\frac{1}{\cos x}-\int \tan x\frac{\sin x}{\cos^2x}dx
      =\frac{\sin x}{\cos^2x}-\int \frac{\sin^2 x}{\cos^3 x}
      =\frac{\sin x}{\cos^2x}-\int \frac{1}{\cos^3x}dx+\int \frac{1}{\cos x}dx
      2\int \frac{1}{\cos^3x}dx =\frac{\sin x}{\cos^2x}+\frac{1}{2}\log\big|\frac{1+\sin x}{1-\sin x}\big|+C
  • \int (\log x)^3dx)
    • t=\log x→\;dt=\frac{1}{x}dx→dx=e^tdt
      =\int t^3e^tdt=t^3e^t-3t^2e^t+6te^t-6e^t+C=(t^3-et^2+6t-6)e^t+C
      =\{(\log x)^3-(\log x)^2+6\log x-6\}x+C

  • Integration by substitution:
  • \int f(x)dx=\int f(g(t))g'(t)dt
    • \int_0^1x^2dx =\int_0^2\frac{1}{4}t^2 \frac{1}{2}dt
      • x=\frac{1}{2}t
      • dx=\frac{1}{2}dt
    • \int_0^1\sqrt{1-x^2}dx...(*)
      let: x=\sin(u),\;dx=\cos udu
      →\sqrt{1-x^2}=\cos(u)
      =\int_0^{\frac{\pi}{2}} \cos^2(u) du =(\frac{u}{2}+ \frac{\sin(2u)}{4}) \big|_0^{\frac{\pi}{2}} =\frac{\pi}{4}
    • \int_a^b(x-a)(x-b)dx =\int_a^b\{(x-a)^2-(b-a)(x-a)\}dx
      =\frac{1}{3}(x-a)^3-(b-a)\frac{1}{2}(x-a)^2\big|_a^b=-\frac{1}{6}(b-a)^3
    • \int \frac{1}{\cos x}dx\; [→*tips]
      =\int \frac{1+t^2}{1-t^2}\frac{2}{1+t^2}dt=\int \frac{2}{1-t^2}dt
      =\int (\frac{1}{1+t}+\frac{1}{1-t})dt=\log |1+t|-\log |1-t|+C =\log |\frac{1+t}{1-t}|+C
      =\log \big|\frac{1+\tan\frac{x}{2}}{1-\tan\frac{x}{2}}\big|+C
    • \int \frac{1}{\sin x}dx\;...(*) [→*tips]
      \frac{1}{\sin x}=\frac{\sin x}{1-\cos^2x},\; t=\cos x→dt=-sin xdx
      (*)=\int \frac{\sin x}{1-t^2}\frac{1}{-\sin x}dt=\int \frac{1}{t^2-1} =\frac{1}{2}\int (\frac{1}{t-1}-\frac{1}{t+1})dt
      =\frac{1}{2}(\log|t-1|-log|t+1|)+C=\frac{1}{2}\log |\frac{t-1}{t+1}|+C
      =\frac{1}{2}\log \big|\frac{1-\cos x}{1+\cos x}\big|+C

  • Rotational body integral: around x-axis:
    V=\pi\int_a^bf(x)^2dx

  • Baumkuchen type integral (shell integration): around y-axis:
    f(x)=g(x)-h(x)
    V=2\pi\int_a^bxf(x)dx

  • Pappus-Guildinus theorem:
    volume of a solid revolution by roatating about an external axis (y):
    • V=2\pi rS\; [r=distance from axis to the gravity, S=space of the graph]
    • x_G=\frac{m_1x_1+m_2x_2+\cdots+m_nx_n}{m_1+m_2+\cdots+m_n}
      =\frac{\int_a^b x\rho f(x)dx}{\int_a^b\rho f(x)dx}\; [\rho=specific density]
      =\frac{\int_a^b xf(x)dx}{S}
    • S=\displaystyle\int_a^b f(x)dx
    • V=\int_a^b 2\pi xf(x)dx\; [Baumkuchen]
    • x_G=\frac{\frac{V}{2\pi}}{S}\therefore\;\boxed{V=2\pi Sx_G}

  • King property:
    • \int_a^b f(x)dx=\int_a^b f(a+b-x)dx
      effective when f(x)+f(a+b-x) is easily calculated.
    • ¶1:\int_{-1}^1\frac{\sin^2(\pi x)}{1+e^x}dx...I
      2I=\{\int_{-1}^1\frac{\sin^2(\pi x)}{1+e^x} +\frac{\sin^2(\pi x)}{1+e^{-x}}\}dx
      =\int_{-1}^1\sin^2(\pi x)dx=\int_{-1}^1\frac{1-\cos (2\pi x)}{2}dx
      =\frac{1}{2}x-\frac{1}{4\pi}\sin(2\pi x)\big|_{-1}^1=1
      \therefore\;I=\frac{1}{2}

23. 積分法:

  • integration by substitution: 置換積分法
  • integration by parts: 部分積分法
  • Basic:
    \sin(\frac{\pi}{2}-x)=\cos x
    \cos(\frac{\pi}{2}-x)=\sin x
    \sin(x+\frac{\pi}{2})=\cos x
    \cos(x+\frac{\pi}{2})=-\sin x
    \sin(x+\pi)=-\sin x
    \cos(x+\pi)=-\cos x

  • Addition formula:
    \sin(a\pm b)=\sin a\cos b\pm \cos a\sin b
    \cos(a\pm b)=\cos a\cos b\mp \sin a\sin b
    \tan(a\pm b)=\frac{\tan a\pm\tan b} {1\mp\tan a\tan b}

  • Double angle formula:
    \sin 2x=2\sin x\cos x
    \cos 2x=\cos^2x-\sin^2x =2\cos^2x-1=1-2\sin^2x
    \tan 2x=\frac{2\tan x}{1-2\tan^2 x}

  • Half angle formula:
    \sin^2x=\frac{1}{2}(1-\cos 2x)
    \cos^2x=\frac{1}{2}(x+\cos 2x)
    \tan^2x=\frac{1-\cos 2x}{1+\cos 2x}
    \tan x=\pm\frac{1-\cos 2x}{\sin 2x}

  • Tripple angle formula:
    \sin 3x=3\sin x-4\sin^3x
    \cos 3x=-3\cos x+4\cos^3x

  • Product-Sum formula:
    \sin a\cos b=\frac{1}{2}\bigl(\sin(a+b) +\sin(a-b)\bigr)
    \sin a\sin b=\frac{1}{2}\bigl(-\cos(a+b) +\cos(a-b)\bigr)
    \cos a\cos b=\frac{1}{2}\bigl(\cos(a+b) +\cos(a-b)\bigr)

  • Sum-Product formula:
    \sin a+\sin b=2\sin\frac{a+b}{2} \cos\frac{a-b}{s}
    \sin a-\sin b=2\cos\frac{a+b}{2} \sin\frac{a-b}{s}
    \cos a+\cos b=2\cos\frac{a+b}{2} \cos\frac{a-b}{2}
    \cos a-\cos b=-2\sin\frac{a+b}{2} \sin\frac{a-b}{2}

  • (\sin^{-1}x)'=\frac{1}{\sqrt{1-x^2}}
    (\cos^{-1}x)'=-\frac{1}{\sqrt{1-x^2}}
    (\tan^{-1}x)'=\frac{1}{1+x^2}

  • >Top Integration by parts: (integral-first!)
  • \int f^0g^0=f^0g^{-1}-\int f^1g^{-1}
    =f^0g^{-1}-f^1g^{-2}+\int f^2g^{-2}
  • integralbyparts.gif

<*Tips>

  • f(\sin x)\cos x→t=\sin x
    →dt=\cos xdx
  • f(\cos x)\sin x→t=\cos x
  • f(\tan x)\frac{1}{\cos^2x} →t=\tan x
    →dt=\frac{1}{\cos^2x}dx
  • f(\sin x,\;\cos x)...(*)
    →t=\tan\frac{x}{2}
    →\sin x=\frac{2t}{1+t^2}
    →\cos x=\frac{1-t^2}{1+t^2}
    →dt=\frac{1+t^2}{2}dx
    \int(*)dx =f(\frac{2t}{1+t^2},\; \frac{1-t^2}{1+t^2}) \frac{2}{1+t^2}dt
  • \int\log xdx→\int 1・\log xdx
  • \int_0^{\pi}xf(\sin x)dx
    =\frac{\pi}{2}\int_0^{\pi}f(\sin x)dx
  • make square of trigonometry!

>Top 24. Multiple integral:

  • Repeated integral:
    • \displaystyle\iint_D f(x,y)dxdy=\displaystyle\int_c^d\displaystyle\int_a^bf(x,y)dxdy
      =\displaystyle\int_a^b\displaystyle\int_c^df(x,y)dydx

  • ¶1: \displaystyle\iint_D(3x-y)dxdy,\;D=\{(xy)|0≤x≤1,\;-1≤y≤2|\}
    • =\displaystyle\int_{-1}^2\displaystyle\int_0^1(3x-y)dxdy
      =\displaystyle\int_{-1}^2\bigl[\frac{3}{2}x^2-yx\bigr]_0^1dy
      =\displaystyle\int_{-1}^2(\frac{3}{2}-y)dy =\bigl[\frac{3}{2}y-\frac{1}{2}y^2\bigr]_{-1}^2=3
    • \displaystyle\int_0^1\displaystyle\int_{-1}^2(3x-y)dydx
      =\displaystyle\int_0^1\bigl[3xy-\frac{1}{2}y^2\bigr]_{-1}^2dx
      =\displaystyle\int_0^1(9x-\frac{3}{2})dx
      =\bigl[\frac{9}{2}x^2-\frac{3}{2}x\bigr]_0^1=3

  • ¶2: \displaystyle\iint_De^{x^2}dxdy,\;D=\{(x,y)|0≤y≤1,\;y≤x≤1\}
    • =\displaystyle\int_0^1\displaystyle\int_y^1e^{x^2}dxdy [antiderivative?]
      =\displaystyle\int_0^1\displaystyle\int_0^xe^{x^2}dydx
      =\displaystyle\int_0^1\bigl[ye^{x^2}\bigr]_0^xdx
      =\displaystyle\int_0^1xe^{x^2}dx=\bigl[\frac{1}{2}e^{x^2}\bigr]_0^1=\frac{1}{2}(e-1)

  • Substitutive integral:
  • ¶1: \displaystyle\iint_D\frac{1}{x^2+y^2}dxdy,\;D=\{(x,y)|1≤x^2+y^2≤4,\;y≥0\}
    • =\displaystyle\int_0^{\pi}\displaystyle\int_1^2\frac{1}{r^2}rdrd\theta
      =\displaystyle\int_0^{\pi}\bigl[\log r\bigr]_1^2d\theta
      =\displaystyle\int_0^{\pi}\log 2\theta=\pi\log 2

  • >Top Jacobian:
    • 0\bigl(x(u,v),\;y(u,v)\bigr)
      A\bigl(x(u+\Delta u,v),\;y(u+\Delta u,v)\bigr)
      B\bigl(x(u,v+\Delta v),\;y(u,v+\Delta v)\bigr)
      • x(u+\Delta u,v)\simeq x(u,v)+\frac{\partial x}{\partial u}\Delta u
    • vec{OA}\simeq \bigl(\frac{\partial x}{\partial u}\Delta u,\;, \frac{\partial y}{\partial u}\Delta u\bigr)
      vec{OB}\simeq \bigl(\frac{\partial x}{\partial v}\Delta v,\;, \frac{\partial y}{\partial v}\Delta v\bigr)
    • \boxed{S\simeq \left|det\pmatrix{\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}}\right|\Delta u\Delta v =J(u,v)\Delta u\Delta v}
    • thus: \displaystyle\iint_D f(x,y)dxdy =\displaystyle\iint_E f\left(x(u,v),\;y(u,v)\right)|J(u,v)|dudv

  • ¶1: J(u,v) on polar coordinates:
    • \cases{x=r\cos\theta\\y=r\sin\theta},\;\cases{u↔r\\v↔\theta}
    • J(u,v)=\|det\pmatrix{\cos\theta&-r\sin\theta\\\sin\theta&r\cos\theta}\|=r
  • ¶2: \displaystyle\iint_D (x-y)e^{x+y}dxdy,\;D=\{(x,y)|0≤x+y≤2,\;0≤x-y≤2\}...(*)
    • let: x+y=u,\;v=x-y;\;\cases{x=\frac{1}{2}(u+v)\\y=\frac{1}{2}(u-v)}
    • J(u,v)=\left|det\pmatrix{\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&-\frac{1}{2}}\right| =\frac{1}{2}
    • (*)=\displaystyle\int_0^2\displaystyle\int_0^2 ve^{u}\frac{1}{2}dudv
      =\frac{1}{2}\displaystyle\int_0^2\bigl[ve^u\bigr]_0^2dv
      =\displaystyle\int_0^2(ve^2-v)dv=\frac{1}{2}\bigl[\frac{1}{2}e^2v^2-\frac{1}{2}v^2\bigr]_0^2=e^2-1

24. 重積分:

  • iterated/repeated integral: 累次積分
  • substitutive integral: 置換積分
  • antiderivative=primitive function: 原子関数
  • Jacobian matrix: ヤコビ行列 J_f
  • doubleintegral.gif
  • repeatedintegral.gif
  • doubleinteger2.gif
  • jacobian.gif

>Top 25. XXXX:

25. XXXX:

>Top 26. YYYY:

26. YYYY:

>Top 27. ZZZZ:

27. ZZZZ:

Comment
  • In studying effectively, it's important to accumulate in the manner of dicrete, heurisitic, empirical, and then formalized knowledge.
  • 効率的に学ぶには、離散的、発見的、実証的、そして形式化した知識を集めることが肝要である。

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