Föreläsning 19

Idag

Flödesintegraler

Exempel

Det elektrostatiska fältet

Elektro={x/(x^2+y^2+z^2)^(3/2),y/(x^2+y^2+z^2)^(3/2),z/(x^2+y^2+z^2)^(3/2)}

VectorPlot3D[Elektro,{x,-1,1},{y,-1,1},{z,-1,1}]

En del av fältet

VectorPlot3D[Elektro,{x,0.1,1},{y,0.1,1},{z,0.1,1}]

Ex

Fig1=Plot3D[If[x^2+y^2<1,x^2+y^2,0],{x,-1,1},{y,-1,1}]

Fig2=VectorPlot3D[{x*y,0,z^2},{x,-1,1},{y,-1,1},{z,0,1}]

Show[Fig1,Fig2]

f[r_,ϕ_]={r*Cos[ϕ],r*Sin[ϕ],r^2}

v1=D[f[r,ϕ],r]

v2=D[f[r,ϕ],ϕ]

v3=Cross[v1,v2]

v3=Simplify[%]

F={r^2*Cos[ϕ]*Sin[ϕ],0,r^4}

Integrate[Integrate[-F.v3,{ϕ,0,2*π}],{r,0,1}]

Ex

Fig1=VectorPlot3D[{x,y,z},{x,-1,1},{y,-1,1},{z,-1,1}]

Fig2=ParametricPlot3D[{Cos[t],Sin[t],s},{t,0,2*π},{s,-1,1}]

Show[Fig1,Fig2]

Ex

Fig1=VectorPlot3D[{y,-x,4},{x,0,1},{y,0,1},{z,0,1}]

Fig2=Plot3D[If[x^2+y^2<1,1-x^2-y^2,0],{x,0,1},{y,0,1}]

Show[Fig1,Fig2]

f[r_,ϕ_]={r*Cos[ϕ],r*Sin[ϕ],1-r^2}

v1=D[f[r,ϕ],r]

v2=D[f[r,ϕ],ϕ]

v3=Cross[v1,v2]

v3=Simplify[%]

F={r*Sin[ϕ],-r*Cos[ϕ],4}

Integrate[Integrate[F.v3,{ϕ,0,π/2}],{r,0,1}]

Ex

Fig1=VectorPlot3D[{2*x/(x^2+y^2),2*y/(x^2+y^2),1},{x,-1,1},{y,-1,1},{z,0,1}]

Fig2=ParametricPlot3D[{s*Cos[t],s*Sin[t], s^2},{s,0,1},{t,0,2*π}]

Show[Fig1,Fig2]

f[s_,t_]={s*Cos[t],s*Sin[t],s^2}

v1=D[f[s,t],s]

v2=D[f[s,t],t]

v3=Cross[v1,v2]

v3=Simplify[%]

F={2*Cos[t]/s,2*Sin[t]/s,1}

Integrate[Integrate[-F.v3,{t,0,2*π}],{s,0,1}]