Funnel

Author

Eric W. Weisstein
June 10, 2014

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/Funnel.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Funnel.html.

©2014 Wolfram Research, Inc. except for portions noted otherwise

Plot

ParametricPlot3D[{u Cos[v],u Sin[v], Log[u]},{u,0,1},{v,0,2π}]

ParametricPlot3D[{u Cos[v],u Sin[v], Log[u]},{u,.1,1},{v,0,2π}]

LiveGraphics3D

WriteLiveForm["funnel.m",Show[ParametricPlot3D[{r Cos[t],r Sin[t],Log[r]},{r,.1,1},{t,0,2Pi},Axes->None,Boxed->False],opts]]

Equations

surf={u Cos[v],u Sin[v],a Log[u]};

Cartesian

Simplify[Total[Take[surf,2]^2]]

Solve[z==a Log[u],u]

z==Log[u]

eqn=x^2+y^2-Exp[2z/a];

FullSimplify[x^2+y^2-Exp[2z/a]/.Thread[{x,y,z}->surf]]

FullSimplify[Log[x^2+y^2]-2z/a/.Thread[{x,y,z}->surf],u>0]

FullSimplify[(1/2)a Log[x^2+y^2]-z/.Thread[{x,y,z}->surf],u>0]

Curvature

Curvatures[{u Cos[v],u Sin[v], a Log[u]},{u,v},Simplify[#,a>0]&]

Integrate[2Pi Sqrt[1+u^2],{u,0,Infinity}]

FullSimplify[GaussianCurvature[eqn,{x,y,z}],z>0]

Metric Properties

surf={u Cos[v],u Sin[v], a Log[u]};

Metric Tensor

MetricTensor[surf,{u,v}]//FullSimplify

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[surf,{u,v}],a>0&&u>0]

Vector Properties

eqn={a u Cos[v],a u Sin[v],Log[u]};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&u∈Reals&&0<v<π;

Vector Length

Assuming[assum,FullSimplify[Norm[eqn]/.rules]]

Normal Vector

Assuming[assum,FullSimplify[NormalVector[eqn,{u,v}]/.rules]]