Paraboloid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Plot

ParametricPlot3D[{a u Cos[v],a u Sin[v],u^2}/.a->1,{v,0,2π},{u,0,2}]

WriteLiveForm["parabol.m",Show[ParametricPlot3D[{u Cos[v],u Sin[v],u^2},{u,-3,3},{v,0,Pi},Boxed->False,Ticks->None,AspectRatio->Automatic,Axes->False],opts]]

Equations

GroebnerBasis[Append[Thread[{x,y,z}=={a u Cos[v],a u Sin[v],u^2}],Cos[v]^2+Sin[v]^2==1],{x,y,z},{u,Cos[v],Sin[v]},MonomialOrder->EliminationOrder]

eqn=-(x^2)-(y^2)+(a^2) z;

Curvatures

Curvatures[{a u Cos[v],a u Sin[v],u^2},{u,v},FullSimplify[#,a>0&&b>0&&0<v<2π&&u>0]&]

GaussianCurvature[-(x^2)-(y^2)+(a^2) z,{x,y,z}]

FullSimplify[PrincipalCurvatures[{a u Cos[v],a u Sin[v],u^2},{u,v}],a>0&&b>0&&0<v<2π&&u>0]

h-(Sqrt[u])parametrization

Curvatures[{a Sqrt[u/h]Cos[v],a Sqrt[u/h]Sin[v],u},{u,v}]

FullSimplify[PrincipalCurvatures[{a Sqrt[u/h]Cos[v],a Sqrt[u/h]Sin[v],u},{u,v}],a>0&&h>0&&0<u<h]

PowerExpand[FullSimplify[Integrate[1/2/h Sqrt[a^4+4a^2 h u],{v,0,2Pi},{u,0,h}]]]

FullSimplify[(((-(a^4)+((a^2)+4 (h^2)) a(Sqrt[(a^2)+4  (h^2)])) π)/(6 (h^2)))]

Simplify[#.#&[{a Sqrt[u/h]Cos[v],a Sqrt[u/h]Sin[v]}]]

Integrate[Pi(((a^2) z)/h),{z,0,h}]

Integrate[Pi(((a^2) z)/h)z,{z,0,h}]

(1/3) (a^2) (h^2) π/((1/2) (a^2) h π)

Metric Properties

u-v

Metric Tensor

MetricTensor[{a u Cos[v],a u Sin[v],u^2},{u,v}]//FullSimplify

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[{a u Cos[v],a u Sin[v],u^2},{u,v}],a>0]

Sqrt[u]-v

MetricTensor[{a Sqrt[u/h]Cos[v],a Sqrt[u/h]Sin[v],u},{u,v}]//FullSimplify

Centroid

{a Sqrt[u/h]Cos[v],a Sqrt[u/h]Sin[v],u}/.u->0

{a Sqrt[u/h]Cos[v],a Sqrt[u/h]Sin[v],u}/.u->h

FullSimplify[(a Sqrt[u/h]Cos[v])^2+(a Sqrt[u/h]Sin[v])^2]

x^2+y^2==a^2 (z/h)

With[{a=1,h=2},RegionPlot3D[z>(x^2+y^2)(h/(a^2))&&0≤z≤h,{x,-a,a},{y,-a,a},{z,0,h},PlotRange->All]]

Assuming[a>0&&h>0,Centroid[z>(x^2+y^2)(h/(a^2))&&0≤z≤h,{x,y,z}]]

v=Assuming[a>0&&h>0,Integrate[Boole[z>(x^2+y^2)(h/(a^2))],{z,0,h},{x,-a,a},{y,-a,a}]]

Assuming[a>0&&h>0,Centroid[z>(x^2+y^2)(h/(a^2))&&0≤z≤h,{z,x,y}]]//Timing

(i=Assuming[a>0&&h>0,Integrate[InertiaTensor[{x,y,z}]Boole[z>(x^2+y^2)(h/(a^2))],{z,0,h},{x,-a,a},{y,-a,a}]]/v)//Timing

Vector Properties

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

Vector Length

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

Normal Vector

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