Cylidrical Coordinates

Author

Eric W. Weisstein
October 26, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CoordinateGeometry/CylindricalCoordinates.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

coords={r Cos[θ],r Sin[θ],z};
vars={r,θ,z};
ranges=0≤θ<2π&&r≥0;

VectorAnalysis Convention

<<Calculus`VectorAnalysis`

SetCoordinates[Cylindrical]

Cylindrical::usage

Scale Factors

FullSimplify[ScaleFactors[coords,vars],ranges]

Unit Vectors

MatrixForm/@(unitvecs=FullSimplify[UnitVectors[coords,vars],ranges])//TraditionalForm

xform=Join[
Thread[unitvecs->ToExpression/@(#<>"hat"&/@(ToString/@vars))],
Thread[-unitvecs->-ToExpression/@(#<>"hat"&/@(ToString/@vars))],
{{0,0,0}->zero}
]

unitvecs/.xform

Derivatives of Unit Vectors

(dunitvecs=Outer[D,unitvecs,vars,1])//MatrixForm//TraditionalForm

Transpose[dunitvecs/.xform]//MatrixForm

Gradients

MatrixForm@(grads=Simplify[Gradients[coords,vars],ranges])

MatrixForm[(grads/.xform)/.x_List?VectorQ:>PolynomialGCD@@x(Cancel[(x/(PolynomialGCD@@x))]/.xform)]

Christoffel Symbols

Misner et al.

MatrixForm/@FullSimplify[ChristoffelSymbols[coords,vars],ranges]

Arfken

MatrixForm/@FullSimplify[ChristoffelSymbols2[coords,vars],ranges]

Cross Products

Subsets[unitvecs,{2}]/.xform

Simplify[Cross@@@Subsets[unitvecs,{2}],ranges]/.xform

Jacobian

Simplify[JacobianDet[coords,vars],ranges]

Gradient

FullSimplify[Grad[coords,vars][f[Sequence@@vars]],ranges]

Laplacian

FullSimplify[Laplacian[coords,vars][f[Sequence@@vars]],ranges]

Properties

<<Calculus`VectorAnalysis`
coord=Cylindrical;

Usage

?Cylindrical

ScaleFactors[coord]

CoordinatesToCartesian[Coordinates[coord],coord]

CoordinatesFromCartesian[{x,y,z},coord]

Div[Through[{F1,F2,F3}@@Coordinates[coord]],coord]//TraditionalForm

Curl[Through[{F1,F2,F3}@@Coordinates[coord]],coord]//TraditionalForm

Grad[f@@Coordinates[coord],coord]//TraditionalForm

Laplacian[f@@Coordinates[coord],coord]//TraditionalForm