Cartesian Coordinates

Author

Eric W. Weisstein
October 18, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/CartesianCoordinates.nb.

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

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

Additional Authors

Original figure by Margherita Barile
July 11, 2002

Initialization

coords={x,y,z};
vars={x,y,z};
ranges={};

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

dunitvecs/.xform

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]

Figure

Show[GraphicsArray[
ParametricPlot[{2+2Cos[t],Sqrt[2]Sin[t]}, {t,0,#1},
Axes->#2,
AxesLabel->TraditionalForm/@{x,y},
PlotStyle->Red,
Ticks->None,
AxesOrigin->{0,0},
Epilog->#3,
PlotRange->{{-.5,4.5},{-2,2}},
AspectRatio->Automatic,
PlotLabel->StyleForm[#4,FontFamily->"Times",FontSlant->"Italic"],
DisplayFunction->Identity]&@@@
{
{2Pi,True,{},"modern"},
{Pi,False,{Red,Line[{{0,0},{4,0}}]},"Descartes"}
}]]