Curl

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/Curl.nb.

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

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

Initialization

f={(Subscript[F, x])[x,y,z],(Subscript[F, y])[x,y,z],(Subscript[F, z])[x,y,z]};

Built-In

<<Calculus`VectorAnalysis`

c=Curl[f,Cartesian[x,y,z]]

Matrix Operator Definition

curl1[f_,{x_,y_,z_}]:=Inner[#1[#2]&,
{
{0&,-D[#,z]&,D[#,y]&},
{D[#,z]&,0&,-D[#,x]&},
{-D[#,y]&,D[#,x]&,0&}
},f]

c1=curl1[f,{x,y,z}]

Determinant Definition

curl2[f_,{x_,y_,z_}]:=Det[{
{hat[x],hat[y],hat[z]},
{dd[x],dd[y],dd[z]},
{f[[1]],f[[2]],f[[3]]}
}]/.dd[n_]g_:>D[g,n]/.{hat[x]->{1,0,0},hat[y]->{0,1,0},hat[z]->{0,0,1}}

c2=curl2[f,{x,y,z}]

Equivalence

c==c1==c2