1######################################################################## 2## 3## Copyright (C) 2009-2021 The Octave Project Developers 4## 5## See the file COPYRIGHT.md in the top-level directory of this 6## distribution or <https://octave.org/copyright/>. 7## 8## This file is part of Octave. 9## 10## Octave is free software: you can redistribute it and/or modify it 11## under the terms of the GNU General Public License as published by 12## the Free Software Foundation, either version 3 of the License, or 13## (at your option) any later version. 14## 15## Octave is distributed in the hope that it will be useful, but 16## WITHOUT ANY WARRANTY; without even the implied warranty of 17## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 18## GNU General Public License for more details. 19## 20## You should have received a copy of the GNU General Public License 21## along with Octave; see the file COPYING. If not, see 22## <https://www.gnu.org/licenses/>. 23## 24######################################################################## 25 26## -*- texinfo -*- 27## @deftypefn {} {[@var{cx}, @var{cy}, @var{cz}, @var{v}] =} curl (@var{x}, @var{y}, @var{z}, @var{fx}, @var{fy}, @var{fz}) 28## @deftypefnx {} {[@var{cz}, @var{v}] =} curl (@var{x}, @var{y}, @var{fx}, @var{fy}) 29## @deftypefnx {} {[@dots{}] =} curl (@var{fx}, @var{fy}, @var{fz}) 30## @deftypefnx {} {[@dots{}] =} curl (@var{fx}, @var{fy}) 31## @deftypefnx {} {@var{v} =} curl (@dots{}) 32## Calculate curl of vector field given by the arrays @var{fx}, @var{fy}, and 33## @var{fz} or @var{fx}, @var{fy} respectively. 34## @tex 35## $$ curl F(x,y,z) = \left( {\partial{d} \over \partial{y}} F_z - {\partial{d} \over \partial{z}} F_y, {\partial{d} \over \partial{z}} F_x - {\partial{d} \over \partial{x}} F_z, {\partial{d} \over \partial{x}} F_y - {\partial{d} \over \partial{y}} F_x \right)$$ 36## @end tex 37## @ifnottex 38## 39## @example 40## @group 41## / d d d d d d \ 42## curl F(x,y,z) = | -- Fz - -- Fy, -- Fx - -- Fz, -- Fy - -- Fx | 43## \ dy dz dz dx dx dy / 44## @end group 45## @end example 46## 47## @end ifnottex 48## The coordinates of the vector field can be given by the arguments @var{x}, 49## @var{y}, @var{z} or @var{x}, @var{y} respectively. @var{v} calculates the 50## scalar component of the angular velocity vector in direction of the z-axis 51## for two-dimensional input. For three-dimensional input the scalar 52## rotation is calculated at each grid point in direction of the vector field 53## at that point. 54## @seealso{divergence, gradient, del2, cross} 55## @end deftypefn 56 57function varargout = curl (varargin) 58 59 fidx = 1; 60 if (nargin == 2) 61 sz = size (varargin{fidx}); 62 dx = (1:sz(2))(:); 63 dy = (1:sz(1))(:); 64 elseif (nargin == 3) 65 sz = size (varargin{fidx}); 66 dx = (1:sz(2))(:); 67 dy = (1:sz(1))(:); 68 dz = (1:sz(3))(:); 69 elseif (nargin == 4) 70 fidx = 3; 71 dx = varargin{1}(1,:); 72 dy = varargin{2}(:,1); 73 elseif (nargin == 6) 74 fidx = 4; 75 dx = varargin{1}(1,:,1)(:); 76 dy = varargin{2}(:,1,1)(:); 77 dz = varargin{3}(1,1,:)(:); 78 else 79 print_usage (); 80 endif 81 82 if (nargin == 4 || nargin == 2) 83 if (! size_equal (varargin{fidx}, varargin{fidx + 1})) 84 error ("curl: size of X and Y must match"); 85 elseif (ndims (varargin{fidx}) != 2) 86 error ("curl: X and Y must be 2-D matrices"); 87 elseif ((length (dx) != columns (varargin{fidx})) 88 || (length (dy) != rows (varargin{fidx}))) 89 error ("curl: size of dx and dy must match the respective dimension of X and Y"); 90 endif 91 92 dFx_dy = gradient (varargin{fidx}.', dy, dx).'; 93 dFy_dx = gradient (varargin{fidx + 1}, dx, dy); 94 rot_z = dFy_dx - dFx_dy; 95 av = rot_z / 2; 96 if (nargout == 0 || nargout == 1) 97 varargout{1} = av; 98 else 99 varargout{1} = rot_z; 100 varargout{2} = av; 101 endif 102 103 elseif (nargin == 6 || nargin == 3) 104 if (! size_equal (varargin{fidx}, varargin{fidx + 1}, varargin{fidx + 2})) 105 error ("curl: size of X, Y, and Z must match"); 106 elseif (ndims (varargin{fidx}) != 3) 107 error ("curl: X, Y, and Z must be 2-D matrices"); 108 elseif ((length (dx) != size (varargin{fidx}, 2)) 109 || (length (dy) != size (varargin{fidx}, 1)) 110 || (length (dz) != size (varargin{fidx}, 3))) 111 error ("curl: size of dx, dy, and dz must match the respective dimesion of X, Y, and Z"); 112 endif 113 114 [~, dFx_dy, dFx_dz] = gradient (varargin{fidx}, dx, dy, dz); 115 [dFy_dx, ~, dFy_dz] = gradient (varargin{fidx + 1}, dx, dy, dz); 116 [dFz_dx, dFz_dy] = gradient (varargin{fidx + 2}, dx, dy, dz); 117 rot_x = dFz_dy - dFy_dz; 118 rot_y = dFx_dz - dFz_dx; 119 rot_z = dFy_dx - dFx_dy; 120 l = sqrt(varargin{fidx}.^2 + varargin{fidx + 1}.^2 + varargin{fidx + 2}.^2); 121 av = (rot_x .* varargin{fidx} + 122 rot_y .* varargin{fidx + 1} + 123 rot_z .* varargin{fidx + 2}) ./ (2 * l); 124 125 if (nargout == 0 || nargout == 1) 126 varargout{1} = av; 127 else 128 varargout{1} = rot_x; 129 varargout{2} = rot_y; 130 varargout{3} = rot_z; 131 varargout{4} = av; 132 endif 133 endif 134 135endfunction 136 137 138%!test 139%! [X,Y] = meshgrid (-20:20,-22:22); 140%! av = curl (2*(X-Y), Y); 141%! assert (all (av(:) == 1)); 142%! [cz,av] = curl (2*(X-Y), Y); 143%! assert (all (cz(:) == 2)); 144%! assert (all (av(:) == 1)); 145%! [cz,av] = curl (X/2, Y/2, 2*(X-Y), Y); 146%! assert (all (cz(:) == 4)); 147%! assert (all (av(:) == 2)); 148%! assert (size_equal (X,Y,cz,av)); 149