Determinant of Jacobian of a displacement field

% J = jacobian(S)
%   Determinant of Jacobian of a displacement field
%
% Herve Lombaert, Jan. 8th, 2013
%
function det_J = jacobian(sx,sy,sz)

    % Gradients
    [gx_y,gx_x,gx_z] = gradient(sx);
    [gy_y,gy_x,gy_z] = gradient(sy);
    [gz_y,gz_x,gz_z] = gradient(sz);
    
    % Add identity
    gx_x = gx_x + 1;
    gy_y = gy_y + 1;
    gz_z = gz_z + 1;
    
    % Determinant
    det_J = gx_x.*gy_y.*gz_z + ...
            gy_x.*gz_y.*gx_z + ...
            gz_x.*gx_y.*gy_z - ...
            gz_x.*gy_y.*gx_z - ...
            gy_x.*gx_y.*gz_z - ...
            gx_x.*gz_y.*gy_z;
end