% I = eye(2,2);
% dP = u * (dF + dF') + l * trace(dF) * I;
function dP = dPdF_fast_linear(global_u, global_l, dF)
    
    [~, lenout, m] = size(dF);
    
    dF_expand  = reshape(dF, 4, []);
    dFT = dF([1,3,2,4], :, :); 

    %% dP1 = u * (dF + dF')
    mu(1,1,:) = global_u(:);
    dP1 = (dF + dFT) .* mu; % 4 * l * m

    %% dP3 = l * trace(dF) * I;
    lambda = global_l(:)';               % 1 * m
    trF = reshape(dF_expand(1,:)+dF_expand(4,:), lenout, []); % l*m
    lm_tr(1,:,:) = lambda .* trF; % 1*l*m
    
    I = reshape(repmat([1;0;0;1], lenout, m), size(dF)); % expand eye(2) to 4*l*m
    dP2 = lm_tr .* I;

    dP = dP1 + dP2; % 4 * l * m  
end