matlab实现曲面积分（surf_integral）

MATLAB语言并未直接提供曲线积分和曲面积分的现成函数。以下是曲面积分函数：

function I = surf_integral(f,vars,t,a,b)
%surf_integral
%第一类曲面积分
%   I = surf_integral(f, z, [x,y], [x_m,x_M], [y_m,y_M])
%   I = surf_integral(f, [x,y,z], [u,v], [u_m,u_M], [v_m,v_M])
%  Examples:
%  计算int_int(x^2*y+z*y^2)dS, 积分曲面如下：
%  x=ucosv, y=usinv, z=v, 0<=u<=a, 0<=v<=2*pi
%  MATLAB求解语句
%  syms u v; syms a positive;
%  x=u*cos(v); y=u*sin(v); z=v; f=x^2*y+z*y^2;
%  I = surf_integral(f,[x,y,z],[u,v],[0,a],[0,2*pi])
%
%第二类曲面积分
%   I = surf_integral([P,Q,R], z, [x,y], [x_m,x_M], [y_m,y_M])
%   I = surf_integral([P,Q,R], [x,y,z], [u,v], [u_m,u_M], [v_m,v_M])
%  注意：I = int_int_S(P*dydz+Q*dxdz+R*dxdy)
%  Examples:
%  计算曲面积分int_int(x*y+z)dxdy, 积分曲面如下：
%  (x/a)^2+(y/b)^2+(z/c)^2=1的上半部，且积分沿椭球面的上面。
%  引入参数方程：x=a*sin(u)*cos(v)，y=b*sin(u)*sin(v)，z=c*cos(u);
%  且0<=u<=pi/2, 0<=v<=2*pi 。
%  MATLAB求解语句
%  syms u v; syms a b c positive;
%  x=a*sin(u)*cos(v); y=b*sin(u)*sin(v); z=c*cos(u);
%  I = surf_integral([0, 0, x*y+z],[x,y,z],[u,v],[0,pi/2],[0,2*pi])

if length(f)==1
    if length(vars)~=1
        E = simplify(sum(diff(vars,t(1)).^2));
        F = sum(diff(vars,t(1)).*diff(vars,t(2)));
        G = simplify(sum(diff(vars,t(2)).^2));
    else
        E = simplify(1+diff(vars,t(1))^2);
        F = diff(vars,t(1))*diff(vars,t(2));
        G = simplify(1+diff(vars,t(2))^2);
    end
    I = int(int(simplify(f*sqrt(E*G-F^2)),t(1),a(1),a(2)),t(2),b(1),b(2));
else
    if length(vars)~=1
        A = diff(vars(2),t(1))*diff(vars(3),t(2)) - diff(vars(3),t(1))*diff(vars(2),t(2));
        B = diff(vars(3),t(1))*diff(vars(1),t(2)) - diff(vars(1),t(1))*diff(vars(3),t(2));
        C = diff(vars(1),t(1))*diff(vars(2),t(2)) - diff(vars(2),t(1))*diff(vars(1),t(2));
    else
        A = - diff(vars,t(1));
        B = - diff(vars,t(2));
        C = 1;
    end
    f = f(:); abc = [A, B, C];
    I = int(int(simplify(abc*f),t(1),a(1),a(2)),t(2),b(1),b(2));
end