surf_integral_surf_matlab_微积分_

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