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2 E! y- n: t5 KMATLAB语言并未直接提供曲面积分的现成函数，因此，此处给出计算曲面积分的函数。
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' H1 t/ {/ h2 H& B. _ |目录
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第一类曲面积分
4 j- }' A- l( S6 C5 h4 e* |0 Y第二类曲面积分
- L9 k4 i" B& ~! P函数实现
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, ~" h# H- B7 u5 b$ D4 }, }& \函数说明
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7 `; M2 |( r, U9 m3 R# F- 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])
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应用举例
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* n( h! j6 w/ M第一类曲面积分, ~. o6 t% x% a5 ^- D
1. 计算
, 积分曲面如下：
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* y! b/ c# P2 F1 g求解方法
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- 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])% }2 E$ I8 S5 Z) v Y( D
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第二类曲面积分
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( p- }2 x! i& ^( X I( o" {2. 计算曲面积分
, 积分曲面如下：
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的上半部，且积分沿椭球面的上面。$ f9 Z a: ^3 Z
注：引入参数方程：
; 且
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0 I! J2 u( P, U+ B; @6 x/ Q( M4 I求解方法
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g x, m% ~/ c- 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])
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函数实现1 j& L5 V. S! h4 j4 `% M1 l# c) U
1 i5 O+ P: |' Z2 n" X3 `- function I = surf_integral(f,vars,t,a,b)
- 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
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发表于 2021-2-1 10:03
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