MATLAB语言并未直接提供曲线积分的现成函数,因此,此处给出计算曲线积分的函数。
目录
函数说明应用举例第一类曲线积分第二类曲线积分 函数实现函数说明
function I = path_integral(F,vars,t,a,b)%path_integral%第一类曲线积分% I = path_integral(f, [x,y], t, t_m, t_M)% I = path_integral(f, [x,y,z], t, t_m, t_M)% Examples:% 计算int_l(z^2/(x^2+y^2))ds, l是如下定义的螺线% x=acost, y=asint, z=at, 0<=t<=2*pi, a>0% MATLAB求解语句% syms t; syms a positive;% x=a*cos(t); y=a*sin(t); z=a*t;% f=z^2/(x^2+y^2);% I=path_integral(f,[x,y,z],t,0,2*pi)%%第二类曲线积分% I = path_integral([P,Q], [x,y], t, a, b)% I = path_integral([P,Q,R], [x,y,z], t, a, b)% I = path_integral(F, v, t, a, b)% Examples:% 曲线积分int_l( (x+y)/(x^2+y^2)*dx - (x-y)/(x^2+y^2)*dy ),% l为正向圆周x^2+y^2=a^2% 正向圆周的参数函数描述: x=acost, y=asint, (0<=t<=2pi)% MATLAB求解语句% syms t; syms a positive;% x=a*cos(t); y=a*sin(t);% F=[ (x+y)/(x^2+y^2), -(x-y)/(x^2+y^2) ];% I=path_integral(F,[x,y],t,2*pi,0)
应用举例
第一类曲线积分
计算 ∫ l ( z 2 x 2 + y 2 ) d s \int_l(\frac{z^2}{x^2+y^2})ds ∫l(x2+y2z2)ds,其中 l l l是如下定义的螺线: x = a c o s t , y = a s i n t , z = a t , 0 ≤ t ≤ 2 ∗ π , a > 0 x=acost, ~~ y = asint, ~~z=at, ~~0 \leq t \leq2*\pi, ~~a>0 x=acost,y=asint,z=at,0≤t≤2∗π,a>0求解方法
syms t; syms a positive;x=a*cos(t); y=a*sin(t); z=a*t;f=z^2/(x^2+y^2);I=path_integral(f,[x,y,z],t,0,2*pi)
第二类曲线积分
计算曲线积分 ∫ l ( x + y x 2 + y 2 d x − x − y x 2 + y 2 d y ) , \int_l( \frac{x+y}{x^2+y^2}dx - \frac{x-y}{x^2+y^2}dy ), ∫l(x2+y2x+ydx−x2+y2x−ydy),其中 l l l为正向圆周 x 2 + y 2 = a 2 x^2+y^2=a^2 x2+y2=a2
注:正向圆周的参数函数描述: x = a c o s t , y = a s i n t , ( 0 ≤ t ≤ 2 p i ) x=acost, y=asint, (0\leq t \leq2pi) x=acost,y=asint,(0≤t≤2pi)
求解方法
syms t; syms a positive;x=a*cos(t); y=a*sin(t);F=[ (x+y)/(x^2+y^2), -(x-y)/(x^2+y^2) ];I=path_integral(F,[x,y],t,2*pi,0)
函数实现
function I = path_integral(F,vars,t,a,b)if length(F)==1I = int(F*sqrt(sum(diff(vars,t).^2)),t,a,b);elseF = F(:).';vars = vars(:);I = int(F*diff(vars,t),t,a,b);end
此函数源文件可前往下面网址下载:
path_integral.m下载通道