《MATLAB微积分运算》PPT课件.ppt

第7章 MATLAB微积分运算,2,7.1 MATLAB微分运算,7.1.1 数值微分,1. 数值差分,例7-1,编制如下命令文件,clear;clc; x=1:9 dx=diff(x) d2x=diff(x,2),运行结果为,x = 1 2 3 4 5 6 7 8 9 dx = 1 1 1 1 1 1 1 1 d2x = 0 0 0 0 0 0 0,3,2. 数值微分,例7-2,编制如下指令文件,clear;clc; x=linspace(0,pi,360); y=sin(x); dx=diff(x); dy=diff(y); dyx=dy./dx; plot(x,y,r-) hold on n=length(x); plot(x(1:n-1),dyx,b-) legend(sin(x),dsin(x)/dx),运行结果如图7-1所示,图 7-1,4,例 7-3,编制如下程序,clear;clc; n = 360 x = linspace(0, 2*pi, n); y = exp(-0.2*x).*(cos(x)+sin(x); dy = diff(y); dx = diff(x); dyx = dy./dx; plot(x,y,r-) hold on plot(x(1:n-1),dyx,b-.) legend(y,dy/dx),运行结果如图7-2所示,图 7-2,5,3. 曲面法线,例 7-4,编制如下命令文件,clear;clc; af=linspace(0, pi, 36) x=2*cos(af); X, Y, Z = cylinder(x, 12); surfnorm(X, Y, Z),运行结果如图7-3所示,图 7-3,6,例 7-5,编制如下程序,clear;clc; x = linspace(0, pi, 30); y = linspace(0, pi, 30); X, Y = meshgrid(x, y); Z = sin(X).*cos(Y); surfnorm(X, Y, Z) box on,运行结果如图7-4所示,图 7-4,7,例 7-6,编制如下程序,clear;clc; X, Y, Z = sphere(20); surfnorm(X, Y, Z) axis square box on,运行结果如图7-5所示,图 7-5,8,7.1.2 符号微分,例 7-7,编制如下命令文件,clear;clc; F=sin(3*x) Fx=diff(F,x) Fxx=diff(F, x, 2),运算结果为,F = sin(3*x) Fx = 3*cos(3*x) Fxx = -9*sin(3*x),9,例 7-8,编制如下程序,clear;clc; Z = exp(a*x+b*y)*(sin(x*y)+sin(x)+cos(y) Zx = diff(Z, x) Zy = diff(Z, y),运行结果为,Z = exp(a*x+b*y)*(sin(x*y)+sin(x)+cos(y) Zx = a*exp(a*x+b*y)*(sin(x*y)+sin(x)+cos(y)+exp(a*x+b*y)*(cos(x*y)*y+cos(x) Zy = b*exp(a*x+b*y)*(sin(x*y)+sin(x)+cos(y)+exp(a*x+b*y)*(cos(x*y)*x-sin(y),10,7.1.3 多元函数的导数,1. 函数梯度,例 7-9,编制如下命令文件,clear;clc; v = -2:0.2:2; x,y = meshgrid(v); z = x .* exp(-x.2 - y.2); px,py = gradient(z); contour(v,v,z), hold on, quiver(v,v,px,py),运行结果如图7-6所示,图 7-6,11,例 7-10,编制如下程序,clear;clc; x = linspace(0, pi, 5); y = x; z = x; X, Y, Z = meshgrid(x, y, z); F = sin(X) + sin(Y) + sin(Z); px,py,pz = gradient(F); quiver3(X,Y,Z, px,py,pz),运行结果如图7-7所示,图 7-7,12,2. Jacobi矩阵,例 7-11,编制如下命令文件,clear;clc; syms x y z F= 3*x-cos(x*y)-5 x2-80*(y+0.5)2+sin(z) exp(-x*y)+20*z+10 JF=jacobian(F,x y z),运行结果为,F = 3*x-cos(x*y)-5 x2-80*(y+1/2)2+sin(z) exp(-x*y)+20*z+10 JF = 3+sin(x*y)*y, sin(x*y)*x, 0 2*x, -160*y-80, cos(z) -y*exp(-x*y), -x*exp(-x*y), 20,13,例 7-12,编制如下程序,clear;clc; syms x y a b F=exp(a*x+b*y); sin(x)+cos(y) JF=jacobian(F,x y),运行结果为,F = exp(a*x+b*y) sin(x)+cos(y) JF = a*exp(a*x+b*y), b*exp(a*x+b*y) cos(x), -sin(y),14,7.2 MATLAB积分运算,7.2.1 一元数值积分,例 7-13,编制如下命令文件,clear;clc; fun=inline(x./(x.2+4),x) F1=quad(fun,0,5) F2=quadl(fun,0,5),运行结果为,fun = Inline function: fun(x) = x./(x.2+4) F1 = 0.9905 F2 = 0.9905,15,例 7-14,编制如下函数文件,function f=fun(x) f=exp(-0.5*x)+x;,在指令窗口中输入,clear;clc; F1=quad(fun,0,5) F2=quadl(fun,0,5),运行结果为,F1 = 14.3358 F2 = 14.3358,16,7.2.2 多元数值积分,例 7-15,编制如下命令文件,clear;clc; fun=inline(x.y+1,x,y) F1=dblquad(fun, 0, 1, 1, 2) F2=dblquad(fun, 1, 2, 0, 1),运行结果为,fun = Inline function: fun(x,y) = x.y+1 F1 = 1.4055 F2 = 2.2293,17,7.2.3 符号积分,例 7-16,编制命令文件,clear;clc; syms a b x fun=cos(a*x+b) F=int(fun, x),运算结果为,fun = cos(a*x+b) F = 1/a*sin(a*x+b),例 7-17,编制命令文件,clear;clc; syms a b x fun=cos(a*x)+b F=int(fun, x, 0, pi),运算结果为,fun = cos(a*x)+b F = (sin(pi*a)+b*a*pi)/a,18,7.3 综合实例,7.3.1 平面图形面积,例7-18,编制函数文件,function y = fun(x) global a b y = b*sqrt(1-x.2/a2);,运行程序,clear;clc; global a b a=2; b=1; S1 = 2*quad(fun,-a,a) x1=linspace(-a,a,101); y1=fun(x1); plot(x1,y1,b-,x1,-y1,b-) hold on a=1.5; b=0.5; S2 = 2*quad(fun,-a,a) x2=linspace(-a,a,101); y2=fun(x2); plot(x2,y2,r-,x2,-y2,r-),19,计算结果为,S1 = 6.2832 S2 = 2.3562,绘制的椭圆图形如图7-8所示,图 7-8,20,7.3.2 点的运动速度,例 7-19,编制如下程序,clear;clc; t=linspace(