MATLAB微积分运算.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=1:9dx=diff(x)d2x=diff(x,2),运行结果为,x=1 2 3 4 5 6 7 8 9dx=1 1 1 1 1 1 1 1d2x=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 onn=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 onplot(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 squarebox 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 zF=3*x-cos(x*y)-5x2-80*(y+0.5)2+sin(z)exp(-x*y)+20*z+10JF=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 bF=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=0.9905F2=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=14.3358F2=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+1F1=1.4055F2=2.2293,17,7.2.3 符号积分,例 7-16,编制命令文件,clear;clc;syms a b xfun=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 xfun=cos(a*x)+bF=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 by=b*sqrt(1-x.2/a2);,运行程序,clear;clc;global a ba=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 ona=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=6.2832S2=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(0,5,101);x=50*t;y=500-5*t.2;plot(x,y)vx=diff(x)./diff(t);vy=diff(y)./diff(t);v=sqrt(vx.2+vy.2);t=t(1:100);figure(2)plot(t,v),运行后得到质点的运动轨迹和速度变化规律图如图7-9和7-10所示,图 7-9,图 7-10,
