好评再送精通matlab2011a符号计算

第5章符号计算写能力的尺度”是相对数值计算的“有限精度、有限空间”而言的。的符Maple。现在，随MuPAD。虽然尽力地保持着形式上的向前兼容，但引擎换装确实导致符号计算环境发生了根本改变。MuPAD引擎的符号计算而写，分三个层面：供的（前台）函数实施的符号计算及仿真。本章第1节比较完整地描述了符号计算的机分析和代数方程。第7节代数状态方程和第8节数据探索的内容，用以表现现代计算能力对传统方法或技巧的冲击。5.9第5章符号计算写能力的尺度”是相对数值计算的“有限精度、有限空间”而言的。的符Maple。现在，随MuPAD。虽然尽力地保持着形式上的向前兼容，但引擎换装确实导致符号计算环境发生了根本改变。MuPAD引擎的符号计算而写，分三个层面：供的（前台）函数实施的符号计算及仿真。本章第1节比较完整地描述了符号计算的机分析和代数方程。第7节代数状态方程和第8节数据探索的内容，用以表现现代计算能力对传统方法或技巧的冲击。5.9节。在这一节中不M函数文件，如何利用符号函数制作用户所需的模块。evalinfevalMuPAD空间完成符号计算。仍可借鉴。值得：随书光盘mbook目录上的“ch05_符号计算.doc”保存有本章全部算例的运MEX文件。5.15.1.1符号对象的产生和识别基本符号对象的创建1定义符号数字和符号常数2定义基本符号变量3定义元符号表达式5.1.2符号计算中的算符和函数指令1符号计算中的算符2符号计算中的函数指令5.1.3符号对象、变量、变量的识别1符号对象的识别】数据对象及其识别指令的使用。（1）clearMn=[a,b;c,d]Ms=sym(Mc)Mn=1324Mc=】数据对象及其识别指令的使用。（1）clearMn=[a,b;c,d]Ms=sym(Mc)Mn=1324Mc=Ms=[a,b][c,d]（2）SizeMn=2SizeMc=1SizeMs=2292（3）CMn=CMc=charCMs=sym（4）isa(Mn,'double')isa(Mc,'char')isa(Ms,'sym')ans=1ans=1ans=1（5）whosMnMcMsNameSizeBytesClassAttributesMcMnMs1x92x22x2183260charsym2符号变量及变量的认定（1）clearsymsabktwxyzXAc=sym('22');f=sym('M*N');ex1=c+f+(exp(-a*t)*sin(w*t)+b*y+k*x+A*X+log(z))ex1=符号变量的机器辨认。log(z)+b*y+k*x+s *w)/exp(a*t)+A*X+M*N+22（2）2符号变量及变量的认定（1）clearsymsabktwxyzXAc=sym('22');f=sym('M*N');ex1=c+f+(exp(-a*t)*sin(w*t)+b*y+k*x+A*X+log(z))ex1=符号变量的机器辨认。log(z)+b*y+k*x+s *w)/exp(a*t)+A*X+M*N+22（2）ans=%<6>[A,M,N,X,a,b,k,t,w,x,y,z]findsym(ex1)ans=（3）ans=%<8>[x,y,w,z,t,k,b,a,X,N,M,A]findsym(ex1,13)%ans=（4）who%<10>%Yourvariablesare:AXaansbcex1fktwx yz（5）iex1db= (ex1,b)dex1dx=1/ziex1db=(y*b^2)/2+(log(z)%<11>%<12>+k*x+s*w)/exp(a*t)+A*X+M*N+22)*b】元符号表达式的定义、符号变量、（1）clearf=sym('M*N');ex2=符号变量的机器辨认。log(z)+b*y+k*x+s*w)/exp(a*t)+A*X+M*N+22（2）symvar(ex2)%<5>ans=[A,M,N,X,a,b,k,t,w,x,y,z]（3）ans=[x,y,w,z,t,k,b,a,X,N,M,A]（4）who%<7>Yourvariablesare:anscex2fans=[A,M,N,X,a,b,k,t,w,x,y,z]（3）ans=[x,y,w,z,t,k,b,a,X,N,M,A]（4）who%<7>Yourvariablesare:anscex2f（5）dex1dx=diff(ex2,z)???Undefinedfunction%<8>variableor'z'.dex1db=(ex2,b)%<9>variable???Undefinedfunctionor'b'.（6）E3=sym('a*sqrt(theta)')%???Errorusing==>sym.sym>convertExpresat2547Error:argumentmustbeof'Type::Arithmetical'[sqrt]Errorsin==>sym.sym>convertCharat2458=convertExpres (x);Errorin==>sym.sym>convertCharWithOptionats=convertChar(x);2441Errorin==>sym.sym>tomupadat2195S=convertCharWithOption(x,a);Errorin==>sym.sym>sym.symat111S.s=tomupad(x,'');E4=sym('a*sqrt(theta123)')E4=a*theta123^(1/2)】symvar确定（1）clear变量是对整个数组进行的。symsawxyzA=[a+b*x,y*s )+w;x*exp(-t),log(z)+v]A=[a+b*x,w+y*s)][x/exp(t), v+log(z)]（2）ans=xsymvar(A,3)ans=[x,y,w]symvar(A,10)ans=[x,y,w,z,v,[x,y,w]symvar(A,10)ans=[x,y,w,z,v,t,b,a]5.1.4符号运算机理和变量假设1符号运算的工作机理2（1）对符号变量的限定性假设（2）3清除变量和撤销假设】syms对变量所作限定性假设的影响。（1）symsxclearrf=solve(f,x)rf=-1/21/41/4-(79^(1/2)*i)/4+(79^(1/2)*i)/4evalin(symengine,'getprop(x)')ans=C_%<4>evalin(symengine,'property::showprops(x);')%<5>ans=[emptysym]（2）symsxrealrfr=-1/2evalin(symengine,'getprop(x)')ans=R_%<8>evalin(symengine,'property::showprops(x);')%<9>ans=[xinR_]（3）clearxsymsxrg=solve(g,x)Warning:Explicitsolutioncouldnotbefound.>Insolveat81rg=[emptysym]（4）symsxclearrgWarning:Explicitsolutioncouldnotbefound.>Insolveat81rg=[emptysym]（4）symsxclearrg=-1/2-(19^(1/2)*i)/2-1/2+(19^(1/2)*i)/2evalin对符号变量进行假设的设定和撤销。（1）clearsymsxf=x^3-5.25*x-2.5;rf=feval(symengine,'solve',f,'x')rf=[-1/2,5/2,-2]%<4>（2）x=sym(x,'itive');rf=feval(symengine,'solve',f,'x')rf=5/2%<6>rfs=solve(f,x)rfs=5/2evalin(symengine,'getprop(x)')ans=%<8>Dom::erval(0,Inf)evalin(symengine,'deletex')ans=C_%<10>（3）evalin(symengine,'ame(x,Type::eger);')%<11>rf=feval(symengine,'solve',f,'x')rf=-2%<12>evalin(symengine,'ameAlso(x>0)')%<13>%<14>%<15>%<16>rf_evc=evalin(symengine,['solve(',char(f),')'])rf_f=feval(symengine,'solve',f,'x')rf_ev=[emptysymrf_evc=[emptysymrf_f=[emptysym]]]evalin(symengine,'ame(x,plex);')%<17>%<18>rf=feval(symengine,'solve',f,'x')rf=[-1/2,5/2,-2]（4）syms%<19>'ans''f''rf''rf_ev''rf_evc''rf_f''rfs''x'5.1.5符号帮助及其他常用指令1符号的帮助体系（[-1/2,5/2,-2]（4）syms%<19>'ans''f''rf''rf_ev''rf_evc''rf_f''rfs''x'5.1.5符号帮助及其他常用指令1符号的帮助体系（1）“直接调用符号计算指令”的求助mfun调用的“特殊函数指令”的求助evalinfeval调用的“MuPAD指令”的求助。laplace,erfc,rec三个指令的求助过程。（1）“可直接调用的符号计算指令”的帮助信息搜索（2）5.1-2帮助浏览器展示全部特殊函数符号计算指令及用法（3）5.1-2帮助浏览器展示全部特殊函数符号计算指令及用法（3）5.1-3MuPAD帮助浏览器展示的rec指令帮助信息2服务于符号的其他指令】各种帮助指令、符号变量罗列指令（1）clearallDa=1.2;Dw=1/3;syms5.1-3MuPAD帮助浏览器展示的rec指令帮助信息2服务于符号的其他指令】各种帮助指令、符号变量罗列指令（1）clearallDa=1.2;Dw=1/3;symssaswsxsysz%<2>%%%<5>%<6>symsABitivesymsCreal（2）whosName%<7>BytesSizeClassAttributesABCDaDwsaswsxsysz1x11x11x11x11x11x11x11x11x11x1606060886060606060symsymsymdoubledoublesymsymsymsymsym（3）syms%<8>'A''B''C''sa''sw''sx''sy''sz'（4）ans=%<9>[A,B,C]（5）clearAsyms%<11>'B''C''ans''sa''sw''sx''sy''sz'evalin(symengine,'anames(Properties)')ans=（3）syms%<8>'A''B''C''sa''sw''sx''sy''sz'（4）ans=%<9>[A,B,C]（5）clearAsyms%<11>'B''C''ans''sa''sw''sx''sy''sz'evalin(symengine,'anames(Properties)')ans=[A,B,C]%<12>（6）symsBclear%<13>syms'B''C''ans''sa''sw''sx''sy''sz'evalin(symengine,'anames(Properties)')ans=[A,C]%<15>5.2数字类型转换及符号表达式操作5.2.1数字类型及转换1三种数字类型及转换指令5.2-1符号、字符、数值间的相互转换2双精度数字向符号数字转换5.2-1】本例目的：准确符号数字的产生；双精度数字转换成符号数字的各种格式；由双精度数字转换而得符号数的误差；vpa的用法；数字类别的判断。（1）clearsa=sym('1/3')sa=1/3sb=pi+2双精度数字向符号数字转换5.2-1】本例目的：准确符号数字的产生；双精度数字转换成符号数字的各种格式；由双精度数字转换而得符号数的误差；vpa的用法；数字类别的判断。（1）clearsa=sym('1/3')sa=1/3sb=pi+5^(1/2)%<2>%<3>（2）formonga=1/3%<5>%<6>%<7>%<8>b=pi+sqrt(5)a=0.333333333333333b=sa_a=0.0sb_b=48593542564188（3）asr=sym(a)bsr=sym(b)sa_asr=vpa(asr=1/3r)r)bsr=sa_asr=0.0sb_bsr=48593542564188（4）ase=sym(a,'e')sa_ase=vpa(sb_bse=vpa(sase=1/3-eps/12e)e)bse=sa_ase=0.0sb_bse=33333333333333*eps48593542564188（5）asf=sym(a,'f')sa_asf=vpa(sb_bsf=vpa(sf)f)asf=6004799503160661/18014398509481984bsf=sa_asf=sb_bsf=4039386113484748593542564188（6）asd=sym(a,'d')sa_asd=vpa(sb_bsd=vpa(sasd=0.bsd=5.sa_asd（5）asf=sym(a,'f')sa_asf=vpa(sb_bsf=vpa(sf)f)asf=6004799503160661/18014398509481984bsf=sa_asf=sb_bsf=4039386113484748593542564188（6）asd=sym(a,'d')sa_asd=vpa(sb_bsd=vpa(sasd=0.bsd=5.sa_asd=d)d)8296162562473904944121092560.0000000000000000sb_bsd=3333332703779219638453173161（7）ans=sym的类别asr的类别')disp([class(a),bl s(4),class(asr),bls(7),class(sa_a)])a的类别double的类别sa_a的类别symsymdoublestr2double的区别。（1）clearformong%<2>ad=1/2+3^(2/3)asym=sym(astr)ad=astr=asym=3^(2/3)+1/2（2）disp('ad astr asym')s(6),' 的数据类别'])disp([bldisp([class(ad),bl s(3),class(astr),blad astr asyms(4),class(asym)])的数据类别chardoublesym（3）asymPLUSda2=asym+0.1asymPLUSad=3^(2/3)+3/5asymPLUSda2=3^(2/3)+的数据类别chardoublesym（3）asymPLUSda2=asym+0.1asymPLUSad=3^(2/3)+3/5asymPLUSda2=3^(2/3)+3/5%<9>%<10>（4）ans=1astr2=asym+'0.1'asym2=3^(2/3)+0.6astr2=3^(2/3)+0.6%<12>%<13>（5）formatformatcompact%<14>%<15>%<16>码值数组')adans=longASCII码值数组48 46 49ad=astr3=ad+'0.1'ad3=%<20>%<21>50.5801astr3=50.580148.580151.580148.580151.5801（6）asym+sym(ad)+sym('0.1')ans=3^(2/3)+3.asym_ad_str=3^(2/3)+3.%<22>%<23>56194766890265619476689026（7）da=double('a')d3=double('3')da=97sda=NaNd3=51sd3=33符号数字向双精度数字转换】double指令的不同作用。（1）clearformongad=1/2+sqrt(2)asym=sym(astr)ad=97sda=NaNd3=51sd3=33符号数字向双精度数字转换】double指令的不同作用。（1）clearformongad=1/2+sqrt(2)asym=sym(astr)ad=astr=1/2+sqrt(2)asym=2^(1/2)+1/2（2）double_asym==addouble_asym=1.914213562373095ans=1（3）formatformatcompactdouble(astr)adans=49475043115113114116405041ad=1.91424符号数字的任意精度表达形式【例5.2-4】本例演示：digits的用法及影响；vpa指定符号数字有效位的影响和含义；符号reset(symengine)。（1）reset(symengine)%<1>sa=sym('1/3+sqrt(2)')sa=2^(1/2)+1/3（2）digits%<3>Digits=32formonga=1/3+sqrt(2)sa_Plus_a=vpa(sa+a,20)a=1.747546895706428sa_Plus_a=%<6>%<7>3.Digits=32formonga=1/3+sqrt(2)sa_Plus_a=vpa(sa+a,20)a=1.747546895706428sa_Plus_a=%<6>%<7>3.sa_Minus_a=-0.0000000000000000969（3）sa32=vpa(sa)digits(48)sa48=vpa(sa)sa32=1.sa5=sa48=1.%<8>%<9>%<10>%<11>13502205754390300520871（4）aa=1.747546895706428b=a5=vpa(a,5),b5=vpa(b,5)a5=b5=a5==b5ans=05.2.2符号表达式的简化操作】各种简化指令的使用示例。（1）symsxy（2）p=x^4-6*x^3+6*x^2-6*x+5;cp=coeffs(p)hp=horn)cp=[5,-6,6,-6,1]hp=x*(x*(x*(x-6)+6)-6)+5sp=(x^2+1)*(x-1)*(x-5)（3）pq=p/qfpq=factor(pq)pq=hp=x*(x*(x*(x-6)+6)-6)+5sp=(x^2+1)*(x-1)*(x-5)（3）pq=p/qfpq=factor(pq)pq=(x^4-6*x^3+6*x^2-6*x+5)/(x^4fpq=(x^2+1)/((x+3)*(x+1))-2*x^3-16*x^2+2*x+15)（4）prod(np)np=2ans=336643831314283（5）f=(x-3)*(x-2)/((x-7)*(x-5))ef=expand(f)eef=expand(ef)f=((x-2)*(x-3))/((x-5)*(xNum=-7))(x-2)*(xDen=(x-5)*(xef=(x^2-5*x-3)-7)+6)/(x^2-12*x+35)eef=6/(x^2-12*x+35)-(5*x)/(x^2-12*x+35)+x^2/(x^2-12*x+35)（6）strif=simple(trif)trif=(tan(x)+tan(y))/(tan(x)*tan(y)-1)strif=-tan(x+y)（7）csx=collect(s)csy=collect(s,y)s=x^2/exp(x)+y*exp(x)+x*y*sin(y)+(x*y)/exp(x)csx=x^2/exp(x)+(y/exp(x)+y*sin(y))*x+y*exp(x)csy=(exp(x)+x/exp(x)+x*sin(y))*y+x^2/exp(x)csexp=(x^2+y*x)/exp(x)+y*exp(x)+x*y*sin(y)（8）fA=factor(A)B=NA./DAA[[=3/2,(3*x)/(x-1)+(x^2+3)/(2*x-1)]3/x+4/x^2,x^2-3*x+4]fA=[3*2^(-1),(x^3+5*x^2-3)/((2*x-1)*(x-x^2-3*x1))]+（8）fA=factor(A)B=NA./DAA[[=3/2,(3*x)/(x-1)+(x^2+3)/(2*x-1)]3/x+4/x^2,x^2-3*x+4]fA=[3*2^(-1),(x^3+5*x^2-3)/((2*x-1)*(x-x^2-3*x1))]+4][(3*x+4)/x^2,NA=[3,x^3+5*x^2-3][3*x+4, x^2-3*x+4]DA=[[B[[2,(2*x-1)*(x-1)]1]x^2,=3/2,(x^3+5*x^2-3)/((2*x+4)/x^2,-1)*(x-x^2-3*x1))]+4]](3*x5.2.3表达式中的置换操作1公因子法简化表达a b5.2-6Ad进行特征向量分解。c（１）clearA=sym('[ab;cd]')[V,D]=eig(A)A=[a,b][c,d]V=[(a/2+d/2-(a^2-2*a*d+d^2+4*b*c)^(1/2)/2)/c+(a^2-2*a*d+d^2+4*b*c)^(1/2)/2)/c-d/c]-d/c,(a/2+d/2[D=[a/20][1,1]+d/2-(a^2-2*a*d+d^2+4*b*c)^(1/2)/2,+d^2+4*b*c)^(1/2)/2]（2）whosigma=%<4>(a^2-ans=[(a/2[[[2*a*d+d^2+4*b*c)^(1/2)+d/2-sigma/2)/c-d/c,(a/2+d/2+sigma/2)/c-d/c]1,1]0]a/2+d/2+sigma/2]a/2+d/2-sigma/2,0,Yourvariablesare:ADVanssigma（3）w=(a^2-2*a*d+d^2Dw=[a/2+（3）w=(a^2-2*a*d+d^2Dw=[a/2+d/2-w/2,+4*b*c)^(1/2)0]a/2+d/2+w/2][0,（4）%<7>RVD=[(a/2[[[w=(a^2-+d/2-w/2)/c-d/c,(a/2+d/2+w/2)/c-d/c]1,a/2+d/2-w/2,0,1]0]a/2+d/2+w/2]2*a*d+d^2+4*b*c)^(1/2)2通用置换指令【例5.2-7】本例目的：用简单算例演示subs的各种置换方式；演示符号计算与数值计算之间的一种转换途径。（1）clearsymsabx;f=b+a*sin(x)（2）%<4>class(f1)f1=b+a*log(y)ans=sym（3）class(f2)f2=b+(311*sin(x))/100ans=sym（4）%<8>class(f3)f3=3^(1/2)+5ans=sym（5）formatformatcompactclass(f4)f4=ans=%<12>（6）class(f5)f5=[b,a+b,(4967757600021511*a)/ans=sym47894502572032+b]（7）t=0:pi/10:2*pi;plot(t,f6)formatcompactclass(f4)f4=ans=%<12>（6）class(f5)f5=[b,a+b,(4967757600021511*a)/ans=sym47894502572032+b]（7）t=0:pi/10:2*pi;plot(t,f6)f6=Columns1through73.0000 3.6180 4.1756Columns8through144.6180 4.1756 3.6180%<17>4.61804.90215.00004.90213.00002.38201.82441.3820Columns151.0979through211.0000 1.09791.38201.82442.38203.0000利用符号表达式变量置换产生的单根曲线（8）k=(0.5:0.1:1)';f6=sub bs(f,{a,b},{k,2}),x,t);%<20>size(f6)plot(t,f6)ans=6212.521.5105.2-3利用两次subs置换产生的多根曲线a bsubsAd特征值、特征相量进行简化。c（１）clearA=sym('[ab;cd]');[V,D]=eig(A);subexpr([V;D]);sigma=%<4>(a^2-2*a*d+d^2+4*b*c)^(1/2)5.2-3利用两次subs置换产生的多根曲线a bsubsAd特征值、特征相量进行简化。c（１）clearA=sym('[ab;cd]');[V,D]=eig(A);subexpr([V;D]);sigma=%<4>(a^2-2*a*d+d^2+4*b*c)^(1/2)（2）Q=sigma^2;Vq=VDq(1:size(V,1),:)Dq=VDq(size(V,1)+1:end,:)%<6>%<7>Vq=[(a/2+d/2-Q^(1/2)/2)/c[Dq=-d/c,1,(a/2+d/2+Q^(1/2)/2)/c-d/c]1][a/2[+d/2-Q^(1/2)/2,0,a/2+d/20]+Q^(1/2)/2]5.3符号微积分5.3.1极限和导数的符号计算32.82.62.42.221.81.61.41.210 1 2 3 4 5 6 71kxsinkt和lim15.3-1】两种重要极限lim。ktxxt0symstxkf=(1-1/x)^(k*x);Lsk=limit(s,0)Lf=limit(f,x,inf)Lsk=1Ls1=1Lf=Lf1=2.7572470937 at3d2f d2f， 。dt2 dtdxdftcosx lnxf求 ，1kxsinkt和lim15.3-1】两种重要极限lim。ktxxt0symstxkf=(1-1/x)^(k*x);Lsk=limit(s,0)Lf=limit(f,x,inf)Lsk=1Ls1=1Lf=Lf1=2.7572470937 at3d2f d2f， 。dt2 dtdxdftcosx lnxf求 ，dxsymsatx;f=[a,t^3;t*cos(x),log(x)];df=di )dfdt2=di ,t,2)dfdxdt=diff(di df=[0, 0][-t*sin(x),1/x]dfdt2=[0,6*t][0, 0]dfdxdt=[0,0][-sin(x),0]f1f1x1x2xx1e25.3-3】求fxx) f2Jacobian矩阵f2。x)x1 22xcos(x)sn(xf12f12331symsx1x2;v=[x1;x2];Jf=%<3>%[exp(x2),x1*exp(x2)][0,1][-sin(x1)*sin(x2),cos(x1)*cos(x2)]5.3-4f(x)sinx（1）clearsymsxsymsditivef_p=sin(x);df_p=limit((subs(f_p,x,x+d)-f_p)/d,d,0)%<5>df_p0=limit((subs(f_p,x,d)-subs(f_p,x,0))/d,d,0)%<6>df_p=cos(x)clearsymsxsymsditivef_p=sin(x);df_p=limit((subs(f_p,x,x+d)-f_p)/d,d,0)%<5>df_p0=limit((subs(f_p,x,d)-subs(f_p,x,0))/d,d,0)%<6>df_p=cos(x)df_p0=1（2）f_n=sin(-x);df_n=limit((f_n-subs(f_n,x,x-d))/d,d,0)%<8>%<9>df_n=-cos(x)df_n0=-1（3）dfdx=di dfdx=cos(abs(x))*sign(x)dfdx0=0%<11>%<12>（4）holdonplot(xnp,subs(f,x,xnp),'k','LineWidth',3)plot(xn,subs(df_n,x,xn),'--r','LineWidth',3)plot(xp,subs(df_p,x,xp),':r','LineWidth',3)gridonholdoff%<14>%<17>函数及其导函数5.3-5】设cos(xsiny)sinydy。dx（1）clearsymsxdgdx=diff(g,x)g=%<3>%<4>dgdx==cos(y(x))*diff(y(x),x)（2）dgdx1=subs(dgdx,'diff(y(x),x)','dydx')%<5>dgdx1=-sin(x+sin(y(x)))*(dydx*cos(y(x))+1)=dydx*cos(y(x))（3）dydx=-sin(x+sin(y(x)))/(cos(y(x))+cos(y(x))*sin(x+sin(y(x))))5.3-6f(x)xexx08函数及其导函数5.3-5】设cos(xsiny)sinydy。dx（1）clearsymsxdgdx=diff(g,x)g=%<3>%<4>dgdx==cos(y(x))*diff(y(x),x)（2）dgdx1=subs(dgdx,'diff(y(x),x)','dydx')%<5>dgdx1=-sin(x+sin(y(x)))*(dydx*cos(y(x))+1)=dydx*cos(y(x))（3）dydx=-sin(x+sin(y(x)))/(cos(y(x))+cos(y(x))*sin(x+sin(y(x))))5.3-6f(x)xexx08Maclaurin级数。（1）symsxpretty(r)r=%<2>x^8/5040+x^7/720+x^6/120+x^5/24+x^4/6+x^3/2+x^2+x8x7x6x5x4x3x210.80.60.40.20-0.2-0.4-0.6-0.8-1-5 -4 -3 -2 -1 0 1 2 3 4 5xsin(abs(x))-cos(x)cos(x)----+---+---+--+--+--+x+x5040 720 120 24 6 2（2）R=evalin(symengine,'series(x*exp(x),x=0,8)')pretty(R)%<4>R=x+x^2+x^3/2+x^4/6+x^5/24+x^6/120+x^7/720+x^8/5040+O(x^9)3x+--2----+---+---+--+--+--+x+x5040 720 120 24 6 2（2）R=evalin(symengine,'series(x*exp(x),x=0,8)')pretty(R)%<4>R=x+x^2+x^3/2+x^4/6+x^5/24+x^6/120+x^7/720+x^8/5040+O(x^9)3x+--24x+--65x+--246x+---1207x+---7208x----504029+O(x)x+x+5.3.2序列/级数的符号求和x2k1(1)kn11k1k1,】求，，。2k1(2k1)2k（1）symsnkf1=1/(k*(k+1));s1=n/(n+1)（2）s2=piecewise([abs(x)<1,atanh(x)])（3）s3=symsum(f3,k,1,inf)s3=[pi^2/8,-log(2)]5.3.3符号积分5.3-8】本例演示：符号积分指令的使用；对符号积分的理解；符号参数的限定假设的影响；BetaGamma函数。（1）clearsymsabxf1=x*log(x)s1= s1=simple(s1)f1=s1=(x^2*(log(x)-1/2))/2s1=x^2*(log(x)/2-1/4)%<5>（2）disp('')disp('The egraloffpretty( (f2))f2=is')[a*x,[1/x,Theegraloffis+-||||||+--+||||||（2）disp('')disp('The egraloffpretty( (f2))f2=is')[a*x,[1/x,Theegraloffis+-||||||+--+||||||-+2ax----,23bx----3log(x),-cos(x)（3）f3=log(x)*log(1-x);s3= (f3,x,0,1)s3=2-pi^2/6（4）symsxns41= (f41,x,0,pi/2)%<14>Warning:Explicit egralcouldnotbefound.s41=+1/2)/2],[Re(n)<=-1,-x^2)^(1/2),x=0..1)])(x^n/(1symsnitive%<15>f42=sin(x)^n;s42= (f42,x,0,pi/2)s42=beta(1/2,n/2+1/2)/2%<17>s42n4vpa=vpa(s42n4)s42n4=(3*pi)/16s42n4vpa=0.%<18>786969262383（5）formongg1=gamma(0.5)g2=gamma(2.5)s4_nn=g1*g2/g3/2g1=1.772453850905516g2=g3=2s4_nn=0.589048622548086x2x2y2x(xyz)dzdydx。xy2 2 2】求三重积分2s4_nn=0.589048622548086x2x2y2x(xyz)dzdydx。xy2 2 2】求三重积分symsx symsyz1F2=((F2=(14912*2^(1/4))/4641-(6072064*2^(1/2))/348075+(64*2^(3/4))/225+1610027357/6563700VF2=224.071376012【例5.3-10】求阿基米德（Archimedes）螺线ra, (a0)在0到间的曲线长度函数，并求出a1,2时的曲线长度。（1）symsarthetaphir=a*theta;x=r*cos(theta);y=r*s heta);dLdth=sqrt(diff(x,theta)^2+diff(y,theta)^2);warningoff%<5>L=simple( (dLdth,theta,0,phi))%<6>L=((asinh(phi)+phi*(phi^2+1)^(1/2))*(a^2)^(1/2))/2（2）L_2pi_vpa=vpa(L_2pi)L_2pi=asinh(2*pi)/2+pi*(4*pi^2+1)^(1/2)L_2pi_vpa=21.702512272566（3）L1=subs(L,a,sym('1'));gridonholdony1=subs(y,a,sym('1'));h1=ezplot(x1,y1,[0,2*pi]);title('')%<14>%<16>-幅角曲线','阿基米德螺线')holdoff螺线长度-幅角曲线阿基米德螺线阿基米德螺线（粗红）和螺线长度函数（细蓝）5.45.4.1微分方程的符号解法符号解法和数值解法的互补作用5.4.2求微分方程符号解的一般指令5.4.3微分方程符号解示例dx】求 y, x的解。dydtdtclearall%<1>disp('')微分方程的解',bldisp([S.x,S.y])S=s(8),'x',bls(20),'y'])y:[1x1x:[1x1微分方程的解[C2*cos(t)xy+C1*s),C1*cos(t)-)]5.4-2yxyy)2的通解和奇解的关系。（1）clearally=dsolve('(Dy)^2-x*Dy+y=0','x')y=%<1>%<2>y420-2-4-6-8-10-12-14-16-5 0 5 10 15 20x螺线长度-幅角曲线阿基米德螺线阿基米德螺线（粗红）和螺线长度函数（细蓝）5.45.4.1微分方程的符号解法符号解法和数值解法的互补作用5.4.2求微分方程符号解的一般指令5.4.3微分方程符号解示例dx】求 y, x的解。dydtdtclearall%<1>disp('')微分方程的解',bldisp([S.x,S.y])S=s(8),'x',bls(20),'y'])y:[1x1x:[1x1微分方程的解[C2*cos(t)xy+C1*s),C1*cos(t)-)]5.4-2yxyy)2的通解和奇解的关系。（1）clearally=dsolve('(Dy)^2-x*Dy+y=0','x')y=%<1>%<2>y420-2-4-6-8-10-12-14-16-5 0 5 10 15 20xx^2/4C3*x-C3^2（2）clf,holdonhy1=ezplot(y(1),[-6,6,-4,8],1);%<4>set(hy1,'Color','r','LineWidth',5)fork=-2:0.5:2ezplot(y2,[-6,6,-4,8],1)endholdoffboxon%<6>%<7>%<9>','通解','Location','Best')ylabel('y')title(['\fontsize{14}微分方程(y0的解'])x^2/4C3*x-C3^2（2）clf,holdonhy1=ezplot(y(1),[-6,6,-4,8],1);%<4>set(hy1,'Color','r','LineWidth',5)fork=-2:0.5:2ezplot(y2,[-6,6,-4,8],1)endholdoffboxon%<6>%<7>%<9>','通解','Location','Best')ylabel('y')title(['\fontsize{14}微分方程(y0的解'])'')^2–xy''+y=5.4-1通解和奇解曲线5.4-3xy3yx2y(1)0y(5)0。（1）y=dsolve('x*D2y-3*Dy=x^2','y(1)=0,y(5)=0','x')y=(31*x^4)/468-x^3/3+125/468（2）xn=-1:6;yn=subs(y,'x',xn)holdontext(1,1,'y(1)=0')text(4,1,'y(5)=0')title(['x*D2y-3*Dy=x^2',',y(1)=0,y(5)=0'])holdoffy微分方程(y2xyy0的解8奇解通解6420-2-4-6 -4 -2 0 2 4 6xyn=Columns10.6667Column814.1132through70.26710-1.3397-3.3675-4.10900.00005.4-2两点边值问题的解曲线5.55.5.1符号变换和符号卷积Fourier变换及其反变换】求（1）Fourier变换。symstwUT=fourier(ut)UT=irac(w)yn=Columns10.6667Column814.1132through70.26710-1.3397-3.3675-4.10900.00005.4-2两点边值问题的解曲线5.55.5.1符号变换和符号卷积Fourier变换及其反变换】求（1）Fourier变换。symstwUT=fourier(ut)UT=irac(w)-i/w（2）SUt=simple(Ut)Ut=(pi+pi*(2*heaviside(t)SUt=heaviside(t)%<5>-1))/(2*pi)（3）t=-2:0.01:2;kk=find(t==0);holdon%<8>x*D2y-3*Dy=x2,y(1)=0,y(5)=054321 y(1)=0 y(5)=00-1-2-3-4-1 0 1 2 3 4 5 6xut(kk)=NaN;plot(t,ut,'-r','LineWidth',3)%<10>holdoffgridonaxis([-2,2,-0.2,1.2])xlabel('\fontsize{14}t'),ylabel('\fontsize{14}ut')title('\fontsize{14}Heaviside(t)')Heaviside(t)定义的阶跃函数/2变换。/2AttyHeaviside函数0（1）symsAtwtaoYw=fourier(yt,t,w)Yw_fy=simplify(Yw)Yw=A*((sin((tao*w)/2)Yw_fy=+cos((tao*w)/2)*i)/w-(-sin((tao*w)/2)+ut(kk)=NaN;plot(t,ut,'-r','LineWidth',3)%<10>holdoffgridonaxis([-2,2,-0.2,1.2])xlabel('\fontsize{14}t'),ylabel('\fontsize{14}ut')title('\fontsize{14}Heaviside(t)')Heaviside(t)定义的阶跃函数/2变换。/2AttyHeaviside函数0（1）symsAtwtaoYw=fourier(yt,t,w)Yw_fy=simplify(Yw)Yw=A*((sin((tao*w)/2)Yw_fy=+cos((tao*w)/2)*i)/w-(-sin((tao*w)/2)+Yw_fy_e=（2）Yt_e=simple(Yt)Yt_fy=simplify(Yt)Yt=%<7>%<8>-(A*(pi*heaviside(t-tao/2)-pi*heaviside(t+tao/2)))/piYt_e=A*heaviside(t+tao/2)-A*heaviside(t-tao/2)Yt_fy=-A*(heaviside(t-tao/2)-heaviside(t+tao/2))（3）utHeaviside(t)10.80.60.40.20-0.2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2tt3=3;tn=-3:0.1:3;yt13=subs(yt,{A,tao},{1,t3})yt13n=subs(yt13,'t',tn);kk=find(tn==-t3/2|tn==t3/2);%<13>plot(tn(kk),yt13n(kk),'.r','MarkerSize',30)holdon%<15>holdoffgridonaxis([-3,3,-0.5,1.5])yt13=heaviside(t+3/2)-heaviside(t-3/2)5.5-2由Heaviside(t)构造的矩形波（4）Yw13=subs(Yw_fy_e,{A,tao},{1,t3});subplot(2,1,1),ezplot(Yw13),gridonsubplot(2,1,2),ezplot(abs(Yw13)),gridon矩形脉冲的频率曲线和幅度频谱3210-1-6 -4 -2 0 2 4 6w3210-6 -4 -2 0 2 4 6w1.510.50-0.5-3 -2 -1 0 1 2 3t3=3;tn=-3:0.1:3;yt13=subs(yt,{A,tao},{1,t3})yt13n=subs(yt13,'t',tn);kk=find(tn==-t3/2|tn==t3/2);%<13>plot(tn(kk),yt13n(kk),'.r','MarkerSize',30)holdon%<15>holdoffgridonaxis([-3,3,-0.5,1.5])yt13=heaviside(t+3/2)-heaviside(t-3/2)5.5-2由Heaviside(t)构造的矩形波（4）Yw13=subs(Yw_fy_e,{A,tao},{1,t3});subplot(2,1,1),ezplot(Yw13),gridonsubplot(2,1,2),ezplot(abs(Yw13)),gridon矩形脉冲的频率曲线和幅度频谱3210-1-6 -4 -2 0 2 4 6w3210-6 -4 -2 0 2 4 6w1.510.50-0.5-3 -2 -1 0 1 2 3e(tx)tx5.5-3f(t)Fouriere(tx)tx5.5-3f(t)Fourierxt是时间变量。tx0（1）clearsymstxwgt=exp(-(t-x));%<3>%<4>（2）F1=(1/exp(w*x*i))/(1+w*i)G1=exp(x)*transform::fourier(1/exp(t),t,-w)（2）F3=simple(fourier(ft))F2=-exp(t^2*i)/(-1+t*i)F3=-(1/exp(t*w*i))/(-1+w*i)5.5.2Laplace变换及其反变换5.5-4】分别求eatsinbt,u(ta,(tb,tnLaplace变换。（1）symstsabF1=laplace(f1,t,s)f1=F1=b/((a+s)^2+b^2)%<2>（2）symaclearf2=heaviside(t-a)ans=af2=heaviside(t-a)F2=laplace(heaviside(t%<4>-a),t,s)symsaitive%<7>F3=laplace(f2)F3=1/(s*exp(a*s))（3）f4=dirac(t-b);F4=piecewise([b<0,0],[0<=b,1/exp(b*s)])f5=dirac(t-a);F5=laplace(f5,t,s)F5=1/exp(a*s)ft_F5=dirac(a-t)%<11>（4）n=sym('n','clear');%<14>F6=laplace(F6=,s)piecewise([-1<f4=dirac(t-b);F4=piecewise([b<0,0],[0<=b,1/exp(b*s)])f5=dirac(t-a);F5=laplace(f5,t,s)F5=1/exp(a*s)ft_F5=dirac(a-t)%<11>（4）n=sym('n','clear');%<14>F6=laplace(F6=,s)piecewise([-1<Re(n),gamma(n+1)/s^(n+1)])n=nitive'),s)%<16>%<17>F6=gamma(n+1)/s^(n+1)5.5.3Z变换及其反变换Z变换、反变换算例。（1）clearsymsnzclearpretty(G)gn=6-6*(1/2)^n%<2>6z22z-3z+1（2）symsnwTzclear%<6>fwn=sin(w*n*T);FW=ztran,n,z);pretty(FW),disp('')inv_FW=iztra,z,n)zsw)2z-2cos(Tw)z+1inv_FW=s （3）symsnzclearf1=1;F1=ztrans(f1,n,z);pretty(F1)%<11>zz-1inv_F1=1（4）clearsymsnzclearKD=ztrans(delta,n,z)inv_KD=iztrans(KD)KD=1（3）symsnzclearf1=1;F1=ztrans(f1,n,z);pretty(F1)%<11>zz-1inv_F1=1（4）clearsymsnzclearKD=ztrans(delta,n,z)inv_KD=iztrans(KD)KD=1inv_KD=kroneckerDelta(n,0)%<16>%<17>0)');（5）symsnzclear%<20>%<21>k=sym('k','itive');FD=ztrans(fd,n,z)inv_FD=iztrans(FD,z,n)FD=piecewise([kinZ_,f(k)/z^k],,0)');[Otherwise,0])inv_FD=piecewise([kinZ_,f(k)*kroneckerDelta(k-n,0)],[Otherwise,0])（6）FD_evalin=evalin(symengine,'ame(k>0):ameAlso(kinZ_):transform::ztrans(f(n)*kroneckerDelta(n,k),n,z):')%<25>FD_evalin=f(k)/z^k（7）symsaznclearGZ=exp(-a/z);gn=(-a)^n/factorial(n)%<28>5.5.4符号卷积5.5-6】已知系统冲激响应h(t)1et/TU(t)，求u(t)etU(t)输入下的输出响应。TsymsTttaout=exp(-t);yt= (subs(ut,t,tao)*subs(ht,t,t-tao),tao,0,t)yt=-(1/exp(t)-1/exp(t/T))/(T-1)Laplace变换和反变换求上例的输出响应。symssyt=simple(ilaplace(yt,s,t))yt=1/(T*(s+1/T)*(s+yt= (subs(ut,t,tao)*subs(ht,t,t-tao),tao,0,t)yt=-(1/exp(t)-1/exp(t/T))/(T-1)Laplace变换和反变换求上例的输出响应。symssyt=simple(ilaplace(yt,s,t))yt=1/(T*(s+1/T)*(s+1))yt=(1/exp(t/T)-1/exp(t))/(T-1)5.6符号矩阵分析和代数方程解5.6.1符号矩阵分析a11a125.6-1A的行列式、逆和特征根。aa2122（1）symsa11a12a21a22A=[a11,a12;a21,a22]A=[a11,a12][a21,a22]DA=a11*a22-a12*a21IA=[a22/(a11*a22-a12*a21),-a12/(a11*a22-a12*a21)][-a21/(a11*a22-a12*a21),a11/(a11*a22-a12*a21)]（2）D=EA=a11/2a11/25.6.2-2*a11*a22+a22^2+4*a12*a21)^(1/2)++a22/2-D/2a22/2+D/2线性方程组的符号解n pn5.6-2】求d22qndqp10qd4pqpn8d1线性方程组的解。（1）A=sym([11/21/2-1;11-11;1-1/4-11;-8-111]);b=sym([0;10;0;1]);X1=A\bX1=1889（2）eq2=sym('d+n-p+q-10');eq3=sym('d-n/4-p+q');%<4>eq4=sym('-8*d-n+p+q-1');disp(['d','n','p','q'])disp([S.d,S.n,S.p,S.q])dnpq[1,8,889（2）eq2=sym('d+n-p+q-10');eq3=sym('d-n/4-p+q');%<4>eq4=sym('-8*d-n+p+q-1');disp(['d','n','p','q'])disp([S.d,S.n,S.p,S.q])dnpq[1,8,8,9]%<8>%<10>5.6.3一般代数方程组的解5.6-3】求方程组uy^2vzw0yzw0yz的解。S=solve('u*y^2+v*z+w=0','y+z+w=0','y','z')%<1>disp('S.y'),disp(S.y),disp('S.z'),disp(S.z)%<2>S=y:[2x1sym]z:[2x1sym]S.y(v+2*u*w+(v^2+4*u*w*v-4*u*w)^(1/2))/(2*u)-w(v+2*u*w-(v^2+4*u*w*v-4*u*w)^(1/2))/(2*u)-wS.z-(v+2*u*w+(v^2+4*u*w*v-4*u*w)^(1/2))/(2*u)-(v+2*u*w-(v^2+4*u*w*v-4*u*w)^(1/2))/(2*u)【例5.6-4】solve指令求dn q，ndqp10，qdnpp的“欠2 24定”方程组解。symsdnpqS=solve(eq1,eq2,eq3,d,n,p,q);%<2>%<3>disp([' S.d',' S.n',' S.p','disp([S.d,S.n,S.p,S.q])S.q'])S.d S.n [z,x8,z11,z21]S.q5.6-5】求(x2x2的解。s=solve('(x+2)^x=2','x')s=%<1>0.26920133106081】求解带限定性假设的代数方程的解。（1）symsxclearEq=x^4-16;X1=2-2%<1>2*i-2*i（2）symsxrealX2=2-2%<4>（3）symsxitive%<6>X3=feval(symengine,'solve',x^4-16,x)X3=2%<7>（4）X4=evalin(symengine,'solve(x^4-16=0,x)X4=2ax>0:')%<8>5.7符号算法的综合应用2*i-2*i（2）symsxrealX2=2-2%<4>（3）symsxitive%<6>X3=feval(symengine,'solve',x^4-16,x)X3=2%<7>（4）X4=evalin(symengine,'solve(x^4-16=0,x)X4=2ax>0:')%<8>5.7符号算法的综合应用5.7.1三维根轨迹和数据探索【例5.7-1】增益放大系数k变化时，图5.7-1所示闭环系统的根轨迹。图中1G(s)。s(s3)(s22s2)5.7-1具有变增益系数K的负反馈系统（1）（2）clearsymsssymskitive%<3>EEC=expand(EC);%<5>（3）kk=[0:0.2:4,4.2:0.001:4.4,4.6:0.2:7.8,8:0.01:8.2,9:12]';forii=1:nk SE=subs(EEC,k,kk(ii)); PE=sym2poly(SE); endRreal=real(Rk);Rimag=imag(Rk);（4）figure(1)%<9>（4）figure(1)%<9>holdonx00=[-3,0.5];plot(x00,zeros(size(x00)),'k-')y00=[-1.5,1.5];plot(zeros(size(y00)),y00,'k-')plot([-2.3,0],[0,1.1],'ro','MarkerSize',15)text(-2.25,0.1,'A'),text(0.05,1,'B')holdoff'),ylabel('根的虚部')title('常规根轨迹')处于数据关联状态的常规根轨迹（5）figure(2)%<20>boxon,gridonview([63,68])根的虚部'),zlabel('增益系数')title('三维根轨迹')处于数据关联状态的三维根轨迹（6）处于数据关联状态的三维根轨迹（6）figure(1)的指定数据源窗figure(2)的指定数据源窗（7）figure(2)的指定数据源窗（7）工作空间的内存变量处于数据关联状态的内存变量（8）（9）（10）关联数据被同时自动标识的效果5.7.2（8）（9）（10）关联数据被同时自动标识的效果5.7.2代数状态方程求符号传递函数1结构框图的代数状态方程解法5.7-1所示某三环系统的传递函数。三环系统的结构框图（1）（2）（3）（4）symsG1G2G3G4H1H2H3A=[ 0, 0, 0, 0, 0,-G2,0,0,0,0,0,0;0,-G1;0;0;0;0;0;0];0];0];G2,0,0,0,0,0,b=[G1;c=[ 0,0,G3,0,0,0,0,0;0,0,0,G4,0,0,0,0;0,0,0,0,（1）（2）（3）（4）symsG1G2G3G4H1H2H3A=[ 0, 0, 0, 0, 0,-G2,0,0,0,0,0,0;0,-G1;0;0;0;0;0;0];0];0];G2,0,0,0,0,0,b=[G1;c=[ 0,0,G3,0,0,0,0,0;0,0,0,G4,0,0,0,0;0,0,0,0,H2,H1,H3,0;1,0,G3,0,0,0,0,0;0,Y2Ua=c*((eye(size(A))-A)\b);为'])disp([bls(5),'传递函数Y2Uapretty(Y2Ua)传递函数Y2Ua为G1G2G3G4G2G3G4H1-G3G4H2+G1G2G3G4H3+12信号流图的代数状态方程解法5.7-1s1s1，1001数状态方程求传递函数；在G，G，G，G1s102s134s44s6s22s12，H s1，H1的情况下，求取参数具体化的传递函数。H1s12s23（1）三环系统的信号流图（2）（3）symsG1G2G3G4H1A=[ 0, 0, 0,G1, 0, 0,0,G2, 0,0, 0,G3,0, 0, 0,b=[ 1; 0; 0;c=[0, 0, 0,H2H30,-H3;0,symsG1G2G3G4H1A=[ 0, 0, 0,G1, 0, 0,0,G2, 0,0, 0,G3,0, 0, 0,b=[ 1; 0; 0;c=[0, 0, 0,H2H30,-H3;0,-H1;0,H2;0, 0;G4, 0];0; 0];0, 1];Y2Ub=c*((eye(size(A))-A)\b);disp([bls(5),'传递函数Y2Ub为'])pretty(Y2Ub)传递函数Y2Ub为G1G2G3G4G2G3G4H1-G3G4H2+G1G2G3G4H3+1（4）symss1),(s+1)/(s+2),1};%<9>[NN,DD]=numden(Y2Uc);NN=expand(NN);disp('参数具体化的传递函数Y2Uc为')pretty(NN/DD)参数具体化的传递函数Y2Uc为%<10>2100s+300s+2005s432+21s+157s+663s+1301s+9103多输入多输出系统传递矩阵的求取5.7-3所示“22输出”系统的传递矩阵。二输入二输出系统的结构框图（1）（2）clearsymsG1G2G3G4H1H2H3symsF1F2F3（1）（2）clearsymsG1G2G3G4H1H2H3symsF1F2F3F4symsWA(1,4)=-G1;B(1,1)=G1;A(2,1)=G2;A(2,9)=G2;A(4,3)=G4;A(6,5)=F2;A(8,7)=F4;B(9,2)=H1;A(11,7)=H3;C(1,3)=1;C(2,7)=1;G=C*((eye(size(A))-A)\B);%<19>[GG,W]=subexpr(G,W);%<20>')')')')disp('W='),disp(simple(W))GG(1,1)=G1*G3*W*(G2-F3*H2*H3+F1*F2*F3*F4*G2)GG(1,2)=-F1*F2*F3*H3-F3*H1*H2*H3+F1*F2*F3*F4*G2*H1)GG(2,1)=GG(2,2)=+F3*H1*H2+F1*F2*F3*G1*G2*G3*G4)W=+1)（3）G_1to1=simple(subs(G(1,1),[H1,H2,H3],[0,0,0]));的传递函数为')pretty(G_1to1)的传递函数为%<26>G1G2G3G1G2G3G4+1G_2to2=simple(subs(G(2,2),[H1,H2,H3],[0,0,0]));%<29>的传递函数为')pretty(G_2to2)的传递函数为F1F2F3F1F2F3F4+15.85.8.1符号计算结果的可视化直接可视化符号表达式1单独立变量符号函数的可视化t23t【例5.8-1】绘制y3e2cos t和它的积分s(t) y(t)dt在[0,4]间的图形（图205.8-1）。symsttaobs( (y,t,0,tao),tao,t)subplot(2,1,1)gridonsubplot(2,1,2)gridon的传递函数为')pretty(G_2to2)的传递函数为F1F2F3F1F2F3F4+15.85.8.1符号计算结果的可视化直接可视化符号表达式1单独立变量符号函数的可视化t23t【例5.8-1】绘制y3e2cos t和它的积分s(t) y(t)dt在[0,4]间的图形（图205.8-1）。symsttaobs( (y,t,0,tao),tao,t)subplot(2,1,1)gridonsubplot(2,1,2)gridontitle('s=\y=y(t)dt')s=1/3-(2*(cos((3^(1/2)*t)/2)/2-(3^(1/2)*sin((3^(1/2)*t)/2))/2))/(3*exp(t/2))ezplot使用示例(2cos((31/2t)/2))/(3exp(t/2))0.60.40.20-0.20 2 4 6 8 10 12ts=y(t)dt0.50.40.30.20 2 4 6 8 10 12t2双独立变量符号函数的可视化】使用球坐标参量画部分球壳（5.8-2）。clfy='cos(s)*s )';z='sin(s)';view(17,40)shadinglight(' ition',[0,0,-10],'style','local')light(' 2双独立变量符号函数的可视化】使用球坐标参量画部分球壳（5.8-2）。clfy='cos(s)*s )';z='sin(s)';view(17,40)shadinglight(' ition',[0,0,-10],'style','local')light(' material([0.5,0.5,0.5,10,0.3])5.8-2ezsurf在参变量格式下绘制的图形5.8.2符号计算结果的数