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#curl clear clc syms x y z f=input("enter the 3d vectir function in the form [f1,f2,f3]"); P(x,y,z)=f(1); Q(x,y,z)=f(2); R(x,y,z)=f(3); crl=curl(f,[x,y,z]) C1(x,y,z)=crl(1); C2(x,y,z)=crl(2); C3(x,y,z)=crl(3); x=linspace(-4,4,10); y=x; z=x; [X,Y,Z]=meshgrid(x,y,z); U=P(X,Y,Z); V=Q(X,Y,Z); W=R(X,Y,Z); CR1=C1(X,Y,Z); CR2=C2(X,Y,Z); CR3=C3(X,Y,Z); figure; subplot(1,2,1); quiver3(X,Y,Z,U,V,W); title('3d view of vector field'); subplot(1,2,2); quiver3(X,Y,Z,CR1,CR2,CR3); title("3d view of curl"); # divergence clear clc; syms x y f=input("enter the funtion [f1,f2] :") div(x,y)=divergence(f,[x,y]) P(x,y)=f(1); Q(x,y)=f(2); x=linspace(-4,4,20); y=x; [X,Y]=meshgrid(x,y); U=P(X,Y); V=Q(X,Y); figure pcolor(X,Y,div(X,Y)) shading interp hold on; quiver(X,Y,U,V,1) axis on hold off; title('vector field of F(x,y)=(f1,f2)'); % gradient clear clc syms x y f=input('enter the function f(x,y) :' ); grad=gradient(f,[x,y]); P(x,y)=grad(1); Q(x,y)=grad(2); x=linspace(-2,2,10); y=x; [X,Y]=meshgrid(x,y); hold on U=P(X,Y); V=Q(X,Y); quiver(X,Y,U,V,1) axis on xlabel("x") ylabel("y") hold on fcontour(f,[-2,2]) #greens clear clc syms x y t F=input("ENTER the vector function M(x,y)I+N(x,y) in the form [M N] :"); M(x,y)=F(1); N(x,y)=F(2); r=input("Enter the parametric form of the curve c as [r1(t) r2(t)]"); r1=r(1); r2=r(2); P=M(r1,r2); Q=N(r1,r2); dr=diff(r,t); F1=sum([P,Q].*dr); T=input("Enter the limits of integration for t [t1,t2] :"); t1=T(1); t2=T(2); LHS=int(F1,t,t1,t2); yL=input("Enter the lmits for y in terms of x: [y1,y2]:"); xL=input("enter the limits for x as constanasts : [x1,x2]:"); y1=yL(1); y2=yL(2); x1=xL(1); x2=xL(2); F2=diff(N,x)-diff(M,y); RHS=int(int(F2,y,y1,y2),x,x1,x2); if LHS==RHS disp('lhs of greens theorem') disp(LHS) disp('rhs of greens theorem') disp(RHS) disp('hence Greens theorem is verified .'); end # alngarqnegs clc clear all syms x y z L f=x*x+y*y+z*z; g=3*x*x+4*x*y+6*y*y-140; F=f+L*g; j=jacobian(F,[x y z]); s=solve(g,j(1),j(2),j(3)); p=double([s.x,s.y,s.z]); h1=p(:,1); h2=p(:,2); h3=p(:,3); X=[h1]; Y=[h2]; Z=[h3]; st_pts=table([X,Y,Z]) [n,m]=size(st_pts); for i=1:n F(i)=subs(f,{x,y,z},{s.x(i),s.y(i),s.z(i)}); end if n>1 F_max=max(F); disp("f_max"); disp(F_max); F_min=min(F); disp("f_min") disp(F_min); end # line integral clc clear all syms x y t f=input("enter the function in form [x,y] "); r=input("enter x,y in the form paramateric form"); g=input("enter limits of t"); a=g(1); b=g(2); dr=diff(r,t); F=subs(f,{x,y},r); Fdr=sum(F.*dr); I=int(Fdr,t,a,b); P(x,y)=f(1); Q(x,y)=f(2); x1=linspace(-2*pi,2*pi,10); y1=x1; [X,Y]=meshgrid(x1,y1); U=P(X,Y); V=Q(X,Y); quiver(X,Y,U,V,1.5) hold on t1=linspace(0,2*pi); x=subs(r(1),t1); y=subs(r(2),t1); plot(x,y,'r') # %%To find the maxima and minima of the single variable function and visualize clc clear all syms x real f=9*x-6*x*x+x^3+1 fx= diff(f,x) fxx= diff(fx,x) c = solve(fx); %z=size(c) c=double(c); for i = 1:length(c) T1 = subs(fxx, x ,c(i)); T1=double(T1); T3= subs(f, x, c(i)); T3=double(T3); if (T1==0) sprintf('The inflection point is x = d',c(i)) else if (T1 < 0) sprintf('The maximum point x is %d', c(i)) sprintf('The maximum value of the function is %d', T3) else sprintf('The minimum point x is %d', c(i)) sprintf('The minimum value of the function is %d', T3) end end cmin = min(c); cmax = max(c); D = [cmin-2, cmax+2]; ezplot(f,D) hold on plot(c(i), T3, 'g*', 'markersize', 15); end %%Definite Integrals and its applications %%To find the area of the regions enclosed by curves and visualize it. clc clear all syms x y real y1=-x*x+4*x; y2=x*x; fg=figure; ax=axes; t=solve(y1-y2); k=double(t) n=length(k) m1=min(k) m2=max(k) ez1=ezplot(y1,[m1-1,m2+1]); hold on ez2=ezplot(y2,[m1-1,m2+1]); TA=0; A=int(y1-y2,t(1),t(2)); TA=abs(A) x1 = linspace(k(1),k(2)); yy1 =subs(y1,x,x1); yy2 = subs(y2,x,x1); x1 = [x1,fliplr(x1)]; yy = [yy1,fliplr(yy2)]; fill(x1,yy,'g') clc clear all syms x y real y1=-x*x+4*x; y2=x*x; fg=figure; ax=axes; t=solve(y1-y2); k=double(t) n=length(k) m1=min(k) m2=max(k) ez1=ezplot(y1,[m1-1,m2+1]); hold on ez2=ezplot(y2,[m1-1,m2+1]); TA=0; if n>2 for i=1:n-1 A=int(y1-y2,t(i),t(i+1)); TA= TA+abs(A) x1 = linspace(k(i),k(i+1)); yy1 =subs(y1,x,x1); yy2 = subs(y2,x,x1); x1 = [x1,fliplr(x1)]; yy = [yy1,fliplr(yy2)]; fill(x1,yy,'g') grid on end else A=int(y1-y2,t(1),t(2)); TA=abs(A) x1 = linspace(k(1),k(2)); yy1 =subs(y1,x,x1); yy2 = subs(y2,x,x1); x1 = [x1,fliplr(x1)]; yy = [yy1,fliplr(yy2)]; fill(x1,yy,'g') end # % To find Maximum and Minimum values (Extreme values) of a function f (x, y) using % MATLAB. clc clear syms x y f(x,y)=2*(x^2-y^2)-x^4+y^4; p=diff(f,x); q=diff(f,y); [ax,ay]=solve(p,q); ax=double(ax);ay=double(ay); r=diff(p,x); s=diff(p,y); t=diff(q,y);D=r*t-s^2; figure fsurf(f); legstr={'Function Plot'};% for Legend for i=1:size(ax) T1=D(ax(i),ay(i)); T2=r(ax(i),ay(i)); T3=f(ax(i),ay(i)); if(double(T1)==0) sprintf('At (%f,%f) further investigation is required',ax(i),ay(i)) legstr=[legstr,{'Case of Further investigation'}]; mkr='ko'; elseif (double(T1)<0) sprintf('The point (%f,%f) is a saddle point', ax(i),ay(i)) legstr=[legstr,{'Saddle Point'}]; % updating Legend mkr='bv'; % marker else if (double(T2) < 0) sprintf('The maximum value of the function is f(%f,%f)=%f', ax(i),ay(i), T3) legstr=[legstr,{'Maximum value of the function'}];% updating Legend mkr='g+';% marker else sprintf('The minimum value of the function is f(%f,%f)=%f', ax(i),ay(i), T3) legstr=[legstr,{'Minimum value of the function'}];% updating Legend mkr='r*'; % marker end end hold on plot3(ax(i),ay(i),T3,mkr,'Linewidth',4); end legend(legstr,'Location','Best'); % Volume of the Solid of Revolution clc clear all syms x f = x^2+2; fL = 0:10; %Enter the interval on which the function is defined yr = 3; %the axis of rotation y = c iL = 0:2; %the integration limits Volume = pi*int((f-yr)^2,iL(1),iL(2)); disp(['Volume is: ', num2str(double(Volume))]) fx = inline(vectorize(f)); xvals = linspace(fL(1),fL(2),201); xvalsr = fliplr(xvals); xivals = linspace(iL(1),iL(2),201); xivalsr = fliplr(xivals); xlim = [fL(1) fL(2)+0.5]; ylim = fx(xlim); figure('Position',[100 200 560 420]) subplot(2,1,1) hold on; plot(xvals,fx(xvals),'-b','LineWidth',2); fill([xvals xvalsr],[fx(xvals) ones(size(xvalsr))*yr],[0.8 0.8 0.8],'FaceAlpha',0.8) plot([fL(1) fL(2)],[yr yr],'-r','LineWidth',2); legend('Function Plot','Filled Region','Axis of Rotation','Location','Best'); title('Function y=f(x) and Region'); set(gca,'XLim',xlim) xlabel('x-axis'); ylabel('y-axis'); [X,Y,Z] = cylinder(fx(xivals)- yr,100); figure('Position',[700 200 560 420]) Z = iL(1) + Z.*(iL(2)-iL(1)); surf(Z,Y+yr,X,'EdgeColor','none','FaceColor','flat','FaceAlpha',0.6); hold on; plot([iL(1) iL(2)],[yr yr],'-r','LineWidth',2); xlabel('X-axis'); ylabel('Y-axis'); zlabel('Z-axis'); view(22,11); clc clear all % 1. Matrix operations A=[1 2;3 4]; B=[4 5;8 12]; C=A+B D=A*B E=inv(A) F=B' % H=A.^2 I= det(A) %2.Circle with centre(1,3) t = linspace(0, 2*pi); x = 1 +2*cos(t); y = 3 +2*sin(t); plot(x,y) axis equal %3.plotting three functions without hold on x = linspace(0,1,101) plot(x,x.^3,'r+',x,sin(x),'b-',x,exp(x),'g.') %4. Solving system of linear equations, say, 4x+5y=7,7x+8y=21 % In general, %X = A\B solves for X in A*X = B and % X = B/A solves for X in X*A = B. A=[4 5;7 8] b=[7 21]' x=A\b %5.solving quadratic equations syms ('x') solve(2*x^2+5*x+12) %6.ezplotting syms x f=sin(2*x)+cos(3*x) ezplot(f) %7.subplot x=0:.1:2*pi; subplot(2,2,1); plot(x,sin(x)); subplot(2,2,2); plot(x,cos(x)); subplot(2,2,3) plot(x,exp(-x)); subplot(2,2,4); %8.symbolic differentiation syms x y f =x^2+2*x*y+y*sin(x) diff(f,x) diff(f,y) %9.Symbolic integration syms x f=x^3+3*x^2+5*x+12 int(f,x,0,2) %%To Plot the Circle syms r a b r= 5; a= 6; b= 4; t = linspace(0, 2*pi); x = a+r*cos(t); y = b+r*sin(t); axis equal plot(x, y) xlabel('x-Coordinate') ylabel('x-Coordinate') title('(x - a)^2 + (y - b)^2 = r^2') %%Multiple plots using Hold on Matlab Code x = linspace(0, 1, 100); plot(x, x.^2,'r', 'LineWidth',2.0) hold on plot(x, cos(x), 'g', 'LineWidth',2.0) hold on plot(x, sin(x),'b', 'LineWidth',2.0) hold on plot(x, exp(x),'c', 'LineWidth',2.0) legend('x2', 'cos(x)' , 'sin(x)' , 'ex') %%Multiple plots without command \hold on" Matlab Code clc clear all x = linspace(0, 1, 200); plot( x, sin(x), x, cos(x), x, x.^3, x, tan(x), 'LineWidth',2.0) legend('sin(x)','cos(x)','x^3', 'tan(x)') %%Graph of a curve and its tangent line in the neighbourhood D of a point. syms x y=x^2; x1 =5; D=[x1-2 x1+2] ezplot(y,D) hold on yd = diff(y,x); slope = subs(yd,x,x1); y1 = subs(y,x,x1); plot(x1,y1,'ko') hold on Tgtline = slope*(x-x1)+y1 ezplot(Tgtline, D) %To Plot the function and its derivatives clc clear all syms x real f= 3*x^2-4; fx= diff(f,x) fxx= diff(fx,x) D = [0, 5]; l=ezplot(f, D) set(l,'color','b'); hold on h=ezplot(fx,D); set(h,'color','r'); e=ezplot(fxx,D); set(e,'color','g'); legend('f','fx','fxx')
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