Kalkulus II.
J\303\241rai Antal
Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak
1. Line\303\241ris algebra
2. T\303\266bbv\303\241ltoz\303\263s f\303\274ggv\303\251nyek
2.1. Hat\303\241r\303\251rt\303\251k \303\251s folytonoss\303\241g
restart;
2.1.1. T\303\241vols\303\241g, k\303\266rnyezetek, korl\303\241toss\303\241g.
verify([3,3,3],[Pi,Pi,Pi],'neighborhood(1)');
verify([1/sqrt(3),1/sqrt(3),1/sqrt(3)],[0,0,0],'neighborhood(1)');
verify([1/sqrt(3),1/sqrt(3),1/sqrt(3)],[0,0,0],'neighborhood(1,closed)');
verify([1/sqrt(3),1/sqrt(3),1/sqrt(3)],[0,0,0],'neighborhood(1,p=1)');
verify([1/sqrt(3),1/sqrt(3),1/sqrt(3)],[0,0,0],'neighborhood(1,p=3)');
2.1.2. Bels\305\221, k\303\274ls\305\221, izol\303\241lt, torl\303\263d\303\241si \303\251s hat\303\241rpontok.
2.1.3. Ny\303\255lt \303\251s z\303\241rt halmazok.
2.1.4. \303\201ll\303\255t\303\241s.
2.1.5. \303\201ll\303\255t\303\241s.
2.1.8. K\303\266vetkezm\303\251ny.
2.1.9. K\303\266vetkezm\303\251ny.
2.1.10. K\303\266vetkezm\303\251ny.
2.1.11. Tartom\303\241ny.
2.1.13. S\305\261r\305\261 halmazok.
2.1.14. Kompakt halmazok.
2.1.16. K\303\266vetkezm\303\251ny: Weierstrass t\303\251tele.
2.1.17.Vektor-skal\303\241r, skal\303\241r-vektor \303\251s vektor-vektor f\303\274ggv\303\251nyek.
with(plots);
spacecurve([cos(t),sin(t),t],t=0..4*Pi);
p:=proc(x,y) sin(x*y) end; plot3d(p,-Pi..Pi,-Pi..Pi);
fieldplot([x,y^2],x=-1..1,y=-1..1);
plot3d([sin(x)+2*sin(0.4*y),cos(x)+2*cos(0.4*y),y],x=0..2*Pi,y=0..10);
fieldplot3d([x-y,x+z,z-y-x],x=-1..1,y=-1..1,z=-1..1);
2.1.18. Folytonoss\303\241g.
2.1.19. P\303\251ld\303\241k.
2.1.21. Seg\303\251dt\303\251tel.
2.1.24. P\303\251ld\303\241k.
*2.1.27. K\303\266vetkezm\303\251ny.
2.1.29. K\303\266vetkezm\303\251ny: Weierstrass t\303\251tele.
2.1.30. Hausdorff t\303\251tele.
*2.1.32. Lebesgue-sz\303\241m.
*2.1.34. Egyenletes folytonoss\303\241g.
*2.1.35. Heine t\303\251tele.
2.1.36. Jobb \303\251s bal oldali folytonoss\303\241g.
2.1.40. Jobb \303\251s bal oldali hat\303\241r\303\251rt\303\251k.
2.1.41. Szakad\303\241sok.
2.1.42. V\303\251gtelen, mint hat\303\241r\303\251rt\303\251k, hat\303\241r\303\251rt\303\251k a v\303\251gtelenben.
2.1.45. T\303\251tel: rend\305\221r-elv.
2.1.50. K\303\266vetkezm\303\251ny: Bolzano-Weierstrass-f\303\251le kiv\303\241laszt\303\241si t\303\251tel.
2.1.51. Cauchy-sorozatok.
2.1.52. Cauchy-f\303\251le konvergenciakrit\303\251rium.
2.1.53. Banach-terek \303\251s Hilbert-terek.
2.1.56. Cauchy-f\303\251le konvergenciakrit\303\251rium.
2.1.57. K\303\266vetkezm\303\251ny.
2.1.58. \303\226szehasonl\303\255t\303\263krit\303\251rium.
2.1.59. K\303\266vetkezm\303\251ny.
2.1.60. Cauchy-f\303\251le gy\303\266kkrit\303\251rium.
2.1.61. d'Alembert-f\303\251le h\303\241nyadoskrit\303\251rium.
2.1.62. Kett\305\221s sor t\303\251tel.
2.1.63. K\303\266vetkezm\303\251ny: sorok \303\241trendez\303\251se.
2.1.64. K\303\266vetkezm\303\251ny.
2.1.65. Ekvivalens norm\303\241k.
*2.1.67. Fixpont \303\251s kontrakci\303\263.
*2.1.68 Banach-f\303\251le fixpontt\303\251tel.
2.2. Differenci\303\241lsz\303\241m\303\255t\303\241s
restart; with(LinearAlgebra); with(VectorCalculus);
2.2.6. Defin\303\255ci\303\263.
Student[VectorCalculus][VectorFieldTutor]();
Student[MultivariateCalculus][DirectionalDerivativeTutor]();
f:=<x^2,x*y,x*z>;
Jacobian(f,[x,y,z]);
diff(f[1],x); diff(f[1],y); diff(f[1],z);
diff(f[2],x); diff(f[2],y); diff(f[2],z);
diff(f[3],x); diff(f[3],y); diff(f[3],z);
f:=<x^2,x*y,x*z,z*y>; Jacobian(f,[x,y,z]);
f:=3*x^2+2*y*z; Gradient(f,[x,y,z]);
Student[MultivariateCalculus][GradientTutor]();
2.2.7. Sima g\303\266rb\303\251k.
2.2.8. \303\226sszegszab\303\241ly.
2.2.9. L\303\241ncszab\303\241ly.
2.2.10. Koordin\303\241taf\303\274ggv\303\251nyek.
2.2.11. Parci\303\241lis<deriv\303\241ltak.
2.2.12. K\303\266z\303\251p\303\251rt\303\251k egyenl\305\221tlens\303\251g.
2.2.13. K\303\266vetkezm\303\251ny.
2.2.14. K\303\266vetkezm\303\251ny.
2.2.16. Magasabbrend\305\261 deriv\303\241ltak.
f:=(x,y,z)->x^2*y*z^3;
D[1](D[1](f)); D[1](D[2](f)); D[2](D[1](f));
f:=<3*x^3+2*y*z*x+x^2*y^2*z^2>; Jacobian(f,[x,y,z]);
convert(%,list); Jacobian(%,[x,y,z]);
convert(%,listlist); map(u->Jacobian(u,[x,y,z]),%);
f:=<x^3,x*y*z,x*z^2,z*y^2>; Jacobian(f,[x,y,z]);
convert(%,listlist); map(u->Jacobian(u,[x,y,z]),%);
2.2.17. Magasabbrend\305\261 deriv\303\241ltak \303\251s parci\303\241lis deriv\303\241ltak kapcsolata.
*2.2.19. Magasabbrend\305\261 deriv\303\241ltak mint multiline\303\241ris lek\303\251pez\303\251sek.
2.2.20. Young t\303\251tele.
2.2.21. Taylor-formula val\303\263s \303\251rt\303\251k\305\261 f\303\274ggv\303\251nyekre.
Student[MultivariateCalculus][TaylorApproximationTutor]();
mtaylor(sin(x^2+y^2),[x,y]);
mtaylor(sin(x^2+y^2),[x=0,y=0]);
mtaylor(sin(x^2+y^2),[x,y],8);
mtaylor(sin(x^2+y^2),[x=1,y=2],3);
mtaylor(g(x,y),[x,y]);
2.2.22. Megjegyz\303\251s.
2.2.23. Implicit f\303\274ggv\303\251ny t\303\251tel.
x:='x';y:='y';z:='z'; implicitdiff(x^2+y^2=1,y(x),x);
implicitdiff(x^2+y^2=z^2,z(x,y),x);
implicitdiff({x^2+y^2+z^2=1,x-y=z},{y(x),z(x)},y,x);
solve({x^2+y^2+z^2=1,x-y=z},{y,z});
2.2.24. Inverz f\303\274ggv\303\251ny t\303\251tel.
*2.2.25. Megjegyz\303\251s.
2.2.26. Sz\303\251ls\305\221\303\251rt\303\251k sz\303\274ks\303\251ges felt\303\251tele.
f:=cos(x+y)*cos(y+z)*cos(x+z); g:=Gradient(f,[x,y,z]);
subs([x=0,y=0,z=0],g); subs([x=Pi/2,y=Pi/2,z=Pi/2],g);
subs([x=Pi/2,y=0,z=Pi/2],g);
2.2.27. Lok\303\241lis sz\303\251ls\305\221\303\251rt\303\251k el\303\251gs\303\251ges felt\303\251tele.
with(Student[MultivariateCalculus]);
SecondDerivativeTest(f,[x,y,z]=[0,0,0]);
SecondDerivativeTest(f,[x,y,z]=[0,0,0],output=hessian);
SecondDerivativeTest(f,[x,y,z]=[Pi/2,Pi/2,Pi/2]);
SecondDerivativeTest(f,[x,y,z]=[Pi/2,Pi/2,Pi/2],output=hessian);
SecondDerivativeTest(f,[x,y,z]=[Pi/2,0,Pi/2]);
SecondDerivativeTest(f,[x,y,z]=[Pi/2,0,Pi/2],output=hessian);
2.2.34. Fel\303\274letek.
2.2.35. Fel\303\274letek el\305\221\303\241ll\303\255t\303\241si m\303\263djai.
2.2.37. Approxim\303\241ci\303\263.
a:='a'; b:='b'; c:='c'; with(CurveFitting);
data:=[[0,13.3],[20,31.6],[50,85.5],[80,169.0],[100,246.0]];
PolynomialInterpolation(data,x);
PolynomialInterpolation(data,x,form='Lagrange');
LeastSquares(data,x,curve=a*x^2+b*x+c);
*2.2.38. Newton-m\303\263dszer.
*2.2.39. Kapcsolat minimumfeladatokkal.
*2.2.40. Az ir\303\241nymenti cs\303\266kkent\303\251s m\303\263dszere.
*2.2.41. Algebrai egyenletek megold\303\241sa.
2.3. Integr\303\241lsz\303\241m\303\255t\303\241s
restart;with(VectorCalculus);
2.3.3. K\303\266vetkezm\303\251ny.
2.3.4. T\303\251tel: az integr\303\241l egy\303\251rtelm\305\261s\303\251ge.
2.3.5. T\303\251tel: az integr\303\241l linearit\303\241sa.
2.3.6. T\303\251tel: integr\303\241l \303\251s koordin\303\241taf\303\274ggv\303\251nyek.
2.3.7. T\303\251tel: az integr\303\241l nemnegativit\303\241sa.
2.3.8. K\303\266vetkezm\303\251ny: az integr\303\241l monotonit\303\241sa.
*2.3.9. T\303\251tel: Cauchy-krit\303\251rium.
*2.3.10. Seg\303\251dt\303\251tel.
2.3.11. T\303\251tel: az integr\303\241l mint halmazf\303\274ggv\303\251ny additivit\303\241sa.
2.3.13. Seg\303\251dt\303\251tel.
2.3.14. Lebesgue-felt\303\251tel.
2.3.15. K\303\266vetkezm\303\251ny.
2.3.17. Megjegyz\303\251s.
2.3.19. Abszol\303\272t integr\303\241lhat\303\263 f\303\274ggv\303\251nyek.
2.3.23. M\303\251rt\303\251k.
2.3.27. P\303\251lda: forg\303\241stest t\303\251rfogata.
2.3.29. Rademacher t\303\251tele.
2.3.30. Integr\303\241ltranszform\303\241ci\303\263s formula.
2.3.31. P\303\251ld\303\241k: pol\303\241r-, henger- \303\251s g\303\266mbi koordin\303\241t\303\241k.
Jacobian(<r*cos(phi),r*sin(phi)>,[r,phi],'determinant'=true);
simplify(%[2],trig);
Jacobian(<r*cos(phi),r*sin(phi),z>,[r,phi,z],'determinant'=true);
simplify(%[2],trig);
Jacobian(<r*cos(phi)*cos(theta),r*sin(phi)*cos(theta),r*sin(theta)>,
[r,phi,theta],'determinant'=true);
simplify(%[2],trig);
with(Student[MultivariateCalculus]);
ChangeOfVariables(x^2+y^2,['cartesian'[x,y],'polar'[r,phi]]);
simplify(%,trig);
ChangeOfVariables(Int(Int(x^2+y^2,x),y),
['cartesian'[x,y],'polar'[r,phi]]);
ChangeOfVariables(x^2+y^2-z^2,['cartesian'[x,y,z],
'cylindrical'[r,phi,z]]);
simplify(%,trig);
ChangeOfVariables(Int(Int(Int(x^2+y^2-z^2,x),y),z),
['cartesian'[x,y,z],'cylindrical'[r,phi,z]]);
2.3.32. T\303\251tel: param\303\251teres integr\303\241lok differenci\303\241l\303\241sa.
2.3.33. P\303\241ly\303\241k.
2.3.36. P\303\251ld\303\241k.
2.3.38. Integr\303\241l\303\241s fel\303\274leteken \303\251s felsz\303\255n.
PathInt(x^2+y^2+z^2,[x,y,z]=Path(<t,t^2,t^3>,t=0..2)); evalf(%);
SurfaceInt(x+y+z,[x,y,z]=Surface(<s,t,4-s*t>,
[s,t]=Triangle(<0,0>,<0,1>,<1,0>))); evalf(%);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn