Calka pow J=calkcalk odS (2,5x+3y+z)ds s:2x+y+z-4=0 x,y,z>=0 rozw z=F(x,y)=4-2x-y F(0,0)=4 z i y=0 wiec 0=4-2x-0 x=2 z i y=0 0=4-2*0-y y=4 rysuje osx=2 osy=4 wych trojkat s:{(x,y,F(x,y); (x,y)nalD} J=calkcalk odD (x,y,F(x,y))*pier(1+F'x^2+F'y^2) dxdy=... D:{(x,y); 0<=x<=2; 0<=y<=4-2x} J=calk od0 do2(calk od0 do4-2x(...)dy)dx=....///kwadraty z=... F(x,y)=... s={...} J=calkcalk odD ...*pier...dxdy=|y=rcosfi x=rsinfi J=r| delta={(r,fi) ..<=r<=...;...<=fi<=...} J=calkcalk oddelta...*J*pier... drdfi=...///Calka pow zor J=calkcalk odS Fds*F(x,y,z)=(y-z,1,x) s:z=6-2x-y x,y,z>=0 cosfi>0 rozw P=y-z Q=1 R=x F(x,y)=6-2x-y F'x=.. F'y=... s={(x,y,F(x,y); (x,y)nalD F(0,0)=6 0=6-y y=6 0=6-2x x=3 rysuje osx=3 osy=6 J=calkcalk poD[R-PF'x-QF'y]dxdy=... x=x y=y z=F(x,y) =... D={(x,y) 0<x<3; 0<y<6-2x} =calk od0 do3 (calk od0 do6-2x ...dy)dx=...///F holomorficzne f(z)=z*Re(z) z=x+iy f(x,y)=(x+iy)x=x^2+ixy u=x^2 V=xy U'x=2x V'y=x 2x=/=x wiec nie holo. zad f=u+iv=? U(x,y)=... rozw U'x=... U''x=... U'y=... U''y=... gradf(x,y)=U''x+U''y=0-ist f h =/=0-nie ist f h jesli ist to u'x=v'y u'y=v'x => v'y=... v'x=... v(x,y)=calk(v'y)dy=...+fi(x) (...+fi(x))'x=v'x+fi'(x) =>fi'(x)=... =>fi(x)=...(C) CnalR odp f(x,y)=U(x,y)+iV(x,y)+Ci CinalR
WZORY calki: x^n=(1/n+1)x^n+1 1/x=ln|x| a^x=(1/lna)a^x pierx=2/3pierx^3 1/pierx=2pierx sinx= -cosx cosx=sinx tgx= -ln |cosx| ctgx=ln|sinx| pochodne: x^n=n x^n-1 pierx=1/2pierx a/x=-a/x^ sinx=cosx cosx=-sinx tgx=1/cos^x ctgx=-1/sin^x a^x=a^x lna lnx=1/x
Calka pow J=calkcalk odS (2,5x+3y+z)ds s:2x+y+z-4=0 x,y,z>=0 rozw z=F(x,y)=4-2x-y F(0,0)=4 z i y=0 wiec 0=4-2x-0 x=2 z i y=0 0=4-2*0-y y=4 rysuje osx=2 osy=4 wych trojkat s:{(x,y,F(x,y); (x,y)nalD} J=calkcalk odD (x,y,F(x,y))*pier(1+F'x^2+F'y^2) dxdy=... D:{(x,y); 0<=x<=2; 0<=y<=4-2x} J=calk od0 do2(calk od0 do4-2x(...)dy)dx=....///kwadraty z=... F(x,y)=... s={...} J=calkcalk odD ...*pier...dxdy=|y=rcosfi x=rsinfi J=r| delta={(r,fi) ..<=r<=...;...<=fi<=...} J=calkcalk oddelta...*J*pier... drdfi=...///Calka pow zor J=calkcalk odS Fds*F(x,y,z)=(y-z,1,x) s:z=6-2x-y x,y,z>=0 cosfi>0 rozw P=y-z Q=1 R=x F(x,y)=6-2x-y F'x=.. F'y=... s={(x,y,F(x,y); (x,y)nalD F(0,0)=6 0=6-y y=6 0=6-2x x=3 rysuje osx=3 osy=6 J=calkcalk poD[R-PF'x-QF'y]dxdy=... x=x y=y z=F(x,y) =... D={(x,y) 0<x<3; 0<y<6-2x} =calk od0 do3 (calk od0 do6-2x ...dy)dx=...///F holomorficzne f(z)=z*Re(z) z=x+iy f(x,y)=(x+iy)x=x^2+ixy u=x^2 V=xy U'x=2x V'y=x 2x=/=x wiec nie holo. zad f=u+iv=? U(x,y)=... rozw U'x=... U''x=... U'y=... U''y=... gradf(x,y)=U''x+U''y=0-ist f h =/=0-nie ist f h jesli ist to u'x=v'y u'y=v'x => v'y=... v'x=... v(x,y)=calk(v'y)dy=...+fi(x) (...+fi(x))'x=v'x+fi'(x) =>fi'(x)=... =>fi(x)=...(C) CnalR odp f(x,y)=U(x,y)+iV(x,y)+Ci CinalR
WZORY calki: x^n=(1/n+1)x^n+1 1/x=ln|x| a^x=(1/lna)a^x pierx=2/3pierx^3 1/pierx=2pierx sinx= -cosx cosx=sinx tgx= -ln |cosx| ctgx=ln|sinx| pochodne: x^n=n x^n-1 pierx=1/2pierx a/x=-a/x^ sinx=cosx cosx=-sinx tgx=1/cos^x ctgx=-1/sin^x a^x=a^x lna lnx=1/x
Calka pow J=calkcalk odS (2,5x+3y+z)ds s:2x+y+z-4=0 x,y,z>=0 rozw z=F(x,y)=4-2x-y F(0,0)=4 z i y=0 wiec 0=4-2x-0 x=2 z i y=0 0=4-2*0-y y=4 rysuje osx=2 osy=4 wych trojkat s:{(x,y,F(x,y); (x,y)nalD} J=calkcalk odD (x,y,F(x,y))*pier(1+F'x^2+F'y^2) dxdy=... D:{(x,y); 0<=x<=2; 0<=y<=4-2x} J=calk od0 do2(calk od0 do4-2x(...)dy)dx=....///kwadraty z=... F(x,y)=... s={...} J=calkcalk odD ...*pier...dxdy=|y=rcosfi x=rsinfi J=r| delta={(r,fi) ..<=r<=...;...<=fi<=...} J=calkcalk oddelta...*J*pier... drdfi=...///Calka pow zor J=calkcalk odS Fds*F(x,y,z)=(y-z,1,x) s:z=6-2x-y x,y,z>=0 cosfi>0 rozw P=y-z Q=1 R=x F(x,y)=6-2x-y F'x=.. F'y=... s={(x,y,F(x,y); (x,y)nalD F(0,0)=6 0=6-y y=6 0=6-2x x=3 rysuje osx=3 osy=6 J=calkcalk poD[R-PF'x-QF'y]dxdy=... x=x y=y z=F(x,y) =... D={(x,y) 0<x<3; 0<y<6-2x} =calk od0 do3 (calk od0 do6-2x ...dy)dx=...///F holomorficzne f(z)=z*Re(z) z=x+iy f(x,y)=(x+iy)x=x^2+ixy u=x^2 V=xy U'x=2x V'y=x 2x=/=x wiec nie holo. zad f=u+iv=? U(x,y)=... rozw U'x=... U''x=... U'y=... U''y=... gradf(x,y)=U''x+U''y=0-ist f h =/=0-nie ist f h jesli ist to u'x=v'y u'y=v'x => v'y=... v'x=... v(x,y)=calk(v'y)dy=...+fi(x) (...+fi(x))'x=v'x+fi'(x) =>fi'(x)=... =>fi(x)=...(C) CnalR odp f(x,y)=U(x,y)+iV(x,y)+Ci CinalR
WZORY calki: x^n=(1/n+1)x^n+1 1/x=ln|x| a^x=(1/lna)a^x pierx=2/3pierx^3 1/pierx=2pierx sinx= -cosx cosx=sinx tgx= -ln |cosx| ctgx=ln|sinx| pochodne: x^n=n x^n-1 pierx=1/2pierx a/x=-a/x^ sinx=cosx cosx=-sinx tgx=1/cos^x ctgx=-1/sin^x a^x=a^x lna lnx=1/x
Calka pow J=calkcalk odS (2,5x+3y+z)ds s:2x+y+z-4=0 x,y,z>=0 rozw z=F(x,y)=4-2x-y F(0,0)=4 z i y=0 wiec 0=4-2x-0 x=2 z i y=0 0=4-2*0-y y=4 rysuje osx=2 osy=4 wych trojkat s:{(x,y,F(x,y); (x,y)nalD} J=calkcalk odD (x,y,F(x,y))*pier(1+F'x^2+F'y^2) dxdy=... D:{(x,y); 0<=x<=2; 0<=y<=4-2x} J=calk od0 do2(calk od0 do4-2x(...)dy)dx=....///kwadraty z=... F(x,y)=... s={...} J=calkcalk odD ...*pier...dxdy=|y=rcosfi x=rsinfi J=r| delta={(r,fi) ..<=r<=...;...<=fi<=...} J=calkcalk oddelta...*J*pier... drdfi=...///Calka pow zor J=calkcalk odS Fds*F(x,y,z)=(y-z,1,x) s:z=6-2x-y x,y,z>=0 cosfi>0 rozw P=y-z Q=1 R=x F(x,y)=6-2x-y F'x=.. F'y=... s={(x,y,F(x,y); (x,y)nalD F(0,0)=6 0=6-y y=6 0=6-2x x=3 rysuje osx=3 osy=6 J=calkcalk poD[R-PF'x-QF'y]dxdy=... x=x y=y z=F(x,y) =... D={(x,y) 0<x<3; 0<y<6-2x} =calk od0 do3 (calk od0 do6-2x ...dy)dx=...///F holomorficzne f(z)=z*Re(z) z=x+iy f(x,y)=(x+iy)x=x^2+ixy u=x^2 V=xy U'x=2x V'y=x 2x=/=x wiec nie holo. zad f=u+iv=? U(x,y)=... rozw U'x=... U''x=... U'y=... U''y=... gradf(x,y)=U''x+U''y=0-ist f h =/=0-nie ist f h jesli ist to u'x=v'y u'y=v'x => v'y=... v'x=... v(x,y)=calk(v'y)dy=...+fi(x) (...+fi(x))'x=v'x+fi'(x) =>fi'(x)=... =>fi(x)=...(C) CnalR odp f(x,y)=U(x,y)+iV(x,y)+Ci CinalR
WZORY calki: x^n=(1/n+1)x^n+1 1/x=ln|x| a^x=(1/lna)a^x pierx=2/3pierx^3 1/pierx=2pierx sinx= -cosx cosx=sinx tgx= -ln |cosx| ctgx=ln|sinx| pochodne: x^n=n x^n-1 pierx=1/2pierx a/x=-a/x^ sinx=cosx cosx=-sinx tgx=1/cos^x ctgx=-1/sin^x a^x=a^x lna lnx=1/x
Calka pow J=calkcalk odS (2,5x+3y+z)ds s:2x+y+z-4=0 x,y,z>=0 rozw z=F(x,y)=4-2x-y F(0,0)=4 z i y=0 wiec 0=4-2x-0 x=2 z i y=0 0=4-2*0-y y=4 rysuje osx=2 osy=4 wych trojkat s:{(x,y,F(x,y); (x,y)nalD} J=calkcalk odD (x,y,F(x,y))*pier(1+F'x^2+F'y^2) dxdy=... D:{(x,y); 0<=x<=2; 0<=y<=4-2x} J=calk od0 do2(calk od0 do4-2x(...)dy)dx=....///kwadraty z=... F(x,y)=... s={...} J=calkcalk odD ...*pier...dxdy=|y=rcosfi x=rsinfi J=r| delta={(r,fi) ..<=r<=...;...<=fi<=...} J=calkcalk oddelta...*J*pier... drdfi=...///Calka pow zor J=calkcalk odS Fds*F(x,y,z)=(y-z,1,x) s:z=6-2x-y x,y,z>=0 cosfi>0 rozw P=y-z Q=1 R=x F(x,y)=6-2x-y F'x=.. F'y=... s={(x,y,F(x,y); (x,y)nalD F(0,0)=6 0=6-y y=6 0=6-2x x=3 rysuje osx=3 osy=6 J=calkcalk poD[R-PF'x-QF'y]dxdy=... x=x y=y z=F(x,y) =... D={(x,y) 0<x<3; 0<y<6-2x} =calk od0 do3 (calk od0 do6-2x ...dy)dx=...///F holomorficzne f(z)=z*Re(z) z=x+iy f(x,y)=(x+iy)x=x^2+ixy u=x^2 V=xy U'x=2x V'y=x 2x=/=x wiec nie holo. zad f=u+iv=? U(x,y)=... rozw U'x=... U''x=... U'y=... U''y=... gradf(x,y)=U''x+U''y=0-ist f h =/=0-nie ist f h jesli ist to u'x=v'y u'y=v'x => v'y=... v'x=... v(x,y)=calk(v'y)dy=...+fi(x) (...+fi(x))'x=v'x+fi'(x) =>fi'(x)=... =>fi(x)=...(C) CnalR odp f(x,y)=U(x,y)+iV(x,y)+Ci CinalR
WZORY calki: x^n=(1/n+1)x^n+1 1/x=ln|x| a^x=(1/lna)a^x pierx=2/3pierx^3 1/pierx=2pierx sinx= -cosx cosx=sinx tgx= -ln |cosx| ctgx=ln|sinx| pochodne: x^n=n x^n-1 pierx=1/2pierx a/x=-a/x^ sinx=cosx cosx=-sinx tgx=1/cos^x ctgx=-1/sin^x a^x=a^x lna lnx=1/x