> ##################### FEUILLE N3 237 ###################################### Exercice 1. plot([cos(3t), sin(2t), t=-pi..pi]);

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1 ##################### FEUILLE N3 37 ###################################### Exercice. plot([cos(3*t), sin(*t), t=-pi..pi]); ###################################### Exercice. restart:plot([*t^4-*t^3,t^-t, t= ]);

2 ###################################### Exercice 3. with(plots): polarplot([sin(*t/3),t,t=0..6*pi],color=blue); Warning, the name changecoords has been redefined

3 ###################################### Exercice 4 with(vectorcalculus): assume(a0): SetCoordinates( 'polar' ); A:=ArcLength( <a*cos(*t), t, t=0..pi/) ; polar A := a~ 3 EllipticE r:=a*cos(*t):simplify(int(sqrt(r^+diff(r,t)^),t)): with(vectorcalculus):assume(a0): SetCoordinates( 'cartesian' ); B:=ArcLength( <*a*cos(t),a*sin(t), t=0..pi/); cartesian B := a~ EllipticE 3

4 simplify(a-b); a~ 3 EllipticE a~ 3 EllipticE with(plots):p:=polarplot([3*sin(*t),t,t=0..4*pi],color=blue): P:=plot([6*cos(t),3*sin(t),t=0..4*Pi],color=red): display(p,p, scaling=constrained); Warning, the name changecoords has been redefined ###################################### Exercice 5. restart:with(plots): P:=polarplot([(cos(*t))^(/), t, t=0..pi/4]): display(p, scaling=constrained); Warning, the name changecoords has been redefined

5 r:=(cos(*t))^(/);rp:=diff(r,t); ds:=simplify((sqrt(r^+rp^)))*dt; r := cos( t ) rp := ds := sin( t ) cos( t ) cos( t ) ########################################### ###################################### Exercice 6. SetCoordinates( 'polar' ); ArcLength( <, t, t=0..*pi) ; polar π dt

6 restart;f:=x^+y^-;fx:=diff(f, x);fy:=diff(f, y); y0:=0.3; x0:=sqrt(-y0^+); F := x + y Fx := x Fy := y y0 := 0.3 x0 := with(plots):b0:=implicitplot(f=0, x=-..,y=-..,color=black, numpoints=000): y0:=0.4; x0:=sqrt(-y0^+); Fx0:=eval( Fx,[x=x0, y=y0]); Fy0:=eval( Fy,[x=x0, y=y0]); l0:=(fx0^+fy0^)^(/); b := arrow(<x0,y0, <-Fy0/l0,Fx0/l0, length=[0.5], width=[0.0, relative], head_length=[0., relative], color=red): b := arrow( <x0,y0, <-Fx0/l0,-Fy0/l0, length=[0.5],width=[0.0,relative], color=blue): display(b0, b, b, scaling=constrained); y0 := 0.4 x0 := Fx0 := Fy0 := 0.8 l0 :=

7 ###################################### Exercice 7. with(vectorcalculus): SetCoordinates( 'cartesian' ): ArcLength( <sin(t),sin(t), t=0..*pi ) ; SetCoordinates( 'cartesian' ):simplify(curvature( <sin(t),sin(t) ))assuming t::real ; 4 0 ###################################### Exercice 8. #Hyperbole restart;f:=x^-y^-;fx:=diff(f, x);fy:=diff(f, y); y0:=; x0:=sqrt(y0^+);

8 F := x y Fx := x Fy := y y0 := x0 := with(plots):b0:=implicitplot(f=0, x=-..,y=-..,color=black, numpoints=000): #y0:=; x0:=sqrt(y0^+); Fx0:=eval( Fx,[x=x0, y=y0]); Fy0:=eval( Fy,[x=x0, y=y0]); l0:=(fx0^+fy0^)^(/); b := arrow(<x0,y0, <-Fy0/l0,Fx0/l0, width=[0.0, relative], head_length=[0., relative], color=red): b := arrow( <x0,y0, <-Fx0/l0,-Fy0/l0,width=[0.0,relative], color=blue): display(b0, b, b, scaling=constrained, axes=framed); Warning, the name changecoords has been redefined Fx0 := Fy0 := - l0 :=

9 ###################################### Exercice 9 #Ellipse restart;f:=*x^+3*y^-;fx:=diff(f, x);fy:=diff(f, y); y0:=.; solve(f,x)=0;x0:=max(eval(solve(f,x), y=y0)); F := x + 3 y Fx := 4 x Fy := 6 y y0 := 0. 6 y +, 6 y + x0 := = 0 with(plots):b0:=implicitplot(f=0, x=-..,y=-..,color=black, numpoints=000):

10 #y0:=; x0:=sqrt(y0^+); Fx0:=eval( Fx,[x=x0, y=y0]); Fy0:=eval( Fy,[x=x0, y=y0]); l0:=(fx0^+fy0^)^(/); b := arrow(<x0,y0, <-Fy0/l0,Fx0/l0, length=[0.5], width=[0.0, relative], head_length=[0., relative], color=red): b := arrow( <x0,y0, <-Fx0/l0,-Fy0/l0, length=[0.5],width=[0.0, relative], head_length=[0., relative], color=blue): display(b0, b, b, scaling=constrained, axes=normal); Fx0 := Fy0 := 0.6 l0 := ###################################### Exercice 0 #Parabole

11 restart;f:=y^-6*x;fx:=diff(f, x);fy:=diff(f, y); y0:=; solve(f,x)=0;x0:=max(eval(solve(f,x), y=y0)); F := y 6 x Fx := -6 Fy := y y0 := y = 0 6 x0 := 6 with(plots): b0:=implicitplot(f=0, x=-..,y=-..,color=black, numpoints=000,coords=cartesian): #y0:=; x0:=sqrt(y0^+); Fx0:=eval( Fx,[x=x0, y=y0]); Fy0:=eval( Fy,[x=x0, y=y0]); l0:=(fx0^+fy0^)^(/); b := arrow(<x0,y0, <Fy0/l0,-Fx0/l0, width=[0.0, relative], head_length=[0., relative], color=red): b := arrow( <x0,y0, <Fx0/l0,Fy0/l0,width=[0.0, relative], head_length=[0., relative], color=blue): display(b0, b, b, scaling=constrained, axes=framed); Warning, the name changecoords has been redefined Fx0 := -6 Fy0 := l0 := 40

12 ###################################### #Une courbe elliptique (cet exemple n'est pas inclus dans la feuille 3) restart;f:=y^-x^3+x;fx:=diff(f, x);fy:=diff(f, y); x0:=-.; solve(f,y)=0;y0:=max(eval(solve(f,y), x=x0)); F := y x 3 + x Fx := 3 x + Fy := y x0 := -0. ( x 3 x, x 3 x ) = 0 y0 :=

13 with(plots):b0:=implicitplot(f=0, x=-..,y=-..,color=black, numpoints=5000): #y0:=; x0:=sqrt(y0^+); Fx0:=eval( Fx,[x=x0, y=y0]); Fy0:=eval( Fy,[x=x0, y=y0]); l0:=(fx0^+fy0^)^(/); b := arrow(<x0,y0, <-Fy0/l0,Fx0/l0, width=[0.0, relative], head_length=[0., relative], color=red): b := arrow( <x0,y0, <-Fx0/l0,-Fy0/l0,width=[0.0, relative], head_length=[0., relative], color=blue): display(b0, b, b, scaling=constrained); Warning, the name changecoords has been redefined Fx0 := 0.97 Fy0 := l0 :=

14 ###################################### Exercice with(vectorcalculus): SetCoordinates( 'cartesian' ); K:=simplify(Curvature( <(+cos(t)^)*sin(t),sin(t)^*cos(t) )) assuming t::real ; Warning, the assigned names <, and < now have a global binding Warning, these protected names have been redefined and unprotected: *, +,., D, Vector, diff, int, limit, series cartesian K := 3 cos( t ) R:=/K; R := 3 cos( t ) x:=(+cos(t)^)*sin(t);y:=sin(t)^*cos(t); xp:=diff(x,t);yp:=diff(y,t); xs:=diff(xp,t);ys:=diff(yp,t); x := ( cos( t ) + ) sin( t ) y := sin( t ) cos( t ) xp := sin( t ) cos( t ) + ( cos( t) + ) cos( t) yp := sin( t ) cos( t ) sin( t ) 3 xs := 6 sin( t ) cos( t) + sin( t ) 3 ( cos( t ) + ) sin( t ) ys := cos( t) 3 7 sin( t ) cos( t ) K:=simplify(((xp*ys-yp*xs)/(xp^+yp^)^(3/))) assuming t::real; K := 3 cos( t ) X:=simplify(x-(yp*(xp^+yp^))/(xp*ys-yp*xs));Y:=simplify(y+xp*( xp^+yp^)/(xp*ys-yp*xs)); X := sin( t) 3 Y := cos( t ) 3 ##################################################### Exercice with(vectorcalculus): SetCoordinates( 'cartesian' ); assume(t0):k:=simplify(curvature( <t-sinh(t)*cosh(t),*cosh(t) )) assuming t::real ; Warning, the assigned names <, and < now have a global binding Warning, these protected names have been redefined and unprotected: *, +,., D, Vector, diff, int, limit, series K := cartesian cosh( t~ ) sinh( t~ )

15 R:=/K; R := cosh( t~ ) sinh( t~ ) x:=t-sinh(t)*cosh(t);y:=*cosh(t); xp:=diff(x,t);yp:=diff(y,t); xs:=diff(xp,t);ys:=diff(yp,t); x := t~ sinh( t~ ) cosh( t~ ) y := cosh( t~ ) xp := cosh( t~ ) sinh( t~ ) yp := sinh( t~ ) xs := 4 sinh( t~ ) cosh( t~ ) ys := cosh( t~ ) K:=simplify(((xp*ys-yp*xs)/(xp^+yp^)^(3/))) assuming t::real; K := cosh( t~ ) sinh( t~ ) X:=simplify(x-(yp*(xp^+yp^))/(xp*ys-yp*xs));Y:=simplify(y+xp*( xp^+yp^)/(xp*ys-yp*xs)); X := 3 sinh( t~ ) cosh( t~ ) + t~ Y := cosh( t~ ) ( cosh( t~ ) ) ################################################################ ###################### ###################################### Exercice 3 r:=tan(t/); x:=r*cos(t);y:=r*sin(t);r := tan(/*t); r := tan t~ x := tan t~ cos( t~ ) y := tan t~ sin( t~ ) r := tan t~ with(vectorcalculus): SetCoordinates( 'polar' ); K:=simplify(Curvature( <tan(t/),t )) assuming t::real ; polar K := ( cos( t~ ) cos( t~ ) ) ( cos( t~ ) + ) ( cos( t~ ) + ) ( 3/ ) rp:=diff(r,t);rs:=diff(rp,t); rp := + tan t~

16 rs := tan t~ + tan t~ xp:=diff(x,t);yp:=diff(y,t); xs:=diff(xp,t);ys:=diff(yp,t); xp := + tan t~ cos( t~ ) tan t~ sin( t~ ) yp := + tan t~ sin( t~ ) + tan t~ cos( t~ ) xs := tan t~ + tan t~ cos( t~ ) + tan t~ sin( t~ ) tan t~ cos( t~ ) ys := tan t~ + tan t~ sin( t~ ) + + tan t~ cos( t~ ) tan t~ sin( t~ ) K:=simplify(((xp*ys-yp*xs)/(xp^+yp^)^(3/))) assuming t::real; ( cos( t~ ) + 3 cos( t~ )) ( cos( t~ ) + ) 3 K := sin( t~ ) ( cos( t~ ) + ) ( 3/ ) X:=simplify(x-(yp*(xp^+yp^))/(xp*ys-yp*xs));Y:=simplify(y+xp*( xp^+yp^)/(xp*ys-yp*xs)); X := sin ( t~ ) ( + cos ( t~ ) ) cos( t~ ) cos( t~ ) Y := ( cos( t~ ) 3 ) cos( t~ ) cos( t~ ) 3 3 cos( t~ ) ###################################### Exercice 4 restart:print('(d/dt)*arcsin(sqrt(t))'=diff(arcsin(sqrt(t)), t)); d arcsin( t ) = dt t t print('(d/dt)*arcsin(sqrt(t))/(sqrt(t*(-t)))'=diff((arcsin(sqrt (t)))/(sqrt(t*(-t))), t)); d arcsin( t ) arcsin( t ) ( t ) = ( 3 / ) dt t ( t ) t t t ( t ) ( t ( t )) simplify(diff((arcsin(sqrt(t)))/(sqrt(t*(-t))), t)); t + t arcsin( t ) t t + arcsin( t ) t ( 3 / ) t t ( 3 / ) ( t ) ( 3/ ) t ( + t ) with(vectorcalculus): SetCoordinates( 'cartesian' ); print('arcsin'(sqrt(t))/(sqrt(t*(-t)))); K:=simplify(Curvature( <t,(arcsin(sqrt(t)))/(sqrt(t*(-t))) ))

17 assuming t::positive : Warning, the assigned names <, and < now have a global binding Warning, these protected names have been redefined and unprotected: *, +,., D, Vector, diff, int, limit, series cartesian arcsin( t ) t ( t) R:=/K: ###################################### Exercice 5 restart;with(plots): plot([t^+t,(t+)*exp(/t), t=0.5..], labels=[x, y]); Warning, the name changecoords has been redefined

18 with(vectorcalculus): SetCoordinates( 'cartesian' ); K:=simplify(Curvature( <t^+t,(t+)*exp(/t) )) assuming t::positive : Warning, the assigned names <, and < now have a global binding Warning, these protected names have been redefined and unprotected: *, +,., D, Vector, diff, int, limit, series cartesian ###################################### Exercice 6 restart;p:=(x-y)/(x^+y^);q:=(x+y)/(x^+y^);py:=diff(p,y); Qx:=diff(Q,x);'Py'-'Qx'=simplify(Py-Qx);F:=int(P,x);'Q'=simplify (diff(int(p,x),y));

19 x y P := x + y x + y Q := x + y ( x y) y Py := x + y ( x + y ) ( x + y) x Qx := x + y ( x + y ) Py Qx = 0 F := ln ( x + y ) arctan x y x + y Q = x + y assume( a 0 ); FP:=evala(subs({x=a, y=-a},f)); FPmoins:=evala(limit((subs({x=a},F), y=0, left))); FPplus:=evala(limit((subs({x=a},F), y=0, right))); FP3moins:=evala(limit((subs({x=a},F), y=0, right))); FP3plus:=evala(limit((subs({x=a},F), y=0, left))); FP := ln( a~ π ) + 4 π FPmoins := ln( a~ ) + π FPplus := ln( a~ ) π FP3moins := ln( a~ ) π FP3plus := ln( a~ ) + print('(f(p[,moins])-f(p[]))+f(p[3,plus])-f(p[,plus])+(f(p[] )-F(P[3,moins]))' =FPmoins-FP+FP3plus-FPplus+FP-FP3moins); F( P, moins ) F( P ) + F( P 3, plus ) F( P, plus ) + F( P ) F( P 3, moins ) = π x:=a*cos(t);y:=a*sin(t); print('p*diff(x,t)+q*diff(y,t)'=p*diff(x,t)+q*diff(y,t)); simplify(p*diff(x,t)+q*diff(y,t)); x := a cos( t ) y := a sin( t )

20 P + = t x Q t y ( a cos( t) a sin( t) ) a sin( t ) a + + ( a cos( t) + a sin( t) ) a cos( t ) cos( t ) a sin( t) a cos( t ) + a sin( t ) print('int(,t=0..*pi)'=int(,t=0..*pi)); 0 π dt = π ###################################### Exercice 7 restart;p:=exp(x)*cos(y)+x*y^;q:=(-exp(x)*sin(y)+x^*y); P := e x cos( y) + x y diff(p,y); diff(q,x); Q := e x sin( y) + x y e x sin( y ) + e x sin( y ) + xy xy F:=exp(x)*cos(y)+x^*y^/;'P'=diff(F,x);'Q'=diff(F,y); F := e x x y cos( y ) + P = e x cos( y) + x y Q = e x sin( y) + x y int(p,x); e x x y cos( y ) + #Soit a;b 0. Calculer int( x^*dy + y^*dx, o u a pour equation cart esienne l'une des # equations suivantes: x^ +y^-a*x=0; (x/a)^ +(y/b)^=; # (x/a)^ +(y/b)^-*(x/a)-*(y/b)=0; assume(a0);assume(b0); x:=(a/)*(cos(t)+);y:=(a/)*(sin(t)); x := a~ ( cos( t ) + ) y := a~ sin( t ) xp:=diff(x,t);yp:=diff(y,t); xp := a~ sin( t ) x^*yp+y^*xp; yp := a~ cos( t ) a ( cos( t ) + ) b cos( t) b ( sin( t ) + ) a sin( t )

21 simplify(int(x^*yp+y^*xp,t)); 4 a~3 ( cos ( t ) sin ( t ) + 3 sin( t ) cos( t ) + 5 sin( t ) + 3 t cos( t ) cos( t )) x^*yp+y^*xp; 8 a~3 ( cos( t ) + ) cos( t ) 8 a~3 sin( t ) 3 simplify(int(x^*yp+y^*xp,t=0..*pi)); a~ 3 π 4 ################################################ x:=a*(cos(t));y:=(b)*(sin(t)); x := a cos( t ) y := b sin( t ) xp:=diff(x,t);yp:=diff(y,t); xp := a sin( t ) yp := b cos( t ) simplify(int(x^*yp+y^*xp,t)); 3 ab( a cos( t ) sin ( t ) + a sin( t ) + 3 b cos( t) b cos( t ) 3 ) x^*yp+y^*xp; a cos( t) 3 b b sin( t ) 3 a simplify(int(x^*yp+y^*xp,t=0..*pi)); 0 ########################################## restart;p:=y^;q:=x^; P:= y diff(p,y); diff(q,x); Q:= x y x ############################################################ restart;((x/a)-)^+((y/b)-)^=; x y + = a b x:=a*(sqrt()*cos(t)+);y:=(b)*(sqrt()*sin(t)+); x := a( cos( t) + ) y := b( sin( t ) + ) xp:=diff(x,t);yp:=diff(y,t);

22 xp := a sin( t ) yp := b cos( t) simplify(int(x^*yp+y^*xp,t)); 3 ab ( a ) sin( t) + 6 a cos( t ) sin( t ) + 7 a sin( t ) + 6 b sin( t ) cos( t ) 6 bt + 9 b cos( t ) + 6 at b cos( t ) ) x^*yp+y^*xp; a ( cos( t ) + ) b cos( t) b ( sin( t ) + ) a sin( t ) simplify(int(x^*yp+y^*xp,t)); 3 ab ( a cos( t ) sin( t) + 6 a cos( t ) sin( t ) + 7 a sin( t ) + 6 b sin( t ) cos( t ) 6 bt + 9 b cos( t ) + 6 at b cos( t ) 3 ) simplify(int(x^*yp+y^*xp,t=0..*pi)); 4 a b π 4 ab π

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