Shared2017-09-29-144956.sagewsOpen in CoCalc
DG
%md
<B> Čelija varijabli </B>

Čelija varijabli
t,x,y,z=var('t,x,y,z')
t,n=var('t,n')
a=1/factorial(n)*(cos(n*pi/2))*t^n
b=1/factorial(n)*(sin(n*pi/2))*t^n
sum(a,n,0,infinity)
sum(b,n,0,infinity)
cos(t) sin(t)
t=var('t')
r=vector((e^t,t^3+t))
der1=diff(r,t);der1
der2=diff(der1,t);der2
der3=diff(der2,t);der3
r=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2+der3(t=0)*t^3/6
r
(e^t, 3*t^2 + 1) (e^t, 6*t) (e^t, 6) (1/6*t^3 + 1/2*t^2 + t + 1, t^3 + t)
t=var('t')
r=vector((e^t,t^3+t))
a1=r
a2=r(t=0)+der1(t=0)*t
a3=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2
a4=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2+der3(t=0)*t^3/6
parametric_plot(a1,(t,-1.5,1.5),color='green')+parametric_plot(a2,(t,-1,1))+parametric_plot(a3,(t,-3,1),color='red')+parametric_plot(a4,(t,-1,1),color='orange')
t=var('t')
r=vector((t^2,t^4,t))
der1=diff(r,t)
der2=diff(der1,t)
der3=diff(der2,t)
der4=diff(der3,t)
a1=r
a2=r(t=0)+der1(t=0)*t
a3=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2
a4=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2+der3(t=0)*t^3/6
a5=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2+der3(t=0)*t^3/6+der4(t=0)*t^4/24
parametric_plot(a1,(t,-1.5,1.5),color='green')+parametric_plot(a2,(t,-1,1),color='red')+parametric_plot(a3,(t,-3,3),color='red')+parametric_plot(a4,(t,-1,1),color='orange')+parametric_plot(a5,(t,-1,1),color='black')
3D rendering not yet implemented
%md
$$\vec{r}(t)=e^t \vec{i}+(t^3+t) \vec{j}$$

$$\vec{r}(t) \approx (\frac{t^3}{6}+\frac{t^2}{2}+t+1) \vec{\imath}+(t^3+t) \vec{\jmath}$$

$$\operatorname{funkcija}$$

a=vector([1,2,-4])
b=vector([2,0,3])
c=vector([1,0,1])
show(a.cross_product(b))
show(a.dot_product(b))
show(a.dot_product(b)*c)
show(a.dot_product(b.cross_product(c)))
show((a.cross_product(b)).dot_product(c))
show((b.cross_product(a)).dot_product(c))
show((a.cross_product(b)).cross_product(c))
show(a.cross_product(b.cross_product(c)))

show((2*a+3*b-5*c).cross_product(a-2*b-4*c))
%md

b*(a.dot_product(c))-a*(b.dot_product(c))
((a.cross_product(b)).cross_product(c))
(-11, -10, 11) (-11, -10, 11)
a.cross_product(b.cross_product(c))
(a.cross_product(b.cross_product(c)))
(4, 0, 1) (4, 0, 1)
%md

2
Vektori su linearno zavisni

t=var('t')
a=vector((1,t,t^2))
b=vector((-t,2*t,t^3))
c=vector((t^2,-1,t))

show(factor(a.dot_product(b.cross_product(c))))

t=var('t')
u=vector((t^3+2*t,sin(t),e^t))
d1=diff(u,t)
d2=diff(d1,t)
d3=diff(d2,t)
show(u,d1,d2,d3)

a,t,T=var('a,t,T')
u=vector((a*cos(t),a*sin(t)))
T=(1+t^2)^0.5
assume(t>0)
d1=diff(u,t)
d2=diff(T,t)
show((d1*1/d2))

t,i,j=var('t,i,j')
u=((cos(t)*i+sin(t)*j))
factor(taylor(u,t,0,10))
-1/3628800*i*t^10 + 1/362880*j*t^9 + 1/40320*i*t^8 - 1/5040*j*t^7 - 1/720*i*t^6 + 1/120*j*t^5 + 1/24*i*t^4 - 1/6*j*t^3 - 1/2*i*t^2 + j*t + i
t=var('t')
y(t)=t
x(t)=1
parametric_plot( (1,t), (t,-5,5),thickness=1 ,color='cyan',aspect_ratio=true)+parametric_plot( (1-(t^2)/2,t), (t,-3,3),thickness=1 ,color='green',aspect_ratio=true)+parametric_plot( (1-(t^2)/2,t-(t^3)/6), (t,-3,3),thickness=1 ,color='orange',aspect_ratio=true)+parametric_plot( (cos(t),sin(t)),(t,0,2*pi),thickness=1,color='red',aspect_ratio=true)+point((1,0),size=30,color='black')+parametric_plot( (1-(t^2)/2+(t^4)/24-(t^6)/720+(t^8)/40320,t-(t^3)/6+(t^5)/120+(t^7)/5040+(t^9)/362880), (t,-4,4),thickness=1 ,color='magenta',aspect_ratio=true)

t,i,j=var('t,i,j')
u=((i*e^t+(t^3+t)*j))
factor(taylor(u,t,0,10))
1/3628800*i*t^10 + 1/362880*i*t^9 + 1/40320*i*t^8 + 1/5040*i*t^7 + 1/720*i*t^6 + 1/120*i*t^5 + 1/24*i*t^4 + 1/6*i*t^3 + j*t^3 + 1/2*i*t^2 + i*t + j*t + i

parametric_plot3d((2*sin(t),2*cos(t),t/2),(t,0,5*pi))
3D rendering not yet implemented
parametric_plot3d(((2*sin(t))^2,4*sin(t)*cos(t),2*cos(t)),(t,0,2*pi))
3D rendering not yet implemented
parametric_plot3d(((cos(t)*e^t,sin(t)*e^t,2*t)),(t,-pi,pi))
3D rendering not yet implemented
t=var('t')
parametric_plot3d((sin(2*t),1-cos(2*t),2*cos(t)),(t,0,2*pi))
3D rendering not yet implemented
t,y,z=var('t,y,z')
parametric_plot3d((sin(2*t),1-cos(2*t),2*cos(t)),(t,0,2*pi))+implicit_plot3d(x^2+(y-1)^2==1,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='yellow')+implicit_plot3d(x^2+(y)^2+z^2==4,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='green')+implicit_plot3d((y-1)+.5*z^2==1,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='brown')
3D rendering not yet implemented
t,i,j=var('t,i,j')
r=((cos(t)*i+(t^2+2*t+1)*j))
factor(taylor(r,t,0,3))
-1/2*i*t^2 + j*t^2 + 2*j*t + i + j
t=var('t')
parametric_plot( (cos(t),t^2+2*t+1), (t,-2,2),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot( (1-0.5*t^2,1+2*t+t^2), (t,-3,3),thickness=1 ,color='green',aspect_ratio=true)
t,y,z=var('t,y,z')
parametric_plot( (t,t^2,t^3), (t,-2,2),thickness=1 ,color='red',aspect_ratio=true)+implicit_plot3d(x^2-y==0,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='yellow')+implicit_plot3d(x^3-z==0,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='green')
3D rendering not yet implemented
t,y,z=var('t,y,z')
parametric_plot( (t,t^2,t^3), (t,-2,2),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot( (t,t^2,0), (t,-2,2),thickness=1 ,color='green',aspect_ratio=true)+parametric_plot( (t,0,t^3), (t,-2,2),thickness=1 ,color='blue',aspect_ratio=true)+parametric_plot( (0,t^2,t^3), (t,-2,2),thickness=1,color='black',aspect_ratio=true)
3D rendering not yet implemented
c=var('c')
r=vector((c*t,c*sqrt(2)*ln(t),c/t))
diff(r,t)
(c, sqrt(2)*c/t, -c/t^2)
(simplify(sqrt(c^2+2*c^2/t^2+c^2/t^4)))
sqrt(c^2 + 2*c^2/t^2 + c^2/t^4)
integrate(sqrt(c^2+2*c^2/t^2+c^2/t^4),t,1,10)
99/10*c
parametric_plot( (3*t-3*t^3,3*t^2,3*t+t^3), (t,-1,1),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot( (3*t,0,3*t), (t,-1,1),thickness=1 ,color='green',aspect_ratio=true)+point((0,0,0),size=10)
3D rendering not yet implemented
r=vector((2*t,ln(t),t^2))
d1=diff(r,t)
d2=diff(d1,t)
d3=diff(d2,t)
d1(t=1),d2(t=1),d3(t=1)
((2, 1, 2), (0, -1, 2), (0, 2, 0))
parametric_plot( (2*t,ln(t),t^2), (t,0.1,3),thickness=1 ,color='red',aspect_ratio=true)
3D rendering not yet implemented
t=var('t')
a=vector([sin(t)^3,cos(t)^3,cos(t)^2])
b=diff(a,t)(t=pi/4)
c=diff(diff(a,t),t)(t=pi/4)
b
c
(3/4*sqrt(2), -3/4*sqrt(2), -1) (3/4*sqrt(2), 3/4*sqrt(2), 0)
(b.cross_product(c)).cross_product(b)
(39/16*sqrt(2), 39/16*sqrt(2), 0)
implicit_plot3d(z==x^2-y^2,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='yellow')+implicit_plot3d(z==x+y-1,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='green')
3D rendering not yet implemented
parametric_plot( (sin(2*t)*cos(t),sin(2*t)*sin(t),cos(2*t)), (t,0,2*pi),thickness=1 ,color='green',aspect_ratio=true)+implicit_plot3d(x^2+y^2+z^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40,color='brown')
3D rendering not yet implemented

Odredite duljinu luka krivulje .

r=vector((t,t^2,2/3*t^3))
a=diff(r,t)
integrate(sqrt(4*(t)^4 + 4*(t)^2 + 1),t,0,2)
22/3
r=vector((t,t^2,2/3*t^3))
a=diff(r,t)
a=a.norm()
integrate(a,t,0,2)
22/3
r=vector((e^t*cos(t),e^t,e^t*sin(t)))
a=diff(r,t)
b=diff(a,t)
c=diff(b,t)
show('$\dot{\\vec{r}}$=',a)
show('$\ddot{\\vec{r}}$=',b)
show('$\dddot{\\vec{r}}$=',c)
=
=
=
d=a.cross_product(b)
show('$\dot{\\vec{r}}\\times \ddot{\\vec{r}}$=',simplify(d))
=
f=(a.norm())^3
d=(d.norm())
h=d/f
show('$\kappa(t)=$',((h(t=5)).n()))
r=vector((10*t,10*sqrt(2)*ln(t),10/t))
a=diff(r,t)
a=a.norm()
integrate(a,t,1,5)
48
r=vector((e^(2*t)*cos(t),e^(2*t)*sin(t),e^(2*t)))
a=diff(r,t);a
b=diff(a,t);b
d=a.norm()
c=a.cross_product(b)
f=c.norm()
l=f/d^3
(l(t=1/2*ln(1/9*sqrt(5)))).n()
(2*cos(t)*e^(2*t) - e^(2*t)*sin(t), cos(t)*e^(2*t) + 2*e^(2*t)*sin(t), 2*e^(2*t)) (3*cos(t)*e^(2*t) - 4*e^(2*t)*sin(t), 4*cos(t)*e^(2*t) + 3*e^(2*t)*sin(t), 4*e^(2*t)) 1.00000000000000
t,i,j,k=var('t,i,j,k')
u=((cos(3*t)*i+(t^2+3*t+1)*j-sin(t)^3*k))
factor(taylor(u,t,0,2))
-9/2*i*t^2 + j*t^2 + 3*j*t + i + j
parametric_plot3d( (t^2+4*t+6,2*t^2+2*t+3,5*t^2+2*t+7), (t,-5,5),thickness=1 ,color='red',aspect_ratio=true)+implicit_plot3d(x-3*y+z-4==0,(x,-100,150),(y,-100,150),(z,-100,150),opacity=.40,color='green')
3D rendering not yet implemented
r=vector((cos(t)+sin(t)^2,sin(t)-sin(t)*cos(t),-cos(t)))
show('$\\vec{r}=$',r)
a=diff(r,t)
b=diff(a,t)
c=diff(b,t)
d=a(t=pi/2)
show('$\\dot{\\vec{r}}=$',a)
show('$\\ddot{\\vec{r}}=$',b)
show('$t=\\frac{\\pi}{2}$')
show('$\\dot{\\vec{r}}(\\frac{\\pi}{2})=$',a(t=pi/2))
d=(d/d.norm())
e=a.cross_product(b)(t=pi/2)
f=e/e.norm()
show('$\\vec{t}_0=$',d)
show('$\\vec{b}_0=$',f)

show('$\\vec{n}_0=$',f.cross_product(d))
parametric_plot3d( (-t^3+3*t,3*t^2,-t^3+3*t), (t,-10,10),thickness=1 ,color='red',aspect_ratio=true)+implicit_plot3d(x-z==0,(x,-970,970),(y,-100,300),(z,-970,970),opacity=.40,color='green')
3D rendering not yet implemented
parametric_plot3d( (t,t^2,t^3), (t,-10,10),thickness=1 ,color='red',aspect_ratio=true)+implicit_plot3d(x==0,(x,-970,970),(y,-100,300),(z,-970,970),opacity=.40,color='green')
3D rendering not yet implemented
x,y,z,t=var('x,y,z,t')
implicit_plot3d(x^2+y^2+4*z^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)+parametric_plot3d( (sin(t),0,0.5*cos(t)), (t,0,pi),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot3d( (cos(t),sin(t),0), (t,0,2*pi),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot3d( (-sin(t),0,0.5*cos(t)), (t,0,pi),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot3d( (sin(t),0,0.5*cos(t)), (t,0,pi),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot3d( (0.6*sin(t),0.8*sin(t),0.5*cos(t)), (t,0,pi),thickness=1 ,color='red',aspect_ratio=true)
3D rendering not yet implemented
parametric_plot((sin(t),sin(t)),(t,0,2*pi))
implicit_plot3d(x^2/sin(pi/50)+y^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)+implicit_plot3d(y^2+4*z^2/(cos(pi/50))==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)
3D rendering not yet implemented
implicit_plot3d(4*z^2-x^2/0.36-1==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40,color='red')+implicit_plot3d(4*z^2-y^2/0.64==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)
3D rendering not yet implemented
implicit_plot3d(x^2+y^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)+implicit_plot3d(x+y==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)
3D rendering not yet implemented
%md

u,v=var('u,v')
parametric_plot3d(((2+cos(u/2)*sin(v)-sin(u/2)*sin(2*v))*cos(u),(2+cos(u/2)*sin(v)-sin(u/2)*sin(2*v))*sin(u),sin(u/2)*sin(v)+cos(u/2)*sin(2*v)),(u,0,2*pi),(v,0,2*pi),color='green')
3D rendering not yet implemented
x,y,z,t=var('x,y,z,t')
implicit_plot3d((x-z)^2+(y-z)^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)+implicit_plot3d(z==0,(x,-2,2),(y,-2,2),(z,-2,2),color='green',opacity=.40)+implicit_plot3d(x^2+y^2==1,(x,-2,2),(y,-2,2),(z,-2,2),color='red',opacity=.40)
3D rendering not yet implemented
x,y,z,t=var('x,y,z,t')
implicit_plot3d(x^2-y==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)
3D rendering not yet implemented
t=var('t')
parametric_plot3d((e^t,e^(-t),sqrt(2)*t),(t,0,1))
3D rendering not yet implemented
%md

Kako glase jednadžbe tangencijalnih ravnina plohe  $\quad\vec{r}(u,v)=\left \{ u,u+v,u^2+v^2 \right \} \quad$ koje prolaze pravcem $\quad \frac{x-2}{2}=\frac{y-3}{3}=\frac{z-1}{10}.$

Kako glase jednadžbe tangencijalnih ravnina plohe koje prolaze pravcem

r=vector((u,u+v,u^2+v^2))
ru=diff(r,u)
rv=diff(r,v)
show('$\\vec{r}(u,v)=$',r)
show('$\\vec{r}_{u}=$',ru)
show('$\\vec{r}_{v}=$',rv)
rurv=ru.cross_product(rv)
show('$\\vec{r}_{u}\\times \\vec{r}_{v}=$',rurv)
p=vector((4,3,10))
show('$\\vec{p}=$',p)
np=rurv.dot_product(p)
show('$\\vec{N}\\cdot\\vec{p}=$',np,'$=0$')
show('$v=4u-5$')
ruu=r(v=4*u-5)
show('$\\vec{r}(u)=$',ruu)
Nu=rurv(v=4*u-5)
show('$\\vec{N}(u)=$',Nu)
show('$A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0$')
show('$T(2,3,10)$')
show('$(6u-10)(2-u)+(-8u+10)(3-5u+5)+1-((4u-5)^2+u^2)=0$')
show(solve((6*u-10)*(2-u)+(-8*u+10)*(3-5*u+5)+1-((4*u-5)^2+u^2),u))
show(((6*u-10)*(x-u)+(-8*u+10)*(y-5*u+5)+z-((4*u-5)^2+u^2))(u=2),'$=0$')
show(((6*u-10)*(x-u)+(-8*u+10)*(y-5*u+5)+z-((4*u-5)^2+u^2))(u=18/17),'$=0$')

[, ]
u,v=var('u,v')
parametric_plot3d((u,u+v,u^2+v^2),(u,-10,10),(v,-10,10),color='green',frame=false)+parametric_plot3d((4*u+2,3*u+3,10*u+1),(u,-10,10),color='red',frame=false)+implicit_plot3d(2*x-6*y+z+13==0,(x,-50,50),(y,-50,50),(z,-50,50),opacity=.40)+implicit_plot3d(-62*x+26*y+17*z+29==0,(x,-50,50),(y,-50,50),(z,-50,50),opacity=.40)
3D rendering not yet implemented