︠58d8e4cc-2cfb-4a13-a38a-2e2a28d1223c︠
%md
<B> Čelija varijabli </B>

︡de58c6a7-0796-404b-b442-25b7b0181984︡{"done":true,"md":"<B> Čelija varijabli </B>"}
︠e18ef417-acff-47ed-84b9-65e838504625s︠
t,x,y,z=var('t,x,y,z')
︡6543a75a-389e-4281-a167-e619917a89d5︡{"done":true}︡
︠a78a10ca-bd07-4927-81ab-f1c3fcb7b07cs︠
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)
︡f67a05e1-b5bc-4dcd-8284-9e67dd9251df︡{"stdout":"cos(t)"}︡{"stdout":"\n"}︡{"stdout":"sin(t)"}︡{"stdout":"\n"}︡{"done":true}︡
︠6a91d2f0-7c17-462a-ac28-ff315d932ee5︠
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
︡4137a6d0-ace1-4eb0-8d06-6899eea062e1︡{"stdout":"(e^t, 3*t^2 + 1)\n"}︡{"stdout":"(e^t, 6*t)\n"}︡{"stdout":"(e^t, 6)\n"}︡{"stdout":"(1/6*t^3 + 1/2*t^2 + t + 1, t^3 + t)\n"}︡{"done":true}︡
︠ee1375fe-413f-4c2c-80dc-a6bfda4d316ds︠
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')
︡71030d5b-17b2-4195-8612-a68181a402cb︡{"file":{"filename":"/home/user/.sage/temp/project-8af3d0b7-e293-4fcc-9728-e847cbb646b4/127/tmp_huyPMi.svg","show":true,"text":null,"uuid":"f4dea853-063f-42b2-b0c9-26611869bd8e"},"once":false}︡{"done":true}︡
︠dc940d83-7f10-4b90-b34c-51552b38f062s︠
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')
︡99212e78-420a-4465-b525-27b5fa6ddb50︡{"file":{"filename":"14f70658-d7df-4de0-9c44-2d92fb6dcce0.sage3d","uuid":"14f70658-d7df-4de0-9c44-2d92fb6dcce0"}}︡{"done":true}︡
︠a9637e9f-0198-4243-8e8c-028c584baac0︠
%md
$$\vec{r}(t)=e^t \vec{i}+(t^3+t) \vec{j}$$

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

$$\operatorname{funkcija}$$
︡e0440354-0eed-4267-bf9b-a290cf57ff06︡{"done":true,"md":"$$\\vec{r}(t)=e^t \\vec{i}+(t^3+t) \\vec{j}$$\n\n$$\\vec{r}(t) \\approx (\\frac{t^3}{6}+\\frac{t^2}{2}+t+1) \\vec{\\imath}+(t^3+t) \\vec{\\jmath}$$\n\n$$\\operatorname{funkcija}$$"}
︠f91b2aae-a44f-43c4-98cf-6de55ef41856i︠
%md
$Domaća \space zadaća \quad 1$

︡c189f1b8-8460-47e3-afe4-42d40fc1d961︡{"done":true,"md":"$Domaća \\space zadaća \\quad 1$\n\nZadatak 01"}
︠dab89c2e-2a69-4681-8105-8c497bb36e46s︠
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)))
︡3914ff73-7c71-4c55-a624-502610bfceec︡{"html":"<div align='center'>$\\displaystyle \\left(6,\\,-11,\\,-4\\right)$</div>"}︡{"html":"<div align='center'>$\\displaystyle -10$</div>"}︡{"html":"<div align='center'>$\\displaystyle \\left(-10,\\,0,\\,-10\\right)$</div>"}︡{"html":"<div align='center'>$\\displaystyle 2$</div>"}︡{"html":"<div align='center'>$\\displaystyle 2$</div>"}︡{"html":"<div align='center'>$\\displaystyle -2$</div>"}︡{"html":"<div align='center'>$\\displaystyle \\left(-11,\\,-10,\\,11\\right)$</div>"}︡{"html":"<div align='center'>$\\displaystyle \\left(4,\\,0,\\,1\\right)$</div>"}︡{"done":true}︡
︠b8fec783-9738-4e58-99e7-5a5882ba53edi︠
%md
︠b4425960-3e68-4304-b05b-4312a2e853d7s︠
show((2*a+3*b-5*c).cross_product(a-2*b-4*c))
︡9d9450d5-e4c7-4f0b-9fa4-a3c0c143242b︡{"html":"<div align='center'>$\\displaystyle \\left(-48,\\,70,\\,34\\right)$</div>"}︡{"done":true}︡
︠2d25a7c5-05e4-4a2a-9ddb-f8076f4345d7︠
%md
b*(a.dot_product(c))-a*(b.dot_product(c))
((a.cross_product(b)).cross_product(c))
︠333f71bb-afee-460b-8c9f-bbdc9928e8c2s︠
a.cross_product(b.cross_product(c))
(a.cross_product(b.cross_product(c)))
︡09a21003-b660-4f0e-b4f8-97cfe150c72b︡{"stdout":"(4, 0, 1)\n"}︡{"stdout":"(4, 0, 1)\n"}︡{"done":true}︡
︠e71b4622-ab90-49d4-a2fd-947d50d3b3c7︠
%md
(a.cross_product(b)).dot_product(c)
show('Vektori su linearno zavisni')
︡1cd5abd9-e1d9-41af-b059-05c5164ab959︡{"stdout":"2\n"}︡{"html":"<div align='center'>Vektori su linearno zavisni</div>"}︡{"done":true}︡
︠936d8e76-2a72-450a-99e2-0533cf5192e4i︠
%md
$Domaća \quad zadaća \quad 2.$

︡f752b0b0-cb96-4248-a978-26957c9c965c︡{"done":true,"md":"$Domaća \\quad zadaća \\quad 2.$\n\nZadatak 01"}
︠3e6e9485-bd35-4246-9371-b0419b1b13bcs︠
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))))
︡8825dfe6-5b75-4a82-bbdf-a8b4db474c75︡{"html":"<div align='center'>$\\displaystyle {\\left(t^{4} - 2 \\, t^{3} + 3 \\, t + 2\\right)} t^{2}$</div>"}︡{"done":true}︡
︠65bc4b2a-0bb9-4eb1-994e-e17d319bcc32i︠
%md
︠96a9fdb5-765c-4b3f-85d6-ee2625c3967bs︠
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)

︡39c32971-0da6-40a3-9cc6-9ae18b452373︡{"html":"<div align='center'>$\\displaystyle \\left(t^{3} + 2 \\, t,\\,\\sin\\left(t\\right),\\,e^{t}\\right)$ $\\displaystyle \\left(3 \\, t^{2} + 2,\\,\\cos\\left(t\\right),\\,e^{t}\\right)$ $\\displaystyle \\left(6 \\, t,\\,-\\sin\\left(t\\right),\\,e^{t}\\right)$ $\\displaystyle \\left(6,\\,-\\cos\\left(t\\right),\\,e^{t}\\right)$</div>"}︡{"done":true}︡
︠7565219c-dfa8-4b78-8a03-f1197a6d83b1i︠
%md
︠ea3b353a-bbed-4bbd-bbf1-e66531eb7a0cs︠
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))
︡4e76a526-25f5-4475-9b8d-a6d1412a2a94︡{"html":"<div align='center'>$\\displaystyle \\left(-\\frac{1.00000000000000 \\, \\sqrt{t^{2} + 1} a \\sin\\left(t\\right)}{t},\\,\\frac{1.00000000000000 \\, \\sqrt{t^{2} + 1} a \\cos\\left(t\\right)}{t}\\right)$</div>"}︡{"done":true}︡
︠b9215abd-3b2d-4f6a-aa13-432ea5af4aa3i︠
%md
︠07865fdc-238b-4647-b9ab-0a5eef68b54es︠
t,i,j=var('t,i,j')
u=((cos(t)*i+sin(t)*j))
factor(taylor(u,t,0,10))
︡28eb03c0-42fa-4da7-8044-db2f2b95c8d3︡{"stdout":"-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\n"}︡{"done":true}︡
︠2426d301-ff89-46d4-a916-fc4e4da6fb02s︠
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)
︡92d4cfff-c8ea-43b8-8f1c-c060b7c3cf48︡{"file":{"filename":"/home/user/.sage/temp/project-8af3d0b7-e293-4fcc-9728-e847cbb646b4/211/tmp_iFc_QO.svg","show":true,"text":null,"uuid":"1ced8ae6-9506-491e-9671-5a3c6b2b7061"},"once":false}︡{"done":true}︡
%md
︠6f42dbaa-c685-4ce0-9197-5434239a18ccs︠
t,i,j=var('t,i,j')
u=((i*e^t+(t^3+t)*j))
factor(taylor(u,t,0,10))
︡5838686f-1cb1-47a4-8c82-8f55b2abaf2d︡{"stdout":"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"}︡{"stdout":"\n"}︡{"done":true}︡
︠01c34050-ea3e-4eaf-b736-3119b92b73e8i︠
%md
$Domaća \quad zadaća \quad 3.$
︡cf8673dc-0629-4899-904b-7112d133be0a︡{"done":true,"md":"$Domaća \\quad zadaća \\quad 3.$"}
︠3551d760-5c85-4704-8580-a1839d241687s︠
parametric_plot3d((2*sin(t),2*cos(t),t/2),(t,0,5*pi))
︠fb7b2b21-eb4a-4996-8261-0b33a45a31a4s︠
parametric_plot3d(((2*sin(t))^2,4*sin(t)*cos(t),2*cos(t)),(t,0,2*pi))
︠62962fd6-e36a-48e0-b6c7-74bb213d0327s︠
parametric_plot3d(((cos(t)*e^t,sin(t)*e^t,2*t)),(t,-pi,pi))
︡86b6e057-2cd1-4357-9b12-b7f9880fe9c8︡{"file":{"filename":"8e378eaa-6865-4b85-a985-95fae69f916f.sage3d","uuid":"8e378eaa-6865-4b85-a985-95fae69f916f"}}︡{"done":true}︡
︠591b93a5-52bd-406f-8eb4-883ecf237eb6s︠
t=var('t')
parametric_plot3d((sin(2*t),1-cos(2*t),2*cos(t)),(t,0,2*pi))
︡f4657d7f-8a85-4fe9-8e1a-7e79386d910a︡{"file":{"filename":"568ef966-ba06-41ed-8d2c-76b1159707e2.sage3d","uuid":"568ef966-ba06-41ed-8d2c-76b1159707e2"}}︡{"done":true}︡
︠c9241b8f-6ac4-443f-a056-5bb419c8fba7s︠
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')
︡f94a6cfd-003d-4e32-b285-c2ef2acc224f︡{"file":{"filename":"1f8403c3-e2c0-4154-b67f-3b6d51a4677b.sage3d","uuid":"1f8403c3-e2c0-4154-b67f-3b6d51a4677b"}}︡{"done":true}︡
t,i,j=var('t,i,j')
r=((cos(t)*i+(t^2+2*t+1)*j))
factor(taylor(r,t,0,3))
︡dcfe4994-bcc3-4b17-a00f-0f19a6eba449︡{"stdout":"-1/2*i*t^2 + j*t^2 + 2*j*t + i + j\n"}︡{"done":true}︡
︠7bef5c2e-95bc-48aa-95a4-50549c633e5ds︠
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)
︡7f30fff7-40cb-497e-a84b-b21506e4842e︡{"file":{"filename":"/home/user/.sage/temp/project-8af3d0b7-e293-4fcc-9728-e847cbb646b4/139/tmp_E7rZGK.svg","show":true,"text":null,"uuid":"5d8832e1-27b2-4b09-848c-73b2357b682d"},"once":false}︡{"done":true}︡
︠518bf06b-a408-4783-97b6-8de8240933fds︠
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')
︡890992db-141b-4354-895b-dc829929543e︡{"file":{"filename":"7e5b69d1-3576-439e-823f-1d388c6ab6f8.sage3d","uuid":"7e5b69d1-3576-439e-823f-1d388c6ab6f8"}}︡{"done":true}︡
︠3baf8c56-e034-4d18-b15d-bc6431bf1b7bs︠
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)
︡f10364e5-cf71-4d3e-ab1e-c8b3335d0793︡{"file":{"filename":"e465488d-d3a8-41f2-86a5-53e23049d207.sage3d","uuid":"e465488d-d3a8-41f2-86a5-53e23049d207"}}︡{"done":true}︡
︠66db940f-ebe7-45e7-a346-2b2918e6e5b6s︠
c=var('c')
r=vector((c*t,c*sqrt(2)*ln(t),c/t))
diff(r,t)
︡b4ea0034-9152-469e-95f3-a5dde9745c99︡{"stdout":"(c, sqrt(2)*c/t, -c/t^2)\n"}︡{"done":true}︡
︠213950d2-78dc-485b-93bb-1e16415d83e1s︠
(simplify(sqrt(c^2+2*c^2/t^2+c^2/t^4)))
︡5912f322-e7a4-4e01-837d-40d983a2ac05︡{"stdout":"sqrt(c^2 + 2*c^2/t^2 + c^2/t^4)\n"}︡{"done":true}︡
︠901a9354-20c1-4aa6-a4ae-2140c32f7071s︠
integrate(sqrt(c^2+2*c^2/t^2+c^2/t^4),t,1,10)
︡04e670d3-36c6-4187-9514-786cc3a56d63︡{"stdout":"99/10*c"}︡{"stdout":"\n"}︡{"done":true}︡
︠a56c423e-82e9-4f9a-8fab-dfa12676fa9fs︠
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)
︡087825f0-cac4-4730-b890-4e07a24055e7︡{"file":{"filename":"26aca7fd-550b-4242-a27d-9128a42e18e0.sage3d","uuid":"26aca7fd-550b-4242-a27d-9128a42e18e0"}}︡{"done":true}︡
︠d42f06c8-f245-4718-af18-32d4fbe1fe0ds︠
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)
︡0ed76fa3-d602-4081-a746-6d3e5dc1b675︡{"stdout":"((2, 1, 2), (0, -1, 2), (0, 2, 0))\n"}︡{"done":true}︡
︠88f336bc-ff58-405c-bc80-8f786087ce1es︠
parametric_plot( (2*t,ln(t),t^2), (t,0.1,3),thickness=1 ,color='red',aspect_ratio=true)
︡29613235-cfcc-4e09-9ebb-55cf33e9af07︡{"file":{"filename":"c4f6de31-f258-4082-a358-7ddd7c3e4a2b.sage3d","uuid":"c4f6de31-f258-4082-a358-7ddd7c3e4a2b"}}︡{"done":true}︡
︠bfc867de-8109-452e-a3e7-8453872829e2s︠
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
︠591ff0b0-4347-430e-b4ab-922e671a213es︠
(b.cross_product(c)).cross_product(b)
︠be530093-ec56-4e56-99de-f6a2a4900774s︠
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')
︡75469953-8fe6-4199-b73c-67c2151e9316︡{"file":{"filename":"a3b7f0b5-cbf1-4ecc-a1fe-c8dced604735.sage3d","uuid":"a3b7f0b5-cbf1-4ecc-a1fe-c8dced604735"}}︡{"done":true}︡
︠4a826460-6838-478a-9a6f-959c04a30f31s︠
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')
︠90b8ed60-5c69-408f-b1fc-64fe5c95c0d9i︠
%md
Odredite duljinu luka krivulje $\vec{r}=t \vec{i}+t^2 \vec{j}+t^3 \vec{k}$.
︡0ee56146-60d6-47ce-b194-130369a310e4︡{"done":true,"md":"Odredite duljinu luka krivulje $\\vec{r}=t \\vec{i}+t^2 \\vec{j}+t^3 \\vec{k}$."}
︠5ed416dc-2ab6-4f1c-ae9b-fdd9b59b0d3as︠
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)
︠e48f5599-723e-4281-9599-db38bd1d7e24s︠
r=vector((t,t^2,2/3*t^3))
a=diff(r,t)
a=a.norm()
integrate(a,t,0,2)
︡b29e03ef-7262-44c5-aedc-68a10ea977c1︡{"stdout":"22/3"}︡{"stdout":"\n"}︡{"done":true}︡
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)
︡6d1cd605-7446-46cd-90b4-1aae4e0f8b4e︡{"html":"<div align='center'>$\\dot{\\vec{r}}$= $\\displaystyle \\left(\\cos\\left(t\\right) e^{t} - e^{t} \\sin\\left(t\\right),\\,e^{t},\\,\\cos\\left(t\\right) e^{t} + e^{t} \\sin\\left(t\\right)\\right)$</div>"}︡{"html":"<div align='center'>$\\ddot{\\vec{r}}$= $\\displaystyle \\left(-2 \\, e^{t} \\sin\\left(t\\right),\\,e^{t},\\,2 \\, \\cos\\left(t\\right) e^{t}\\right)$</div>"}︡{"html":"<div align='center'>$\\dddot{\\vec{r}}$= $\\displaystyle \\left(-2 \\, \\cos\\left(t\\right) e^{t} - 2 \\, e^{t} \\sin\\left(t\\right),\\,e^{t},\\,2 \\, \\cos\\left(t\\right) e^{t} - 2 \\, e^{t} \\sin\\left(t\\right)\\right)$</div>"}︡{"done":true}︡
︠2464ee17-6b6a-4526-8177-f87c04ca59eds︠
d=a.cross_product(b)
show('$\dot{\\vec{r}}\\times \ddot{\\vec{r}}$=',simplify(d))
︡1bb3a19d-95c4-4cac-91f0-f475bf451128︡{"html":"<div align='center'>$\\dot{\\vec{r}}\\times \\ddot{\\vec{r}}$= $\\displaystyle \\left(2 \\, \\cos\\left(t\\right) e^{\\left(2 \\, t\\right)} - {\\left(\\cos\\left(t\\right) e^{t} + e^{t} \\sin\\left(t\\right)\\right)} e^{t},\\,-2 \\, {\\left(\\cos\\left(t\\right) e^{t} - e^{t} \\sin\\left(t\\right)\\right)} \\cos\\left(t\\right) e^{t} - 2 \\, {\\left(\\cos\\left(t\\right) e^{t} + e^{t} \\sin\\left(t\\right)\\right)} e^{t} \\sin\\left(t\\right),\\,{\\left(\\cos\\left(t\\right) e^{t} - e^{t} \\sin\\left(t\\right)\\right)} e^{t} + 2 \\, e^{\\left(2 \\, t\\right)} \\sin\\left(t\\right)\\right)$</div>"}︡{"done":true}︡
︠015d8488-11f8-49ef-82a9-f9c1216ca6d0s︠
f=(a.norm())^3
d=(d.norm())
h=d/f
show('$\kappa(t)=$',((h(t=5)).n()))
︡b5278a98-c1db-49de-8c9e-a22c7e89e3c1︡{"html":"<div align='center'>$\\kappa(t)=$ $\\displaystyle 0.00317629867621925$</div>"}︡{"done":true}︡
︠2a4c90b0-6185-4f02-9765-c765e2b3e38cs︠
r=vector((10*t,10*sqrt(2)*ln(t),10/t))
a=diff(r,t)
a=a.norm()
integrate(a,t,1,5)
︡daafe5fd-ba3f-4b0d-8195-8b57c3348603︡{"stdout":"48"}︡{"stdout":"\n"}︡{"done":true}︡
︠4b860f39-f3fd-407c-9376-f2d3892f60ces︠
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()
︡8dd65b57-4cdf-45c7-9bea-c143113f30fc︡{"stdout":"(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))\n"}︡{"stdout":"(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))\n"}︡{"stdout":"1.00000000000000\n"}︡{"done":true}︡
︠be26bbe9-fe48-479c-bc16-6d1842128e46s︠
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))
︡f58fcba9-3a28-4465-b7d3-82cd6e4df347︡{"stdout":"-9/2*i*t^2 + j*t^2 + 3*j*t + i + j\n"}︡{"done":true}︡
︠5474000e-acf5-401f-8da3-32eccf401ba8s︠
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')
︡e6130fed-21da-4b3a-b94f-a1c022d42219︡{"file":{"filename":"cfb8b358-0cd8-4798-ab28-7386b29f90b9.sage3d","uuid":"cfb8b358-0cd8-4798-ab28-7386b29f90b9"}}︡{"done":true}︡
︠f7aba151-2a6d-4356-9234-d3106bc53da5s︠
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))
︡b06d8f09-131d-4bc8-9043-80d4f106ddb7︡{"html":"<div align='center'>$\\vec{r}=$ $\\displaystyle \\left(\\sin\\left(t\\right)^{2} + \\cos\\left(t\\right),\\,-\\cos\\left(t\\right) \\sin\\left(t\\right) + \\sin\\left(t\\right),\\,-\\cos\\left(t\\right)\\right)$</div>"}︡{"html":"<div align='center'>$\\dot{\\vec{r}}=$ $\\displaystyle \\left(2 \\, \\cos\\left(t\\right) \\sin\\left(t\\right) - \\sin\\left(t\\right),\\,-\\cos\\left(t\\right)^{2} + \\sin\\left(t\\right)^{2} + \\cos\\left(t\\right),\\,\\sin\\left(t\\right)\\right)$</div>"}︡{"html":"<div align='center'>$\\ddot{\\vec{r}}=$ $\\displaystyle \\left(2 \\, \\cos\\left(t\\right)^{2} - 2 \\, \\sin\\left(t\\right)^{2} - \\cos\\left(t\\right),\\,4 \\, \\cos\\left(t\\right) \\sin\\left(t\\right) - \\sin\\left(t\\right),\\,\\cos\\left(t\\right)\\right)$</div>"}︡{"html":"<div align='center'>$t=\\frac{\\pi}{2}$</div>"}︡{"html":"<div align='center'>$\\dot{\\vec{r}}(\\frac{\\pi}{2})=$ $\\displaystyle \\left(-1,\\,1,\\,1\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{t}_0=$ $\\displaystyle \\left(-\\frac{1}{3} \\, \\sqrt{3},\\,\\frac{1}{3} \\, \\sqrt{3},\\,\\frac{1}{3} \\, \\sqrt{3}\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{b}_0=$ $\\displaystyle \\left(\\frac{1}{14} \\, \\sqrt{14},\\,-\\frac{1}{7} \\, \\sqrt{14},\\,\\frac{3}{14} \\, \\sqrt{14}\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{n}_0=$ $\\displaystyle \\left(-\\frac{5}{42} \\, \\sqrt{14} \\sqrt{3},\\,-\\frac{2}{21} \\, \\sqrt{14} \\sqrt{3},\\,-\\frac{1}{42} \\, \\sqrt{14} \\sqrt{3}\\right)$</div>"}︡{"done":true}︡
︠466fa63e-e948-4d62-982e-3fffab6309e2s︠
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')
︡730ce790-f5cd-4148-8437-53e3ddb5690d︡{"file":{"filename":"c3e52d1b-358b-4018-b593-622553896e08.sage3d","uuid":"c3e52d1b-358b-4018-b593-622553896e08"}}︡{"done":true}︡
︠cf259375-f30c-4915-9899-8ac89dd5a229s︠
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')
︡4b86b949-1e90-4dd5-9c41-1dcf1276fb62︡{"file":{"filename":"6aa9c5f8-aaaa-4bab-97bc-3d5c3d8a9da0.sage3d","uuid":"6aa9c5f8-aaaa-4bab-97bc-3d5c3d8a9da0"}}︡{"done":true}︡
︠3a9a9caa-4793-4dec-8896-61e93e73cb94s︠
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)
︡1e10ec1f-a35a-4c91-885b-402c26f1edf2︡{"file":{"filename":"a0681cab-028a-4f42-b739-f0bd35c60ca8.sage3d","uuid":"a0681cab-028a-4f42-b739-f0bd35c60ca8"}}︡{"done":true}︡
︠3a4f8fc0-cbcb-466e-a416-38ab90b26b4es︠
parametric_plot((sin(t),sin(t)),(t,0,2*pi))
︡6ed2106c-c5b9-495a-8fde-2b7ddc6bb26a︡{"file":{"filename":"/home/user/.sage/temp/project-8af3d0b7-e293-4fcc-9728-e847cbb646b4/138/tmp_ZbzpQP.svg","show":true,"text":null,"uuid":"e9dae3dc-304d-429b-9b67-394c280a63c0"},"once":false}︡{"done":true}︡
︠4af63fd7-881b-4d28-a312-576a336d096fs︠
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)
︡5e9e0125-4f6b-4cbe-a5f9-6a9b9bb7815e︡{"file":{"filename":"ae6cd2be-9256-459b-a342-346b4e2219e3.sage3d","uuid":"ae6cd2be-9256-459b-a342-346b4e2219e3"}}︡{"done":true}︡
︠0bacd931-000f-4128-a90a-b23c5c76c1ccs︠
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)
︠4d83bdea-d918-4379-9175-487b0dfc4f4as︠
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)
︡b1feb9ee-fd0f-4d98-912e-70b7a453fe23︡{"file":{"filename":"ba62fbdd-937c-4907-a7b7-4bd1763d3563.sage3d","uuid":"ba62fbdd-937c-4907-a7b7-4bd1763d3563"}}︡{"done":true}︡
︠3bb3e4fd-a896-4900-9c9e-2b3781095d2d︠
%md

︡382b4461-af74-4512-a664-b6a99d62bbf7︡
︠8e63d8fe-dd01-46c8-a5cf-ed74594f9da5s︠
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')
︡41daf307-0ef3-4118-a11e-762f8b9208e4︡{"file":{"filename":"bf835bbd-b60f-4518-9b97-8109123ae510.sage3d","uuid":"bf835bbd-b60f-4518-9b97-8109123ae510"}}︡{"done":true}︡
︠4ec4d5a7-5790-4b7e-9383-e38fb324481fs︠
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)
︡993679c4-8f7b-4ec9-886a-18735da6f0fa︡{"file":{"filename":"c3a5cee3-9a84-44d6-bc3a-c7092f9e392d.sage3d","uuid":"c3a5cee3-9a84-44d6-bc3a-c7092f9e392d"}}︡{"done":true}︡
︠23eef7a1-76b3-4f83-b772-ddbaf72c40a0s︠
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)
︠bd01dfa2-4284-4a34-8c1c-0fc012ca9937s︠
t=var('t')
parametric_plot3d((e^t,e^(-t),sqrt(2)*t),(t,0,1))
︡bdbc6265-fae4-464b-8cf4-69c510e5967a︡{"file":{"filename":"f5734fd6-ae69-42e5-88c6-eda5f4a25f51.sage3d","uuid":"f5734fd6-ae69-42e5-88c6-eda5f4a25f51"}}︡{"done":true}︡
︠50375dbf-f239-4272-b41a-8e962b5bde70︠
%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 $\quad \frac{x-2}{2}=\frac{y-3}{3}=\frac{z-1}{10}.$
︡947146e4-a4fe-4e15-90d1-0f07a7873125︡{"done":true,"md":"\nZadatak 2.\n\nKako glase jednadžbe tangencijalnih ravnina plohe  $\\quad\\vec{r}(u,v)=\\left \\{ u,u+v,u^2+v^2 \\right \\} \\quad$ koje prolaze pravcem $\\quad \\frac{x-2}{2}=\\frac{y-3}{3}=\\frac{z-1}{10}.$"}
︠7309b5c0-4792-48eb-9f4a-bb26180cf1f3s︠
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),'$=0$')

︡f0e553c7-0953-4eb3-b3f9-dc3e2422a187︡{"html":"<div align='center'>$\\vec{r}(u,v)=$ $\\displaystyle \\left(u,\\,u + v,\\,u^{2} + v^{2}\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{r}_{u}=$ $\\displaystyle \\left(1,\\,1,\\,2 \\, u\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{r}_{v}=$ $\\displaystyle \\left(0,\\,1,\\,2 \\, v\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{r}_{u}\\times \\vec{r}_{v}=$ $\\displaystyle \\left(-2 \\, u + 2 \\, v,\\,-2 \\, v,\\,1\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{p}=$ $\\displaystyle \\left(4,\\,3,\\,10\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{N}\\cdot\\vec{p}=$ $\\displaystyle -8 \\, u + 2 \\, v + 10$ $=0$</div>"}︡{"html":"<div align='center'>$v=4u-5$</div>"}︡{"html":"<div align='center'>$\\vec{r}(u)=$ $\\displaystyle \\left(u,\\,5 \\, u - 5,\\,{\\left(4 \\, u - 5\\right)}^{2} + u^{2}\\right)$</div>"}︡{"html":"<div align='center'>$\\vec{N}(u)=$ $\\displaystyle \\left(6 \\, u - 10,\\,-8 \\, u + 10,\\,1\\right)$</div>"}︡{"html":"<div align='center'>$A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0$</div>"}︡{"html":"<div align='center'>$T(2,3,10)$</div>"}︡{"html":"<div align='center'>$(6u-10)(2-u)+(-8u+10)(3-5u+5)+1-((4u-5)^2+u^2)=0$</div>"}︡{"html":"<div align='center'>[$\\displaystyle u = \\left(\\frac{18}{17}\\right)$, $\\displaystyle u = 2$]</div>"}︡{"html":"<div align='center'>$\\displaystyle 2 \\, x - 6 \\, y + z + 13$ $=0$</div>"}︡{"html":"<div align='center'>$\\displaystyle -\\frac{62}{17} \\, x + \\frac{26}{17} \\, y + z + \\frac{29}{17}$ $=0$</div>"}︡{"done":true}︡
︠75dce355-e744-4277-8e03-38e27f34ea71s︠
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)
︡d337ed92-517d-4a78-8f85-c7732904fb74︡{"file":{"filename":"5fc84aff-e061-421e-9cc8-2d01294a83fe.sage3d","uuid":"5fc84aff-e061-421e-9cc8-2d01294a83fe"}}︡{"done":true}︡
︠fc741df2-e77c-48c6-9050-2cbe7ef4f002︠