Formule za pismeni i usmeni dio ispita iz Matematike 4

Duljina luka krivulje

α…… (Overscript[r, →]) = (Overscript[r, →]) (t)
	s = (∫_((Subscript[t, 1]))^((Subscript[t, 2])))(Sqrt[((((dx/dt)))^2)+((((dy/dt)))^2)+((((dz/dt)))^2)])dt

Fleksija i torzija

α... (Overscript[r, ⇀])=(Overscript[r, ⇀])(s) = x(s)(Overscript[i, ⇀])+y(s)(Overscript[j, ⇀])+z(s)(Overscript[k, ⇀])
	Fleksija:	κ(s) = |((d  (Overscript[t, ⇀]))/ds)| = |((Overscript[r, ⇀])^″)(s)| = (Sqrt[((((x^″)))^2)+((((y^″)))^2)+((((z^″)))^2)])
	Torzija:	τ(s) = - (((Overscript[n, ⇀]))^o) · (((Overscript[b, ⇀]))^o)'= \(\(\(\ \)\((r\&⇀\^′, 
                r\&⇀\^″, \(r\&⇀\^′\)\^″)\)\)\/\(\(\(\(|\)\(r\)\)\&⇀\^″\)\( | \^2\)\)\) = ((|({{(x^′), (y^′), (z^′)}, {(x^″), (y^″), \(z\^″\)}, {(((x^″))^′), \(\(y\^″\)\^′\), (((z^″))^′)}})|)/(((((x^″)))^2)+((((y^″)))^2)+((((z^″)))^2)))	
α...  (Overscript[r, ⇀])=(Overscript[r, ⇀])(t) = x(t)(Overscript[i, ⇀]) + y(t)(Overscript[j, ⇀]) + z(t)(Overscript[k, ⇀])
	Fleksija:	κ(t) = \(\(\(|\)\(\(\(r\&⇀\)\& . ×\(r\&⇀\)\&¨\)\(|\)\)\)\/\(\(\(\(\(|\)\(r\)\)\&⇀\)\& . \)\( | \^3\)\)\)=(Sqrt[((((((\(x\& . \)))^2)+(((\(y\& . \)))^2)+(((\(z\& . \)))^2))(((Overscript[x, ¨])^2)+((Overscript[y, ¨])^2)+((Overscript[z, ¨])^2))-(((\(x\& . \)(Overscript[x, ¨])+\(y\& . \)(Overscript[y, ¨])+\(z\& . \)(Overscript[z, ¨])))^2))/((((((\(x\& . \)))^2)+(((\(y\& . \)))^2)+(((\(z\& . \)))^2)))^3))])=\(TraditionalForm\`\@\(\(\(\(r\&⇀\)\& . \^2·\(r\&⇀\)\&¨\^2 - \((\(\(r\&⇀\)\& . \) \(r\&⇀\)\&¨)\)\^2\)\/\((\(r\&⇀\)\& . \^2)\)\^3\)\(
\)
        \)\)		
	Torzija:		τ(t)=\(TraditionalForm\`\((\(r\&⇀\)\& . , \(r\&⇀\)\&¨, \(\(r\&⇀\)\& ... \))\)\/\(\(\(\(\(|\)\(r\)\)\&⇀\)\& . ×\(r\&⇀\)\&¨\)\( | \^2\)\)\)=((|({{\(x\& . \), \(y\& . \), \(z\& . \)}, {(Overscript[x, ¨]), (Overscript[y, ¨]), (Overscript[z, ¨])}, {\(x\& ... \), \(y\& ... \), \(z\& ... \)}})|)/((((\(x\& . \))^2)+((\(y\& . \))^2)+((\(z\& . \))^2))(((Overscript[x, ¨])^2)+((Overscript[y, ¨])^2)+((Overscript[z, ¨])^2))-(((\(x\& . \)(Overscript[x, ¨])+\(y\& . \)(Overscript[y, ¨])+\(z\& . \)(Overscript[z, ¨])))^2)))=\(\((\(r\&⇀\)\& . , \(r\&⇀\)\&¨, \(\(r\&⇀\)\& ... \))\)\/\(\(r\&⇀\)\& . \^2·\(r\&⇀\)\&¨\^2 - \((\(\(r\&⇀\)\& . \) \(r\&⇀\)\&¨)\)\^2\)\)

Tangencijalna   ravnina  plohe

({{\(Jednadžba\ plohe\), Tangencijalna ravnina}, {F(x, y, z) = c, (Subscript[(((∂F)/(∂x))), o])(x-(Subscript[x, o]))+(Subscript[(((∂F)/(∂y))), o])(y-(Subscript[y, o]))+(Subscript[(((∂F)/(∂z))), o])(z-(Subscript[z, o]))=0}, {z = f(x, y), z-(Subscript[z, o])=(Subscript[(((∂z)/(∂x))), o])(x-(Subscript[x, o]))+(Subscript[(((∂z)/(∂y))), o])(y-(Subscript[y, o]))}, {x = x(u, v)
y = y(u, v)
z = z(u, v)
,                      det(({{x-(Subscript[x, o]), y-(Subscript[y, o]), z-(Subscript[z, o])}, {(Subscript[(((∂x)/(∂u))), o]), (Subscript[(((∂y)/(∂u))), o]), (Subscript[(((∂z)/(∂u))), o])}, {(Subscript[(((∂x)/(∂v))), o]), (Subscript[(((∂y)/(∂v))), o]), (Subscript[(((∂z)/(∂v))), o])}}))=0,  ili
(Subscript[(((∂(y, z))/(∂(u, v)))), o])(x-(Subscript[x, o]))+(Subscript[(((∂(z, x))/(∂(u, v)))), o])(y-(Subscript[y, o]))+(Subscript[(((∂(x, y))/(∂(u, v)))), o])(z-(Subscript[z, o]))=0}, {(Overscript[r, ⟶])(u, v)=x(u, v)(Overscript[i, ⟶])+
+y(u, v)(Overscript[j, ⟶])+z(u, v)(Overscript[k, ⟶]), ((Overscript[ρ, ⇀])-(Subscript[(OverVector[r]), o]))·((Subscript[(((∂(OverVector[r]))/(∂u))), o])× (Subscript[(((∂(OverVector[r]))/(∂v))), o]))=0
ili
((Overscript[ρ, ⇀])-(Subscript[(OverVector[r]), o]))·(OverVector[(Subscript[N, o])])=0}})

Duljina luka krivulje na plohi

s = (∫_((Subscript[t, 1]))^((Subscript[t, 2])))(Sqrt[ E ((((du/dt)))^2) + 2 F (du/dt) (dv/dt) + G ((((dv/dt)))^2)])dt
	
Kut između  dviju krivulja na plohi 
	α……(Subscript[(Overscript[r, ⟶]), 1])(t) =(Subscript[(Overscript[r, ⟶]), 1])(u(t), v(t)),		u = u(t),   v = v(t),  i
	β……(Subscript[(Overscript[r, ⟶]), 2])(t) =(Subscript[(Overscript[r, ⟶]), 2])(\(TraditionalForm\`u\&_\)(t), \(TraditionalForm\`\(\(v\)\(\ \)\)\&_\)(t)),	u = \(TraditionalForm\`u\&_\)(t),   v = \(TraditionalForm\`\(\(v\)\(\ \)\)\&_\)(t),  na plohi  (Overscript[r, ⇀]) = (Overscript[r, ⇀]) (u,v): 
	cosω = ((\(E\ du\ d  u\&_\) + F( \(du\ d  v\&_\ \  + d  u\&_\ dv\))+\(G\ dv\ d  v\&_\))/((Sqrt[E (du^2)+2F du dv  +  G (dv^2)])(Sqrt[E d((\(u\&_\))^2)+2F d\(u\&_\) d\(v\&_\)  + G d((\(v\&_\))^2)])))
	
Površina područja (K) na plohi
	S = (Underscript[∫∫, (K)])dS = (Underscript[∫∫, D]) |(Subscript[(Overscript[r, ⇀]), u])×(Subscript[(Overscript[r, ⇀]), v])| du dv = (Underscript[∫∫, D]) (Sqrt[EG - (F^2)])du dv

Jednadžba za dobivanje glavnih zakrivljenosti

((Subscript[K, n])^2) - ((EN-2MF+GL)/(EG-(F^2)))(Subscript[K, n]) + ((LN - (M^2))/(EG-(F^2)))= 0

Glavni smjerovi

|({{(μ^2), -μ, 1}, {G, F, E}, {N, M, L}})|=0

Glavne krivulje zakrivljenosti

|({{(du^2), -dudv, (dv^2)}, {G, F, E}, {N, M, L}})|=0     ili   (FL - EM)(du^2)+ (GL - EN)dudv + (GM - FN)(dv^2) = 0

Weingartove  i Gaussove derivacione  formule

Weingartove: 	({{(Overscript[(Subscript[N, u]^o), ⟶]) =((FM  -  GL)/((W^2)))(Overscript[((Subscript[r, u])), ⟶])+((FL  -  EM)/((W^2)))(Overscript[((Subscript[r, v])), ⟶])}, {(Overscript[(Subscript[N, v]^o), ⟶]) =((FN  -  GM)/((W^2)))(Overscript[((Subscript[r, u])), ⟶])+((FM  -  EN)/((W^2)))(Overscript[((Subscript[r, v])), ⟶])}})		
        
Gaussove: 	(Overscript[(Subscript[r, ij]), ⟶])=((((Overscript[((Subscript[r, u])), ⟶]) (Overscript[(Subscript[r, ij]), ⟶]))G - ((Overscript[((Subscript[r, v])), ⟶]) (Overscript[(Subscript[r, ij]), ⟶]))F)/((W^2)))(Overscript[((Subscript[r, u])), ⟶]) + ((((Overscript[((Subscript[r, v])), ⟶]) (Overscript[(Subscript[r, ij]), ⟶]))E - ((Overscript[((Subscript[r, u])), ⟶]) (Overscript[(Subscript[r, ij]), ⟶]))F)/((W^2)))(Overscript[((Subscript[r, v])), ⟶]) + ((((Overscript[((Subscript[r, u])), ⟶]) (Overscript[((Subscript[r, v])), ⟶]) (Overscript[(Subscript[r, ij]), ⟶])))/W)(Overscript[( (N^o)), ⟶]),
                                                                                              i  =  u, v ;   j  =  u, v.

Christoffelovi  simboli  prve i druge  vrste

[({{({{i, j}})}, {k}})] = (Subscript[Γ, ij, k]) = (Overscript[(Subscript[r, ij]), ⟶])·(Overscript[((Subscript[r, k])), ⟶])	
	[({{({{i, j}})}, {k}})] = (1/2)[((∂( (Overscript[(Subscript[r, j]), ⟶])·(Overscript[((Subscript[r, k])), ⟶]) ))/(∂i))+((∂( (Overscript[(Subscript[r, i]), ⟶])·(Overscript[((Subscript[r, k])), ⟶]) ))/(∂i))-((∂( (Overscript[(Subscript[r, i]), ⟶])·(Overscript[((Subscript[r, j])), ⟶]) ))/(∂i))],    i,j,k  =  u,v
	
	{({{({{i, j}})}, {k}})} = (Subscript[Γ, ij]^k),		 i,j,k = u,v .	
	{({{({{i, j}})}, {u}})} = (Subscript[Γ, ij]^u)= (( [({{({{i, j}})}, {u}})]G -  [({{({{i, j}})}, {v}})]F)/(EG-(F^( 2))))= ((((Overscript[((Subscript[r, v])), ⟶])\(\(\(\ \)\(N\^o\)\)\&⟶\) (Overscript[(Subscript[r, ij]), ⟶])))/W) ,
                                                                                       i  =  u,v ;   j  =  u,v 
{({{({{i, j}})}, {v}})} = (Subscript[Γ, ij]^v)= (( [({{({{i, j}})}, {v}})]E -  [({{({{i, j}})}, {u}})]F)/(EG-(F^( 2))))= ((((Overscript[((Subscript[r, u])), ⟶]) (Overscript[(Subscript[r, ij]), ⟶])\(\(\(\ \)\(N\^o\)\)\&⟶\)))/W),

Christoffelovi simboli prve vrste izraženi  pomoću Gaussovih veličina  prvog reda i njihovih derivacija

(Overscript[(Subscript[r, uu]), ⟶])·(Overscript[((Subscript[r, u])), ⟶]) =  [({{({{u, u}})}, {u}})] = (Subscript[Γ, uu, u]) =  (1/2)[((∂( (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, u])), ⟶]) ))/(∂u))+((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, u])), ⟶]) ))/(∂u))-((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, u])), ⟶]) ))/(∂u))] =(1/2) (Subscript[E, u]),
	(Overscript[(Subscript[r, uu]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) =  [({{({{u, u}})}, {v}})] = (Subscript[Γ, uu, v]) =  (1/2)[((∂( (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂u))+((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂u))-((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, u])), ⟶]) ))/(∂v))] =(Subscript[F, u]) - (1/2) (Subscript[E, v]),
	(Overscript[(Subscript[r, uv]), ⟶])·(Overscript[((Subscript[r, u])), ⟶]) =  [({{({{u, v}})}, {u}})] = (Subscript[Γ, uv, u]) =  (1/2)[((∂( (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂u))+((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, u])), ⟶]) ))/(∂v))-((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂u))] =(1/2) (Subscript[E, v]),
	(Overscript[(Subscript[r, uv]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) =  [({{({{u, v}})}, {v}})] = (Subscript[Γ, uv, v]) =  (1/2)[((∂( (Overscript[(Subscript[r, v]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂u))+((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂v))-((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂v))] =(1/2) (Subscript[G, u]),
	(Overscript[(Subscript[r, vv]), ⟶])·(Overscript[((Subscript[r, u])), ⟶]) =  [({{({{v, v}})}, {u}})] = (Subscript[Γ, vv, u]) =  (1/2)[((∂( (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂v))+((∂(  (Overscript[(Subscript[r, u]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂v))-((∂(  (Overscript[(Subscript[r, v]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂u))] =(Subscript[F, v])(1/2) (Subscript[G, u]),
	(Overscript[(Subscript[r, vv]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) =  [({{({{v, v}})}, {v}})] = (Subscript[Γ, vv, v]) =  (1/2)[((∂( (Overscript[(Subscript[r, v]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂v))+((∂( (Overscript[(Subscript[r, v]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂v))-((∂( (Overscript[(Subscript[r, v]), ⟶])·(Overscript[((Subscript[r, v])), ⟶]) ))/(∂v))] =(1/2) (Subscript[G, v]).

Christoffelovi simboli druge vrste izraženi  pomoću Gaussovih veličina  prvog reda i njihovih derivacija

{({{({{u, u}})}, {u}})} = (Subscript[Γ, uu]^u)= ((G (Subscript[E, u])-  2F (Subscript[F, u])+ F (Subscript[E, v]))/(2(EG-(F^( 2))))),
 	{({{({{u, u}})}, {v}})} = (Subscript[Γ, uu]^v)= ((-F (Subscript[E, u])+  2E (Subscript[F, u])- E (Subscript[E, v]))/(2(EG-(F^( 2))))),
 	{({{({{u, v}})}, {u}})} = (Subscript[Γ, uv]^u)= ((G (Subscript[E, v])-  F (Subscript[G, u]) )/(2(EG-(F^( 2))))),
 	{({{({{u, v}})}, {v}})} = (Subscript[Γ, uv]^v)= ((E (Subscript[G, u])-  F (Subscript[E, v]))/(2(EG-(F^( 2))))),
 	{({{({{v, v}})}, {u}})} = (Subscript[Γ, vv]^u)= ((- F (Subscript[G, v])+  2G (Subscript[F, v])- G (Subscript[G, u]))/(2(EG-(F^( 2))))),
 	{({{({{v, v}})}, {v}})} = (Subscript[Γ, vv]^v)= ((E (Subscript[G, v])-  2F (Subscript[F, v])+ F (Subscript[G, u]))/(2(EG-(F^( 2))))).

 		(Subscript[(Overscript[r, ⟶]), uu])={({{({{u, u}})}, {u}})}(Overscript[((Subscript[r, u])), ⟶]) + {({{({{u, u}})}, {v}})}(Overscript[((Subscript[r, v])), ⟶]) + L(Overscript[( (N^o)), ⟶]),  	        
	(Subscript[(Overscript[r, ⟶]), uv])={({{({{u, v}})}, {u}})}(Overscript[((Subscript[r, u])), ⟶]) + {({{({{u, v}})}, {v}})}(Overscript[((Subscript[r, v])), ⟶]) + M(Overscript[( (N^o)), ⟶]),                        
	(Subscript[(Overscript[r, ⟶]), vv])={({{({{v, v}})}, {u}})}(Overscript[((Subscript[r, u])), ⟶]) + {({{({{v, v}})}, {v}})}(Overscript[((Subscript[r, v])), ⟶]) + N(Overscript[( (N^o)), ⟶]),  	         
	({{(Overscript[(Subscript[N, u]^o), ⟶]) =((FM  -  GL)/((W^2)))(Overscript[((Subscript[r, u])), ⟶])+((FL  -  EM)/((W^2)))(Overscript[((Subscript[r, v])), ⟶]),}, { (Overscript[(Subscript[N, v]^o), ⟶]) =((FN  -  GM)/((W^2)))(Overscript[((Subscript[r, u])), ⟶])+((FM  -  EN)/((W^2)))(Overscript[((Subscript[r, v])), ⟶]).}})                                           		
 	 Gauss-Weingartenove derivacione formule.

"Theorema  egregium"

K =(1/(W^4))(|({{E, F, (Subscript[F, v]) - (1/2) (Subscript[G, u])}, {F, G,  (1/2) (Subscript[G, v])}, { (1/2) 
                      (Subscript[E, u]),  (Subscript[F, u]) - (1/2) (Subscript[E, v]),   (Subscript[F, uv])-(1/2)(Subscript[G, uu])-(1/2)(Subscript[E, vv])}})| - |({{E, F,  (1/2)(Subscript[E, v])}, {F, G,  (1/2)(Subscript[G, u])}, { (1/2)(Subscript[E, v]),  (1/2)(Subscript[G, u]), 0}})| )
ili
K = -(1/(4  (W^4))) |({{E, (Subscript[E, u]), (Subscript[E, v])}, {F, (Subscript[F, u]), (Subscript[F, v])}, { G, (Subscript[G, u]), (Subscript[G, v])}})| \(+\(1\/\(2  
                    W\)\)[\(∂\/∂u\) \((\(F\_v - G\_u\)\/W)\) + \(∂\/∂v\) \((\(F\_u - E\_v\)\/W)\)]\)

Diferencijalna   jednadžba   geodetskih  linija

|({{(u^′),    (u^″)+{({{({{u, u}})}, {u}})}(((u^′))^2)+2{({{({{u, v}})}, {u}})}(u^′)(v^′)+{({{({{v, v}})}, {u}})}(((v^′))^2)}, {(v^(′ )),   (v^″)+{({{({{u, u}})}, {v}})}(((u^′))^2)+2{({{({{u, v}})}, {v}})}(u^′)(v^′)+{({{({{v, v}})}, {v}})}(((v^′))^2)}})|=0,	α…(Overscript[r, ⇀])=(Overscript[r, ⇀])(u(s), v(s))	
												
|({{\(u\& . \),    (Overscript[u, ¨])+{({{({{u, u}})}, {u}})}\(u\& . \^2\)+2{({{({{u, v}})}, {u}})}\(u\& . \)\(v\& . \)+{({{({{v, v}})}, {u}})}\(v\& . \^2\)}, {\(v\& . \^\ \),   (Overscript[v, ¨])+{({{({{u, u}})}, {v}})}\(u\& . \^2\)+2{({{({{u, v}})}, {v}})}\(u\& . \)\(v\& . \)+{({{({{v, v}})}, {v}})}\(v\& . \^2\)}})|=0,		α…(Overscript[r, ⇀])=(Overscript[r, ⇀])(u(t), v(t))

Torzija krivulje  na plohi

(1^o))	α......(Overscript[r, ⟶])(s) = (Overscript[r, ⟶])(u(s), v(s))
	torzija...τ(s) = - (((Overscript[n, ⇀]))^o) · (((Overscript[b, ⇀]))^o)'= \(\(\(\ \)\((r\&⇀\^′, 
                r\&⇀\^″, \(r\&⇀\^′\)\^″)\)\)\/\(\(\(\(|\)\(r\)\)\&⇀\^″\)\( | \^2\)\)\)
	α......(Overscript[r, ⟶])(t) = (Overscript[r, ⟶])(u(t), v(t))
	torzija...τ(t) = τ(t) = \(\((\(r\&⇀\)\& . , \(r\&⇀\)\&¨, \(\(r\&⇀\)\& ... \))\)\/\(\(\(\(\(|\)\(r\)\)\&⇀\)\& . ×\(r\&⇀\)\&¨\)\( | \^2\)\)\)

(2^o))	τ=((Overscript[(t^( o)),, ⟶])(Overscript[(N^( o)), ⟶]),(Overscript[(N^( o)), ⟶])')+((dθ)/(d s))

Geodetska torzija T

T=((Overscript[(t^( o)),, ⟶])(Overscript[(N^( o)), ⟶]),(Overscript[(N^( o)), ⟶])')
	ili
	T =(1/(Sqrt[E G - (F^2)]))|({{(Subscript[E, (u^′)]) + 
                  (Subscript[F, (v^′)]), (Subscript[E, (u^′)]) + 
                  (Subscript[F, (v^′)])}, {(Subscript[E, (u^′)]) + 
                  (Subscript[F, (v^′)]), (Subscript[E, (u^′)]) + 
                  (Subscript[F, (v^′)])}})|

sadržaj