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atemtica II Unidad IV: Curvas en el

Espacio

6

Unidad IV: Curvas en el

Espacio

Ejemplo: Sea

[ ]0,2t),esen2t,ecos2t,(e(t)f t7-t7-t7- =

una funcin vectorial de

variable real. Hallar:

a) La grafica de la curva C en

[ ]2,0t

b) Los vectores unitarios

B,N,Ten t 0= 0

c) Los planos:P0,PNyPren t0=0

d) Curvatura y Torsin en t0=0

Solucin:

a) En forma paramtrica:

Si

[ ]

>=+

=

=

=

0z

zyx

ez

0,2tsen2t,ey

cos2tex

:C222

7t

7t

7t

Cono Circular

a!ulacin:

t 0

4

2

4

3

4

5

2

3

4

7 2

tex t 2cos7= 1 0

2

7

e

0 7e 0 221

e

0 14e

tseney t 27= 0

47

e

04

21

e

04

35

e

04

49

e

0

tez7= 1

47

e 27

e 421

e

7e

435

e 221

e 449

e

14e

"rfica de la Curva C:

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X

Y

Z


Espacio

6

Clculo de los valores

)(f t

)('f t

)(''f t

*

)('''f t

en t+,

Valor de la funcin

fen t

o

+ ,

:

)esen2t,ecos2t,(e(t)fS 7t-7t-7t-=

. Entonces:

===

========

=

==

1eez(0)

0(1)(0)sen(0)esen2(0)ey(0)

1(1)(1)cos(0)ecos2(0)ex(0)

ez(t)

sen2tey(t)

cos2tex(t)

07(0)

07(0)

07(0)

7t

7t

7t

Entonces:

(1,0,1))]0(),0(),0([(0)f ==

zyx

or tanto:

(1,0,1)(0)f =

$rimera derivada de

fen t+o:

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Espacio

6

)esen2t,ecos2t,(e(t)fS7t-7t-7t-=

. Entonces:

==

+=+(==

=+(==

=

=

=

7t-7t

7t-7t-7t7t7t

-7t-7t7t7t7t

7t

7t

7t

-7e)'e((t)z'

cos2t2esen2t-7e)(sen2t)'(e2t)()'esen2t]'e[(t)y'

sen2t2e-cos2t-7e)(cos2t)'(ecos2t)()'ecos2t]'e[(t)x'

ez(t)

sen2tey(t)

cos2tex(t)

sen

Entonces:

==

=+=

==

=

+=

=

7-7e(0)z'

2cos2(0)2esen2(0)-7e(0)y'

-7sen2(0)2e-cos2(0)-7e(0)x'

-7e(t)z'

cos2t2esen2t-7e(t)y'

sen2t2e-cos2t-7e(t)x'

7(0)-

7(0)-7(0)-

-7(0)-7(0)

7t-

7t-7t-

-7t-7t

Entonces:

)e,-7cos2t2esen2t7e-,sen2t2e-cos2t(-7e)]('),(')('[)t('f7t-7t-7t-7t-7t- +==

tztytx

Entonces:

)e,-7cos2t2esen2t7e-,sen2t2e-cos2t(-7e)t('f 7t-7t-7t-7t-7t- +=

Entonces

)7-,2,7-()]0('),0(',)0('[)0('f ==

zyx

$or tanto:

)7-,2,7-()0('f =

Se%unda derivada

fen t+o:

Si)e,-7cos2t2esen2t7e-,sen2t2e-cos2t(-7e)t('f 7t-7t-7t-7t-7t- +=

.Entonces:

=

+=

=

7t-

7t-7t-

-7t-7t

-7e(t)z'

cos2t2esen2t-7e(t)y'

sen2t2e-cos2t-7e(t)x'

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Espacio

6

===

=+==

++===

7t-7t-

7t-7t-7t-7t-7t-7t-

-7t-7t-7t-7t-7t-7t

e49)'(-7e]'(t)'[(t)'z'

sen2t4e-cos2te14-cos2t14e-sen2te49cos2t]'2esen2t[-7e]'(t)'[(t)'y'

cos2t4e-sen2t14esen2t14ecos2te49sen2t]'2e-cos2t[-7e]'(t)x'[(t)'x'

z

y

Entonces:

==

==

=+=

=

=

+=

49e49(0)'z'

-2!cos2(0)2!e-sen2(0)e45(0)'y'

45sen2(0)e2!cos2(0)e45(0)'x'

e49(t)'z'

cos2t2!e-sen2te45(t)'y'

sen2te2!cos2te45(t)'x'

7(0)-

7t-7(0)-

-7(0)-7(0)

7t-

7t-7t-

-7t-7t

Entonces:

)e49,cos2te2!-sen2te45,sen2t2!ecos2te45(

(t)]'z'(t),'y'),(''[)t(''f

7t-7t-7t-7t-7t- +=

==

tx

Entonces:

)e49,cos2te2!-sen2te45,sen2t2!ecos2te45()t(''f7t-7t-7t-7t-7t- +=

Luego:

)492!,-45,((0)]'z'(0),'y'),0(''[)0(''f ==

x

Por tanto:

)492!,-45,()0(''f =

ercera /erivada

fen t+o:

Si

)e49,cos2te2!-sen2te45,sen2t2!ecos2te45()t(''f7t-7t-7t-7t-7t- +=

.

Entonces:

=

=

+=

7t-

7t-7t-

-7t-7t

e49(t)'z'

cos2t2!e-sen2te45(t)'y'

sen2te2!cos2te45(t)'x'

==

==

+==

]'[49e]'(t)''[(t)''z'

cos2t]'e2!-sen2te[45]'(t)''[(t)''y'

sen2t]'2!ecos2te45[]'(t)'x'[(t)''x'

7t-

7t-7t-

-7t-7t

z

y

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Espacio

6

Entonces:

=

+++=

+=

7t-

7t-7t-7t-7t-

-7t-7t-7t-7t

e-343(t)''z'

t2sene5"cos2te19"t2cose90t2sene-315(t)''y'

t2cose5"sen2te19"-t2sene90-t2cose-315((t)''x'

Entonces:

==

=+=

==

=

+=

=

343e-343(0)''z'

2"!(0)2cose2!"2(0)sene-259(0)''y'

-2592(0)sene2!"-(0)2cose259-(0)''x'

e-343(t)''z'

t2cose2!"t2sene-259(t)''y'

t2sene2!"-t2cose259-(t)''x'

7(0)-

7(0)-7(0)-

-7(0)-7(0)

7t-

7t-7t-

-7t-7t

Entonces

)e343-,t2cose2!"t2sene259-,sen2te2!"-t2cose259-(

(t)]''z'(t),''y'),('''[)t('''f

7t-7t-7t-7t-7t- +=

=

tx

Entonces:

)e343-,t2cose2!"t2sene259-,t2sene2!"-t2cose259-()t('''f 7t-7t-7t-7t-7t- +=

Entonces:

)343-,2!",259-((0)]''z'(0),''y'),0('''[)0('''f ==

x

or tanto:

)343-,2!",259-()0('''f =

b) Clculo de los Vectores unitarios

!t)T

#

(t)$

y

(t)%

:

Vector an%ente Unitario

!t)T

en t = t0:

)!t"f

)!t"f)!tT

0

00

=



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Espacio

6

)e,-7cos2t2esen2t7e-,sen2t2e-cos2t(-7e

)e,-7cos2t2esen2t7e-,sen2t2e-cos2t(-7e

(t)'f

(t)'f(t)&

7t-7t-7t-7t-7t-

7t-7t-7t-7t-7t-

+

+==

),-7cos2t2sen2t7-,sen2t2-(-7cos2te

),-72cos2tsen2t7-,sen2t2-(-7cos2te

7t-

-7t

+

+=

),-7cos2t2sen2t7-,sen2t2-(-7cos2t

),-72cos2tsen2t7-,sen2t2-(-7cos2t

++

=

222(-7))cos2t2sen2t7-()sen2t2-7cos2t-(


+++

+=

492tcos42tsen2tcos2!-2tsen492tsen42tsen2tcos2!2tcos49

)7-,t22cost2sen7-,t2sen2-t27cos(-2222 +++++

+=

49)2tcos2tsen(4)2tsen2tcos(49


2222 ++++

+

=

49449


++

+=

102

),-72cos2tsen2t7-,sen2t2-(-7cos2t)(

+=

tT

En t = t0=0:102

),-72,(-7

(0)T

$or tanto:

102

),-72,(-7(0)T

Vector Unitario 2ormal

)!t"T

)!t"T)!t#

0

00

=

en t = t0



7/12


Espacio

6

Si10$

)%&'$cos$tsen$t'&%sen$t$&!&'cos$t +=

)(tT

Entonces:10$

)0%t$sen(&t$cos1(&%t$cos(&$t!1(sen=

)(' tT

.

En t = t0= 0:102

)0,14-,4-((0)'& =

Luego:

*

)0%'&%$&!

)0%'%$!10$

$

)0%'&%$&!10$

$

10$)0%1(&%(&!

10$

)0%1(&%(&!

!0)"T

!0)"T!0)# ====

$or tanto:

*

)0%'&%$&!!0)# =

Vector Unitario 3inormal:

)(t$)x(t&)(t% 000

=

en t = t0

540"

)53,14,49-(

07-2-

7-27

53102

1

53

)0,7-,2-(

102

)7-,2,(-7(0)$(0)x&(0)% ====

x

$or tanto:

540"

)53,14,49-((0)% =

c) Clculo de los $lanos PN;Po;Pr:

Plano NormalPN:

0)(tf-z)y,(x,).(tT 00 =

en t = t0:

En t0= 0:



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Espacio

6

PN:

0(0)f-z)y,(x,(0)& =

PN:

[ ] 0)1%0%1!&+)y%!,%.10$

)%&'$%!&'=

PN:01)&+y%1%&!,).%&'$%!&' =

PN:0'+'&y$','& =+++

PN:014z7-y2x7- =++

$or tantoPN: 45 -* 7 48 &0 + ,

Plano RectificantePr :

0tf-z)y,(x,N 0 =

)().(: 0t

en t = t0

En t0= 0

Pr :

0)0(f-z)y,(x,)0($: =

Pr :

[ ] 01)0,1,(-z)y,(x,53

)0,7-,2-(: =

Pr :

01)-zy,1,-(x)0,7,-2-( =

Pr :

07y-22x- =+

$or tantoPr :-5 7 4* - + ,

Plano Oc!la"orP0:

0tf-z)y,(x,B 0 =

)().( 0t

en t = t0



9/12


Espacio

6

En t0= 0

P0:

0)0(f-z)y,(x,)0(% =

P0:

[ ] 0)1,0,(1-z)y,(x,540"

)53,14,49-( =

P0:

01)-zy,1,-(x)53,14,(-49 =

P0:

053-z53y1449x49- =+++

P0:

04-z53y14x49- =++

$or tantoP0: 0' 5 &0 * 1.8 7 0 + ,

") #$lc!lo "e la #!r%at!ra y Tori&n "e la c!r%a #:

C*+c,+o -e +. #!r%at!ra en t = t0:

)t('f

)t(''f)xt('f

)/(t3

0

000

=

En t0= 0

33

)7-,2,7-(

)49,2!-,45()x7-,2,7-(

)0('f

)0(''f0)x('f

/(0) ==

-nde:

)10",2!,9!(

492!45

727-

)49,2!-,45()x7-,2,7-()0(''f0)x('f =

==

Entonces:

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10/12


Espacio

6

20142747252"01

53

10"120!

21"24

)727(

10"2!9!

)7-,2,7-(

)10",2!,9!-(/(0)

3222

222

3 ===

++

++==

20142747252"01

53/(0) ==

$or tanto: ;,) + ,


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Espacio

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Cuarta $rctica de

atemtica II

Unidad IV: Curvas en el

Espacio

=pellidos * 2om!res:>>>>>>>>>>>>>>>>>>>>>>>>>>>>

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