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Calculo Vectorial

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Calculo Vectorial

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Para mi

Created by

Joaquin Michelena
Joaquin Michelena
Uruguay

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Calculo Vectorial
 

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Calculo VectorialOnline version

Para mi

by Joaquin Michelena
1

Una curva C es simple si admite una parametrización

2

Una curva C es cerrada si admite una parametrización

3

Una curva C es regular si alfa es y 0

4

Recta tangente : alfa debe ser y en t0 es ( ) + ( ) ( - )

5

Alfa C1 , su Longitud de arco es : Integral entre de

6

Una parametrización es por longitud de arco si es y alfa es

7

Alfa :
T = /
N = /
B = &
B' = N

8

f : U C a campo escalar continuo y C una curva contenida en U , alfa [ a , b ] a U de clase C1 e en [ a , b )

C ? fds = ? ( ( ) x )

9

F : U C a continua , alfa [ a , b ] a Rn de clase C1 e , alfa con a la de C

C ? Fds = ? [ ( ) ) , ] d t

10

Sea F : U C a continuo
F es de si existe f : U C Rn a R tal que =
f se llama

11

U y , F : U C Rn a Rn continuo
1 ) es de
2 ) Integral de linea en es
3 )

12

F : U C a campo vectorial C1 , F = ( P , , )
rotor ( F ) = ( - , - , - )
Si rotor ( F ) = 0 , F es

13

F : U C a , C1 , F = ( P , Q , R )
div ( F ) = + Qy +
si div ( F ) = 0 , F es
( ( F ) ) = 0 si F es

14

F : U C a
U simplemente
F es de si y solo si es

15

Cilindro : J : [ , 2pi ] x [ , ] a R3
J ( u , v ) = ( cosu , , )

Esfera : J : [ , ] x [ , ] a R3
J ( u , v ) = ( , , )

Cono positivo : J : C a R3
J ( u , v ) = ( , , + )
donde D = ( ( u , v ) e R2 : + ? )

Paraboloide : J : a R3
J ( u , v ) = ( , , + )

16

Area ( J ) = D ? ? & dud v

17

Sea J : D C a parametrización de S , J de clase y
f : G C a continua tal que G C S
Definimos la integral de f sobre J :

J ? ? fds = D ? ? ( ( , ) ) & dud v

18

Sea J : D C a de clase ,
F : G C a continuo tal que G C S
Definimos la integral del campo vectorial sobre F sobre J :

J ? ? Fds = J ? ? [ , ] ds

Donde es & / & N

19

Teorema de Green :
Sea C C una curva , , a , según
Sea D C la acotada tal que = C
Sea F : G C a de clase tal que G C D U BD

C ? Fds = D ? ? - dxdy

20

Teorema de Stokes :
Sea S C R3 superficie , regular , con borde tal que BS es una a y
Considero S y BS orientadas compatiblemente
Sea F : G C a R3 de clase tal que S U BS C G

BS ? Fds = S ? ? ( ) d s

21

Teorema de Gauss :
Sea V un limitado por una superficie S con orientación dada por la n a S
Si X es un campo vectorial de clase es un que contiene a V U S entonces tenemos :

V ? ? ? ( ) dV = S ? ? dS

 
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