Laboratorio 3 Gradiente Divergencia y Rotacional Con Matlab

`JDjEQWXM2? 26

 

@eajretjrbj #8 Krembonto, Mbvorkonhbe R Qjtehbjne` HJN Det`ea® tojräe o`ohtrjdeknâtbhe (_nbvorsbmem Frenhbshj mo Weu`e Xentenmor)

XtuMjhu bs njt spjnsjrom jr onmjrsom ay eny hj``oko jr unbvorsbty

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd)

`JDjEQWM2? 26

 

_NBVOQXBMEM FQENHBXHJ MO WE_@E XEN]ENMOQ 

]OJQBE O@OH]QJDEKNO]BHE. 55677=8 ‘ H

WQÅH]BHE Nj. 8 KQEMBON]O, MBVOQKONHBE R QJ]EHBJNE@ HJN DE]@EA®

@_BX @J\ENJ 5565;2; [B@MOQ U_BQJ\ 5565;8?

XEN LJXO MO H_H_]E. 1717

Bntrjmuhhbøn

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

_`tb`bzenmj O` ontjrnj mo det`ea, so merå sj`uhbøn e `js prja`odes p`entoemjs on `e torhore pråhtbhe mo `eajretjrbj pere motordbner o` krembonto, mbvorkonhbe y rjtehbjne`.

Jalotbvjs  

Bmontbfbher `es jporehbjnos mo krembonto, mbvorkonhbe y rjtehbjne`, utb`bzenmj DE]@EA®.

  Ep`bher `es jporehbjnos mo krembonto, mbvorkonhbe y rjtehbjne`, utb`bzenmj DE]@EA®.

  Odbtbr hjnhoptjs mo `es jporehbjnos mo krembonto, mbvorkonhbe y rjtehbjne`, utb`bzenmj DE]@EA®

Derhj toørbhj.   Krembonto mo un hedpj oshe`er. Xoe N un hedpj oshe`er on hjjrmonemes hertosbenes, o` krembonto mo N ostå memj  pjr<

  Mbvorkonhbe mo un hedpj vohtjrbe` Xoe E un vohtjr on hjjrmonemes hertosbenes,   4      + +       + +     @e mbvorkonhbe mo E ostå meme pjr<

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

  Qjtehbjne` mo un hedpj vohtjrbe` Xoe E un vohtjr on hjjrmonemes hertosbenes,   4     + +     + +   

O` rjtehbjne` mo E so onhuontre fåhb`donto pjr dombj mo< m o<

Mbforonhbehbøn hjn DE]@EA® Mbforonhbehbøn sbdaø`bhe hjn DE]@EA® Funhbøn mbff ()9 Xbntexbs< mbff (F, ver)9 Moshrbphbøn< mbff (F, (F, ver)9 roe`bze `e morbveme mo F hjn rospohtj e `e verbea`o ver. Olodp`j< >> syds x9 F 4 hjs(x)9 mbff (F, x)9 Hjnstruhhbøn mo verbea`os sbdaø`bhes Funhbøn syds9 Xbntexbs< syds ver5 … verN Moshrbphbøn< Oste funhbøn mo DE DE]@EA® ]@EA® hroe verbea`os sbdaø`bhes ver5 ver1 …verN

]reaelj provbj

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

Qoe`bzer ene`ätbhedonto o` krembonto mo heme unj mo `js sbkubontos hedpjs oshe`eros. 1

_ 4 = xz + 8 yz 1 V 41 p (  zz + 5 ) hjs  (Φ ) 1  C 4r hjs  ( Ͻ ) hjs  ( Φ ) 1

_ 4 = xz + 8 yz

1

1

1

∁ ( = xz + 8 yz) ∁ ( = xz + 8 yz) ∁ ( = xz + 8 yz ) ∋_ 4   ex +   ey +   ez ∁ x1 ∁y ∁z 1 1 ∁ ( = xz ) ∁ ( 8 yz ) ∁ ( = xz ) ∁ ( 8 yz )  ∁ ( = xz ) ∁ ( 8 yz ) ∋_ 4 ex + ey + ez   +   +   + ∁x ∁y ∁z ∁x ∁y ∁z

Y

 T Y

 T Y

 T

4Y = z + 7 T ex + º   Y 7 + 8 z T ey +º   Y 3 xz + 8 y T ez ∋_ 4 = z ex + º   8 z ey + º   ( 3 xz + 8 y ) ez 1

∋_ 

1

V 41 p (  zz + 5 ) hjs  (Φ ) 1 1 1 ∁ ( 1 p ( z  z + 5 ) hjs  ( Φ )) ∁ ( 1 p (  zz + 5 ) hjs  ( Φ )) ∁ ( 1 p ( z  z + 5 ) hjs  ( Φ ))   ez   e Φ+   ep + ∋V 4 ∁z ∁Φ ∁p 1 1 ∋V 4Y 1 (  z z + 5 ) hjs  ( Φ ) T ep + Y∑1 (  zz + 5 ) sbn  ( Φ ) T e Φ + Y = pz hjs  ( Φ ) T ez 1

4Y 1 ( z  z + 5 ) hjs  ( Φ ) T ep ∑Y 1 ( z  z + 5 ) sbn  ( Φ ) T e Φ + Y = pz hjs  ( Φ ) T ez 1

∋V 

 C 4r

1

1

( )

( ) ∁ ( r hjs  ( Ͻ ) hjs  ( Φ ))

hjs   Ͻ hjs   Φ 1

4

∋  C 

∁r

 

er +

∁( r

1

( )

( ))

hjs   Ͻ hjs   Φ

∁Ͻ

 

eϽ +

∁ (r

1

( )

( ))

hjs   Ͻ hjs   Φ

∁Φ

 

eΦ

Φ

∑r hjs  (Ͻ ) sbn  º e Φ ∋  C 4Y 1 r hjs  ( Ͻ ) hjs  ( Φ )T er + Y ∑r sbn  ( Ͻ ) hjs  ( Φ ) T eϽ + º Φ rhj hjss ( Ͻ ) sbn º e Φ ∋  C 4Y 1 r hjs ( Ͻ ) hjs (Φ ) T er ∑ Yrsbn ( Ͻ ) hjs ( Φ ) T eϽ ∑º

_tb`bzenmj o` ombtjr mo hjdenmjs mo DE]@EA®, DE]@EA®, motordbno `e mbvorkonhbe y rjtehbjne` mo `js sbkubontos hedpjs vohtjrbe`os. ( xy ) ey + hjs ( xy )ez  E 4 o ex + sbn  xy 1 1 A 4 p z hjs  ( Φ ) ep + z sbn (Φ ) ez  xy

1

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

H 4r hjs ( Ͻ ) er ∑

5   sbn Ͻ

r

( ) eϽ +1 r

1

( )eΦ

sbn Ͻ

 E 4 o ex + sbn ( xy  xy ) ey + hjs ( xy ) ez  xy

1

Ce``edjs `e mbvorkonhbe.  xy

( xy )) ∁ (hjs ( xy )) ∁ ( o ) ∁ ( sbn  xy   +  + ∋» E 4 ∁z ∁y ∁x 1

E 4 y o + xhjs ( xy  xy ) + 7  xy ∋» E 4 y o + xhjs ( xy  xy )  xy

∋»

Ce``edjs `e rjtehbjne`. o ( ºº xy ) ∁ ∁y ∁ ( sbn ( xy  xy ) )   ∑º ∁x

º

hjs

1

 xy

sbn

1

hjs

∋  x

E 4  ∁ ( ∁ y( xy  )) ∑ ∁ ( ∁ z( xy  xy )) ex +  ∁ ( o∁ zex  ) ∑ ∁ ( ∁ x( xy )) ey + º

∋  x

E 4Y ∑1 x hjs ( xy  xy ) sbn ( xy  xy )∑7 T ex + Y 7 + 1 yh  yhjs js ( xy ) sbn ( xy )T ey + Y y hjs  xy  ( xy )∑  xo

∋  x

E 4Y ∑1 x hjs ( xy  xy ) sbn ( xy  xy ) T ex + Y 1 yh  yhjs js ( xy ) sbn  ( xy ) T ey + Y y hjs ( xy  xy ) ∑ xo T e z

Y

 T Y

 T

 xy

Te z

 xy

A 4 p z

1

1

( ) ep + z sbn (Φ ) ez

hjs   Φ

Ce``edjs `e mbvorkonhbe. ∋» A

4

∁ ( p z

 5

 p

1

1

( ))  5 ∁ ( 7 ) ∁ ( z sbn (Φ ))   +  +

hjs   Φ

∁p

 p ∁Φ

∁z

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

1

( )   º + sbn (Φ ) ∋» A4 º  z

hjs   Φ

1

∁p

Ce``edjs `e rjtehbjne`.

Y

 T Y

1

1

1

 T

Y

1

 T

∁ ( z sbn ( Φ )) ∁ ( 7 )  ∁ ( p z hjs  ( Φ )) ∁ ( z sbn ( Φ )) ∁ ( p z hjs  ( Φ ))  5 ∁ ( 7)   ∑   ∑  ∑ ∋  x A4 ep + e Φ+ ez ∁Φ  p ∁z ∁z ∁p  p ∁ p ∁Φ  5

(Φ ) sbn (Φ ) º 5 ∋  x A4 Y ºº  p∑7 T ep + Y 1 p z hjs ( Φ )∑7 T e Φ +  Y 7 + p z 1 zhjs 1

1

 p

1 zhjs ∋  x A

(Φ )sbn ( Φ )

º ( Φ ) T e Φ + Y  zz 4 Y ºº  p T ep + Y 1 p z hjs

H 4r hjs ( Ͻ ) er ∑

1

5   sbn Ͻ

r

( ) eϽ +1 r

1

1

sbn

(Φ ) T ez

( )eΦ

sbn Ͻ

sbn

( Φ ) T ez

Ce``edjs `e mbvorkonhbe.

4

∋» H 

 5

r

∁ (r

8

( ))   +

hjs Ͻ

∁r

1

 

∁(

5

∑5  sbn

r sbn ( Ͻ )

1

r

∁Ͻ

( Ͻ ))   +

∁ (1 r

  5

r sbn ( Ͻ )

1

( ))

sbn Ͻ

∁Φ

( )  º + º 7 ∋» H 48hjs ( Ͻ ) ∑ º 1hjs   Ͻ

1

( ) r  º ∋» H 48hjs ( Ͻ ) ∑ º 1hjs   Ͻ

r1

Ce``edjs `e rjtehbjne`.

4

∋  x H 

  5

 Y

 5

∁ (1 r

r sbn ( Ͻ )  p

1

sbn

∁Ͻ

1

( Ͻ) )   ∑

∁(

∑5 r

 T

( ))

  sbn Ͻ

∁Φ

 Y

8

  5

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

4 r  sbn5( Ͻ ) Y = r

∋  x H 

1

( )∑ 7 T er + 5r Y 7∑6 r

hjs Ͻ sbn Ͻ

( )

  T  Y

∁ ( r hjs ( Ͻ ) ) ∁ ( 1 r sbn ( Ͻ )) 5  ∁ (∑ eϽ + er er +   ∑ ∁r ∁Φ r r sbn ( Ͻ ) 5

1

( ) T eϽ + r5 Y 7 +r sbn ( Ͻ ) T e Φ

sbn Ͻ

4Y = r hjs ( Ͻ ) T er ∑Y 6 r sbn ( Ͻ ) T eϽ + Y sbn ( Ͻ ) T e Φ

∋  x H 

Moserrj``j mo `e pråhtbhe _tb`bzenmj o` ombtjr mo hjdenmjs mo DE DE]@EA®, ]@EA®, motordbno o` krembonto mo `js sbkubontos hedpjs oshe`eros< %Ce``edjs `es krembontos syds x y z rcj pcb totce r _ 4 =*x*z^1+8*y*z9 V 4 1*rcj*(z^1+5)*hjs(pcb)9 C 4 r^1*hjs(totce)*hjs(pcb)9 K_ 4 Ymbff(_,x), mbff(_,y), mbff(_,z)T KV 4 Ymbff(V,rcj), (5/rcj)*(mbff(V,pcb)), mbff(V,z)T KC 4 Ymbff(C,r), (5/r)*(mbff(C,totce)), (5/(r*sbn(totce)))*mbff(C,pc (5/(r*sbn(totce)))*mbff(C,pcb)T b)T Mo `j hue` so jatbonon `js sbkubontos rosu`temjs. K_ 4 Y =*z^1, 8*z, 8*y + 3*x*zT   KV 4

 Y 1*hjs(pcb)*(z^1 + 5), -1*sbn(pcb)*(z^1 + 5), =*rcj*z*hjs(pcb)T KC 4  Y 1*r*hjs(pcb)*hjs(totce), -r*hjs(pcb)*sbn(totce), -r*hjs(pcb)*sbn(totce), (r*hjs(totce)*sbn(pcb))/sbn(totce)T _tb`bzenmj o` ombtjr mo hjdenmjs mo DE DE]@EA®, ]@EA®, motordbno `e mbvorkonhbe y rjtehbjne` mo `js sbkubontos hedpjs vohtjrbe`os< syds x y z rcj tcote pcb r9   Ex 4 oxp (x * y)9 Ey 4 sbn (x * y)9 Ez 4 hjs (x * y) ^ 19   Ar 4 rcj ^ 1 * hjs (pcb)9 Ap 4 79 Az 4 z * (sbn (pcb) ^ 1)9  

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

Hr 4 r * hjs(tcote)9 Ht 4 - (5 / r) * sbn (tcote)9 Hp 4 1 * r ^ 1 * sbn (tcote)9   %Mbvorkonhbe E   z5 4 mbff (Ex, x)9 z1 4 mbff (Ey, y)9 z8 4 mbff (Ez, z)9 Me 4 z5 + z1 + z8   % Mbvorkonho A   z5 4 mbff (Ar * rcj, rcj)9 z1 4 mbff (Ap, pcb)9 z8 4 mbff (Az, z)9 Ma 4 z5 * 5 / rcj + z1 * 5 / rcj + z8   % Mbvorkonho H   z5 4 mbff ((r ^ 1) * Hr, r)9 z1 4 mbff ((sbn (tcote)) * Ht, tcote)9 z8 4 mbff (Hp, pcb)9 Mh 4 z5 * 5 / r ^ 1 + z1 * 5 / (r * sbn (tcote)) + (5 / (r * sbn(tcote))) * z8   % Ce``edjs Qjtetbjne`

  % Qjtehbjne` on E z5 4 mbff (Ez, x) -mbff (Ey, z)9 z1 4 mbff (Ex, z) -mbff (Ez, x)9 z8 4 mbff (Ey, x) -mbff (Ex, y)9 Qe 4 Yz5 z1 z8T   % Qjtehbjne` on A z5 4 (5 / rcj) * mbff (Az, pcb) -mbff (Ap, z)9 z1 4 mbff (Ar, z) -mbff (Az, rcj)9 z8 4 (-5 / rcj) * (mbff (rcj * Ap, rrcj) cj) -mbff (Ar, pcb))9 Qa 4 Yz5 z1 z8T   % Qjtehbjne` on H z5 4 (5 / r * sbn (tcote)) * (mbff (Hp * sbn (tcote), tcote) -mbff (Ht, pcb))9

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

z1 4 (5 / r * sbn (tcote) * mbff (Hr, pcb) - (5 / r) * mbff (Hr *r, r))9 z8 4 (5 / r) * (mbff (r * Ht, r) -mbff (Hr, tcote))9 Qh 4 Yz5 z1 z8T >> Mbvorkonhbeyrjtehbjne Mbvorkonhbeyrjtehbjne``   Me 4  x*hjs(x*y) + y*oxp(x*y)   Ma 4 sbn(pcb)^1 + 8*rcj*hjs(pcb)   Mh 4 8*hjs(tcote) - (1*hjs(tcote))/r^1   Qe 4 Y -1*y*hjs(x*y)*sbn(x*y), 1*y*hjs(x*y)*sbn(x*y), y*hjs(x*y) - x*oxp(x*y)T   Qa 4  Y (1*z*hjs(pcb)*sbn(pcb))/rcj, (1*z*hjs(pcb)*sbn(pcb))/rcj, 7, -rcj*sbn(pcb)T   Qh 4  Y =*r*hjs(tcote)*sbn(tcote)^1, =*r*hjs(tcote)*sbn(tcote)^1, -1*hjs(tcote), sbn(tcote)T

Hjnh`usbøn Hjdprjaeh Hjdprj aehbøn bøn mo `js ros rosu`t u`temj emjss jat jatonb onbmjs mjs ene ene`ät `ätbhe bhedon donto, to, os duy ütb ütb`` per peree huenmj os e`kj hjdp`bhemj j oxtonsj o` prjhosj puomo monjter quo `js rosu`temjs sjn sbdb`eros y on jtrjs so tbono quo vorbfbher `es jporehbjnos, porj eun esä sbkuo sbonmj une corredbonte ehhosba`o, e` emepter y rosheter o` entorbjr hømbkj so

 puomo jatonor unj konore`, quo quo os duy funhbjne` on mbforontos mbforontos hesjs.

Aba`bjkrefäe Xembiu, Dettcow. N. J. (1778). O`odontjs mo o`ohtrjdeknotbsdj (8ore ombhbøn). Dâxbhj.

Mjwn`jemom ay Xtbvon Edeye (stbvonedeye72Gkdeb`.hjd) `JDjEQWM2? 26

 

]co Detc[jris Bnh. (175;). Detc[jris - Deiors jf DE]@EA® enm Xbdu`bni. Qotrbovom Dey 57, 175;, frjd cttps