Calculus: Derivation of the laplace equation in spherical coordinates

[Wyatt Mathewson]

CALCULUS: DERIVATION OF THE LAPLACE EQUATION IN SPHERICAL COORDINATES

Derivation of Laplace Equation in Spherical Coordinates is a challenging task, not that it is difficult but rather; the list of terms to be added in the process become very exhaustive to the extent that they can fill completely an entire page of an A4 sheet, whether in Portrait or Landscape orientation. In this work however, I have devised my own way of adding up such list of exhaustive terms using a method I have named as "THE TABULAR METHOD".

Under this method, each of the three transformed Cartesian components (i.e. x, y and z) has its corresponding spherical polar component, FIRST added together in its own, separate table and eventually, all the three components combined and added together in a single, FINAL table

The most important thing to note in this work is that someone perhaps new to the derivation, may somehow find it difficult to follow the procedures involved in the process, if at all s/he is not accustomed to working with 3x3 Matrices. I have therefore (right after the last page of the derivation process), included some simple practical reviews of 3x3 matrices examples, more inclined to finding inverses of 3x3 matrices

This work is part of the manual I'm currently working on, which will be made available to you (and so is to the public), immediately after its completion.

Please see more in the attachment, and let me have your feedback, if any!

Hii ngoma nilisoma chuo kikuu School of Engineering!

Lazima kichwa kitanuke!

 

[Makanyaga]

Hii ngoma nilisoma chuo kikuu School of Engineering!

Lazima kichwa kitanuke!

Siyo ngumu, ila ni ndefu!

Ulishajaribu kutafuta integral ya square root of tangent x, (i.e. integrate sqrt(tanx) kwa kutumia njia ya partial fractions? Kama hujawahi ijaribu halafu ukishindwa nijulishe nitakupa solution.

By the way, what is the squareroot of complex number i, where ixi=-1?

 

[Wyatt Mathewson]

Siyo ngumu, ila ni ndefu!

Ulishajaribu kutafuta integral ya square root of tangent x, (i.e. integrate sqrt(tanx) kwa kutumia njia ya partial fractions? Kama hujawahi ijaribu halafu ukishindwa nijulishe nitakupa solution.

By the way, what is the squareroot of complex number i, where ixi=-1?

Partial fractions is another long long process

Sio ngumu ila ni long kisenge

Unaweza pata Simultaneous Equations kama nane na uknowns nane...

Thats stupidity!

 

[Makanyaga]

Partial fractions is another long long process

Sio ngumu ila ni long kisenge

Unaweza pata Simultaneous Equations kama nane na uknowns nane...

Thats stupidity!

Let I=integral sqrt(tanx) and u= sqrt(tanx)

then

I=integral (2u^2)/(1+u^4) (after proper transformations of the derivatives and variables from x to u)

One of the methods of solving this polynomial rational function in "u" is by using partial fractions. However there other different ways as well

 

[NostradamusEstrademe]

Nilisoma physics UDSM hadi leo sijaelewa polar coordinates na cylindrical coordinates nilibaki na cartesian tu.Ila nilifaulu.
Tatizo nilisoma pcb ndio maana.
Kusoma pcb chuo kikuu usome physics ni mateso

Sent using Jamii Forums mobile app

 

[Wyatt Mathewson]

Let I=integral sqrt(tanx) and u= sqrt(tanx)

then

I=integral (2u^2)/(1+u^4) (after proper transformations of the derivatives and variables from x to u)

One of the methods of solving this polynomial rational function in "u" is by using partial fractions. However there other different ways as well

Kuna kitu inaitwa “Advanced Complex Integrals” husolve kienyeji enyeji namna hiyo ya kulet “u” mzee!

Ushacheck “MIT Integral Bee”...mashindano ya integral MIT?

Yapo Youtube mzee angalia uone watu wanavyoteseka na integral!

Naona unapeleka peleka kirahisi rahisi sana mzee!

Huwenda njia zako ni rahisi kweli ila bado sijaamini,nidhihirishie zaidi labda!

 

[Makanyaga]

Kuna kitu inaitwa “Advanced Complex Integrals” husolve kienyeji enyeji namna hiyo ya kulet “u” mzee!

Ushacheck “MIT Integral Bee”...mashindano ya integral MIT?

Yapo Youtube mzee angalia uone watu wanavyoteseka na integral!

Naona unapeleka peleka kirahisi rahisi sana mzee!

Huwenda njia zako ni rahisi kweli ila bado sijaamini,nidhihirishie zaidi labda!

See the attachment below, if you should be able to follow!

 

[Wyatt Mathewson]

See the attachment below, if you should be able to follow!

Ni vizuri kujipa practice namna hii kama freestyle

Shida inakuja pale unapewa chaki na ubao unapewa dakika 3 usolve complex advanced integral problem

Inaitwa at “room temperature”,lazima ukae sawa mzee!

Ile jamaa ana”let” mpaka ubao unaisha halafu giza mzee!

 

[Makanyaga]

Nilisoma physics UDSM hadi leo sijaelewa polar coordinates na cylindrical coordinates nilibaki na cartesian tu.Ila nilifaulu.
Tatizo nilisoma pcb ndio maana.
Kusoma pcb chuo kikuu usome physics ni mateso

Sent using Jamii Forums mobile app

Angalia hiyo michoro kwenye attachment halafu kwa sasa utaelewa uhusiano uliopo kati ya hizi coordinates. Kipindi unazisoma ukiwa chuo, ulikuwa unasoma ili uweze kujibu swali kwenye mtihani na kuelewa haikuwa kwa sana. Kipindi hiki utazisoma ukiwa una nia ya kuzielewa tu basi. Angalia hiyo michoro pamoja na hizo x, y, na z tu halafu mengine achana nayo. Ukifanya hivyo nina hakika utaelewa uhusiano uliopo kati ya coordinates hizi. Ubarkiwe

 

[Makanyaga]

Nilisoma physics UDSM hadi leo sijaelewa polar coordinates na cylindrical coordinates nilibaki na cartesian tu.Ila nilifaulu.
Tatizo nilisoma pcb ndio maana.
Kusoma pcb chuo kikuu usome physics ni mateso

Sent using Jamii Forums mobile app

Ukiona idea yoyote unashindwa kuielewa, jua kuwa umeshindwa kuikamata LOGIC ya kitu hicho. Darasani kawaida tunapokuwa tunafundishwa, ili uweze kuelewa kile unachofundishwa, lazima upate logic yake KWANZA halafu ndiyo kuelewa kunafuata . Tofauti na hivyo, itabidi kukariri. Hata hivyo kwa namna ambayo mimi huwa ninaona, hiki kitu ni natural principal katika maisha ya kila siku ya mwanadamu kwamba ili uweze kuelewa chochote kile katika masiha, lazima upate logic kwanza. Tuseme labda mtu anakuja kwako halafu anaanza kukueleza kitu, ili uweze kumwelewa lazima anachokueleza kilete logic kwako kwanza. Kwa mfano anakuambia "nisaidie hata elfu 5 tu nimeishiwa kabisa, halafu kesho kutwa nina mpango wa kwenda kununua kiwanja cha million 50 Bunju ! Think of this scenerio.
Hata hivyo kuna exception kwenye rule hii kwamba kwa mambo ambayo yana-operate kwenye ulimwengu wa roho, this rule DOES NOT NECESSARILY HOLD, though it may sometimes hold, but there is no guarantee that IT WILL ALWAYS HOLD. Kwenye ulimwenu ambao myths na mysteries ni kitu cha kawaida, logic huwa inafanya kazi partially