xrandr

Posted on September 30, 2024 by rpflee

Quantum Mechanics: End Game

physical definition

~ define microscopic events in terms of observable physical phenomena, such as the changes in readings of the measuring device

~ define unobservable events in terms of observable events

— Me@2022-01-31 08:33:01 AM

I give you the final answer to this topic. You should continue to study this topic until you understand that.

A mixed state exists due to the lack of information.

A superposition state exists not due to that.

A superposition state exists due to the lack of a physical definition.

An eigenstate and a superposition state are both pure states.

A superposition state is neither an AND nor an OR because it is a pure state, meaning that it is one single quantum state, not a mixture of two.

A mixed state is an OR.

xrandr --output DP-0 --auto --output HDMI-0 --left-of DP-0

In the realm of knowledge, I share with you,
The final answer, clear and true.
Study deeply, let curiosity flow,
Until this truth, you truly know.

A mixed state emerges, shrouded in doubt,
Born from the shadows where facts are left out.
Yet a superposition, distinct in its grace,
Doesn’t arise from that uncertain place.

It finds its definition in the absence of form,
A concept elusive, where ideas transform.
Both eigenstates and superpositions, pure,
In the quantum dance, their essence is sure.

Neither an AND nor an OR, it stands alone,
One single state, in the vastness, it’s grown.
Not a blend of two, but a singular hue,
While a mixed state proclaims, “I am an OR,” too.

— Me@2024-08-19 03:23:59 PM

— Me@2024-09-30 03:40:25 PM

DSE 數學補邊個好？（神器四）

Posted on September 29, 2024 by rpflee

這篇文章繼續討論，選擇數學補習老師的一些準則。而這裡的數學，是指香港中學文憑試的數學科，包括核心，M1 及 M2。

如果你問我物理補習推介，特別是物理補習天王的話，我會答：「補 physics，補 Ken Chan！」但是，如果你問我數學補習天王的話，恕沒有推介，因為本人沒有遇過。

那樣，數學科補習界如果有 Ken Chan，他又必須具備，什麼條件呢？

上文提到：

1. 背誦篇

理科是文科，但不是一般的文科，而是極超巨大級的文科。

1.1 他會提醒你，數學其實是，極超大量的背誦。

……

1.2 極超大量的背誦，來自極端深刻的理解。

……

2. 年月時間篇：歷屆試題

2.1 他會提議你，考試前要操練 10 年、20 年或 30 年的歷屆試題。

……

2.1.2 他會告訴你「八二法則」——頭八十分，只需兩成的付出；而尾二十分，則需佔八成的努力。亦即是話，付出和收穫，不成比例。

……

2.2 最近五年的題目，則從來不要看，因為你要保留它們，在公開試前的休假月時，來作模擬考試用。

2.3 最近五年以前的題目，則可以在課程的那三年間，隨時隨地做。

留意，做試題有兩大主要方法：「分課題」或「分年份」。

……

2.4 而歷屆試題的來源是：……

3. 星期時間篇

他會告訴你，如何從平衡到駕馭，各科溫習時間之爭。

其方法是，用自動時間表——透過預先設計好的時間程式，每天自動生成的時間表。每天你毋須花額外時間，人手編配時間表，考慮溫什麼。

該機械化時間表，會自動編排和分配，各科的溫習時間和長度。同一天會由，不同類型科目組合而成，導致腦部的不同部分，可以輪流休息，即使整體的工作時數很多。

該系統化時間表，會有適當分割，導致溫習不會辛苦。

有關詳情，請點擊本綱誌之文章分類

in-time-時來運到，

然後於網址結尾，加以下字串：

?order=asc

記住，刻苦不能長久，因為你的潛意識，會無時無刻，找機會「逃走」。那就是「疲倦」的真正動機。

你工作系統的目標，並不是「控制自己」——控制不想讀書的自己讀書，勉強自己做不想做的事；而是「釋放自己」——釋放學習動機，爆發學術潛能。

頭腦清醒時，靈感才會自來，天份才能發揮。

4. 分秒時間篇

……

— Me@2024-09-28 11:11:52 AM

Where are you? 2.3

Posted on September 22, 2024 by rpflee

Utopia | 何有之鄉, 2.3

除了微觀短期，可不可遷就外，還要考慮宏觀長期。去或留的大原則是，你要問，究竟該職位，是令你的財富穩步上升，還是穩步下降。

同理，考慮應否與心儀對象，發展成情侶，乃至於夫妻時，你可以問，有了她，與她開始交往以後，你覺得人生吃力了很多，還是輕鬆了一點；反之亦然——她也應該要有，同樣的考慮。

更甚的話，你更可以問，自從認識她以後，你的財富穩步上升，還是穩步下降。亦即是話，你要一早判斷，究竟她是成家妻，還是敗家婆。

工作和愛情的主要分別是，一個職位的理想壽命，是兩至三年；而愛情的目標時長，則是一生一世。其他方面，則有很多地方相同。所以，你可以把一些選擇工作的原則，改編翻譯成愛情的版本。

選擇戀愛對象的原則，並不是找「理想伴侶」，而是找厭惡得來，仍然可以遷就到的人。反過來說，必須排除那些，「不可能遷就到」的可能對象。可以遷就到的，就是缺點；想遷就也遷就不到的，就為之人格問題。

— Me@2024-09-05 03:33:08 PM

Euler problem 23.1.1

Posted on September 21, 2024 by rpflee

(defmacro sum (lst)
  `(reduce #'+ ,lst))

(defun proper-divisors (n)
  (when (> n 1)  
    (let ((divisors '())
          (limit (floor (sqrt n))))  
      (loop :for i :from 1 :to limit
            :when (zerop (mod n i))  
              :do (progn
                    (push i divisors)  
                    (when (/= n (floor n i))  
                      (push (floor n i)
                            divisors))))  
      (remove-duplicates (sort divisors #'<)
                         :test
                         #'equal))))

(defmacro sum-proper-divisors (n)
  `(sum (proper-divisors ,n)))

(defmacro is-abundant (n)  
  `(> (sum-proper-divisors ,n) ,n))

(defmacro gen-list (min max)
  `(loop :for n :from ,min :to ,max
         :collect n))

(defmacro filter (predicate list)
  `(remove-if-not ,predicate ,list))

(defmacro macro-to-function (macro-name)
  `(lambda (xs) (,macro-name xs)))

(defmacro db (x)
  `(* 2 ,x))

((lambda (xs) (db xs)) 2.1)

(funcall (macro-to-function db) 2.1)

;; Evaluation Context:

;; ((macro-to-function db) 2.1): This line attempts to call the result of the macro macro-to-function directly as if it were a function. However, since macro-to-function returns a lambda expression, this results in an "illegal function call" error because the macro is not expanded in the context of a function call.

;; (funcall (macro-to-function db) 2.1): In this line, funcall is used to invoke the lambda function returned by macro-to-function. This correctly evaluates the lambda and applies it to 2.1, allowing the macro to be expanded properly.

(defmacro abundant-numbers (limit)
  `(filter (macro-to-function is-abundant) (gen-list 1 ,limit)))

— Me@2024-09-20 11:53:30 PM

Ex 3.2 Verification, 1

Posted on September 18, 2024 by rpflee

Functional Differential Geometry

A one-form is a concept from differential geometry and tensor analysis. It is defined as an order 1 covariant tensor field. Here are the key points about one-forms:

1. Definition: A one-form is a mathematical object that can be thought of as a linear functional that takes a vector as input and produces a scalar. In simpler terms, it is a way to assign a number to each vector in a vector space.

2. Mathematical Representation: One-forms can be represented in local coordinates as:

$\omega = f_1 \, dx^1 + f_2 \, dx^2 + \ldots + f_n \, dx^n$

where $f_i$ are smooth functions and $dx^i$ are the differentials of the coordinates.

3. Applications: One-forms are used in various fields, including physics, particularly in the context of differential forms, which are essential in the formulation of concepts like integration on manifolds and in the study of electromagnetic fields.

4. Geometric Interpretation: Geometrically, one-forms can be visualized as fields that assign a scalar value to each tangent vector at a point on a manifold, allowing for the measurement of various properties like lengths and angles.

Conclusion

In summary, a one-form is a fundamental concept in differential geometry, serving as a linear functional that maps vectors to scalars, and plays a crucial role in various mathematical and physical applications.

— AI

Turning point, 4

Posted on September 17, 2024 by rpflee

Almost every trivial [matter] can be a turning point.

— Me@2011.12.19

超整理法, 3

Posted on September 17, 2024 by rpflee

魔法師是怎樣煉成的, 2

Posted on September 16, 2024 by rpflee

這段改編自 2023 年 6 月 23 日的對話。

有同情，就沒有愛情，因為，愛情的先決條件是：她傾慕你，而你又欣賞她。

把妻子比喻成你的「另一半」，可以用來強調，對方的重要性。但是要留意，那詞亦會引起誤會。

大部分人以為，「另一半」的單位是「人」，導致二人中的任何一方，都只是「半個人格」，不完整之人也。例如，市面上有個雖則浪漫，但嚴重錯誤的講法：

~~雖然你現在的人生千瘡百孔，但是只要找到，你的未來太太，你就可以被她彌補所有遺撼，人生達致圓滿。~~

錯誤漏洞，起碼有二：

一、既然千瘡百孔，何以吸引未來之人？
二、如果單身時一個不開心，雙身時則只會變成，兩個不開心。

正確的講法應該是：

只要現在的單身生活圓滿，你找到未來太太的機會，就會大大提高。起碼要做到，雖然未達圓滿，但是尚算完整。

「另一半」的單位，並不是「人」，因為，並不是說，只有半個人格，就可吸引到人。

「另一半」的單位，是「夫妻」。一對人的一半，就是一個人，即是一個原本已百分百完整之人，起碼在人格和生活的基礎設施上而言。

情侶二人，是兩個人，而不是兩半人。

完整的人格不需要愛情，但可以選擇愛情。不需要而可以選，才是自由意志，才有長久愛情。

— Me@2024-08-13 04:04:35 PM

Sub ExportCalendarToOrgMode()

Posted on September 15, 2024 by rpflee

Sub ExportCalendarToOrgMode()
    Dim objFolder As Outlook.Folder
    Dim objAppointment As Outlook.AppointmentItem
    Dim strOutput As String
    Dim objFile As Object
    Dim strFilePath As String
    Dim totalAppointments As Integer

    strFilePath = "Z:\media\d\Outl.org"

    Set objFolder = Application.Session.GetDefaultFolder(olFolderCalendar)

    For Each objAppointment In objFolder.Items
        strOutput = strOutput & "* " & objAppointment.Subject & vbCrLf
        strOutput = strOutput & "SCHEDULED: <" & Format(objAppointment.Start, "yyyy-MM-dd>") & vbCrLf
        strOutput = strOutput & "DESCRIPTION: " & objAppointment.Body & vbCrLf
        strOutput = strOutput & vbCrLf

        totalAppointments = totalAppointments + 1
    Next objAppointment

    Set objFile = CreateObject( "Scripting.FileSystemObject" ).CreateTextFile(strFilePath, True)
    objFile.Write strOutput
    objFile.Close

    MsgBox "Calendar exported to: " & strFilePath & vbCrLf & "Total appointments exported: " & totalAppointments, vbInformation
End Sub

Sub ShiftCalendarEvents()
    Dim calFolder As Object
    Dim calItems As Object
    Dim calItem As Object
    Dim startDate As Date
    Dim endDate As Date
    Dim shiftDays As Integer

    shiftDays = 7

    ' 9 represents olFolderCalendar in Outlook 2010
    Set calFolder = Application.Session.GetDefaultFolder(9)

    Set calItems = calFolder.Items

    For Each calItem In calItems
        If calItem.Class = 26 Then ' 26 represents olAppointment in Outlook 2010
            calItem.Start = DateAdd("d", shiftDays, calItem.Start)
            calItem.Duration = 60
            calItem.Save
        End If
    Next calItem

    MsgBox "Calendar events have been shifted by " & shiftDays & " days."
End Sub

— Me@2024-09-14 12:47:16 PM

Ex 1.31 Velocity transformation

Posted on September 14, 2024 by rpflee

Structure and Interpretation of Classical Mechanics

Use the procedure Gamma-bar to construct a procedure that transforms velocities given a coordinate transformation. Apply this procedure to the procedure p->r to deduce (again) equation (1.65).

~~~

(define ((Gamma-bar f-bar) path-q-local-tuple)
  (let* ((tqva path-q-local-tuple)
         (t (time tqva))         
         (O-tqva (osculating-path tqva)))   
    ((f-bar O-tqva) t)))

(define (F->C F)
  (define (f-bar q-prime)
    (define q
      (compose F (Gamma q-prime)))
    (Gamma q))
  (Gamma-bar f-bar))

(define (F->v F)
  (compose velocity (F->C F)))

(show-expression 
 ((F->v p->r)
  (->local 't
           (up 'r 'theta)
           (up 'rdot 'thetadot))))

— Me@2024-09-13 07:17:24 PM

Detexify

Posted on September 13, 2024 by rpflee

2024.09.13 Friday ACHK

Split-brain, 2.3

Posted on September 12, 2024 by rpflee

montyf on May 13, 2018 | parent | next [–]

Otherwise though, I found that his personas largely reflected his current emotional state. One persona would almost always be “out” when he was upset with a friend, or struggling with some stress. Another was more childlike and playful. He described them as always there, and even though he appeared to allow one of them to be “in charge” as he put it, he said the voices were always in the back of his mind, directing his thoughts.

That doesn’t sound like a disorder. The abnormal thing about your friend is that he is more aware than the average person.

The illusion of a single, unified self is just that. When you talk to people in your dreams, who are you talking to? Their responses are quite intelligent, if you listen. When you say embarrassing things under anesthesia with no memory of it, who said those things? When you get irrationally angry, who the hell is it that comes out? Same with the alcoholic who swears one day he’ll quit and forgets about it the next day — different people.

Moreover, there are huge areas of your brain that work quietly and subconsciously (from your perspective). Not only are they conscious, they’re smarter than you. They prove it through intuition and gut feeling, if you listen.

while his condition lent itself to occasional mood swings, he made a point of respecting his voices, allowed them to become a part of him, and I feel had largely learned to cope.

I’m sure you didn’t mean it this way, but I found your tone somewhat condescending. It’s not his alternate selves that he needs to “cope” with, but society’s notion that something is wrong with him. This might sound too new-age for people, but the more aware you are, the more you realize just how sick everyone else is. We literally have conscious beings imprisoned within us.

He had figured himself out for better or worse, …

Probably figured himself out better than the rest of us ever will.

— The Sound of Madness

— Hacker News

2024.09.12 Thursday ACHK

一見中情

Posted on September 11, 2024 by rpflee

這段改編自 2021 年 12 月 16 日的對話。

任何情況下，「一見鍾情」都應視為假。正如，雖然有人可以靠，彩票中頭獎為生，但機會微乎其微，不應考慮。

視「一見鍾情」為真，是不負責任的，因為：

一、你還不知道，她的性格如何。

二、你未得對方同意，就擅自標籤對方，為自己的真命天女。

一見鍾情者，一見終情也。

假設你還單身，正確的心態是，當有「一見鍾情」這錯覺時，你就企圖去認識對方。如果對方同意，就開始交往作朋友。至於相處後，會否不斷互相吸引，從而發展至情侶，則受客觀因素的影響，不受任何一方之控制。

— Me@2024-09-10 07:49:01 PM

Euler problem 22.2

Posted on September 10, 2024 by rpflee

import Data.Char (ord)
import Data.List (sort)

score word
  = sum $ map (\char -> ord char - ord 'A' + 1) word

nameScore filename = do
  content <- readFile filename
  let    
    names = read $ "[" ++ content ++ "]"       
  return $
    sum $ zipWith (*) [1..] $ map score $ sort names

λ> nameScore "names.txt"
871198282
λ>

— Me@2024-09-09 03:40:11 PM

Quick Calculation 13.3.0

Posted on September 8, 2024 by rpflee

A First Course in String Theory

Physical Meaning of Virasoro Operators

The Virasoro operators are fundamental in the context of two-dimensional conformal field theory (CFT) and string theory. They arise from the study of the symmetries of two-dimensional surfaces, particularly in how these surfaces can be mapped conformally (i.e., preserving angles) onto one another.

Conformal Symmetry

The Virasoro operators are associated with the Virasoro algebra, which is an infinite-dimensional Lie algebra that extends the algebra of diffeomorphisms on a circle. This algebra captures the symmetries of two-dimensional conformal transformations. In physical terms, these transformations are crucial for understanding how physical theories behave under changes of coordinates on the worldsheet of strings or in two-dimensional quantum field theories.

Role in String Theory

In string theory, the Virasoro operators are derived from the quantization of the string’s motion. They correspond to the modes of oscillation of the string and are denoted as $L_n$ . The operator $L_0$ is particularly significant because it acts as the Hamiltonian for the system, determining the energy levels of the string states. The commutation relations among these operators encode important physical information, such as the constraints on physical states (the Virasoro constraints) that must be satisfied for a consistent theory.

Physical States and Constraints

The Virasoro constraints arise from the requirement that physical states must be invariant under the action of the Virasoro operators. This leads to the notion of physical states in the theory, which are those that satisfy the condition $L_n | \text{physical state} \rangle =0$ for $n>0$ . This condition ensures that the states are not affected by the higher modes of the Virasoro algebra, which correspond to unphysical degrees of freedom.

Applications in CFT

In the context of conformal field theory, the Virasoro operators play a crucial role in the conformal bootstrap approach, where they help in constructing correlation functions and understanding the structure of the theory. The Virasoro conformal blocks, which are special functions derived from these operators, are essential for calculating observables in CFTs.

Summary

In summary, the Virasoro operators are central to the mathematical formulation of two-dimensional conformal field theories and string theory. They encapsulate the symmetries of these theories, dictate the physical state conditions, and facilitate the computation of physical observables. Their significance extends beyond mathematics into the core of theoretical physics, influencing our understanding of fundamental interactions and the structure of spacetime.

— AI

2024.09.08 Sunday ACHK

Chrono Cross 3

Posted on September 7, 2024 by rpflee

Chrono Trigger 5

Master Gogh is an unfortunate man.

In his youth, the master was a promising artist, but since her ladyship had a frail constitution, she fell ill after the birth of the young master.

Back then, the master was very poor… Without money to buy medicine, he could only watch her ladyship pass away. He changed that day.

He started working constantly, without a moment’s rest.

My master’s devotion to collecting all those works of art must be due to dreams he couldn’t fulfill when he was younger.

— Chrono Cross

2024.09.07 Saturday ACHK

DSE 數學補邊個好？（神器三）

Posted on September 6, 2024 by rpflee

這篇文章繼續討論，選擇數學補習老師的一些準則。而這裡的數學，是指香港中學文憑試的數學科，包括核心，M1 及 M2。

那樣，數學科補習界如果有 Ken Chan，他又必須具備，什麼條件呢？

上文提到：

1. 背誦篇

理科是文科，但不是一般的文科，而是極超巨大級的文科。

1.1 他會提醒你，數學其實是，極超大量的背誦。

……

1.2 極超大量的背誦，來自極端深刻的理解。

……

2. 年月時間篇：歷屆試題

2.1 他會提議你，考試前要操練 10 年、20 年或 30 年的歷屆試題。

……

2.1.2 他會告訴你「八二法則」——頭八十分，只需兩成的付出；而尾二十分，則需佔八成的努力。亦即是話，付出和收穫，不成比例。

原因很簡單：由 80 分至 100 分的那 20 分，是最深的題目。所以，奪 B (5*) 相比取 A (5**)，只需一半時間，因為，凡是最深，最花時間的部分，你都可以忽略。同理，奪 C (5) 相比取 B (5*)，亦只需一半時間。

具體執行的版本是，計劃奪 A 的話，就操練 30 年的歷屆試題；奪 B 就 15 年；奪 C 則 8 年。

留意，還有兩年時間準備的話，你可以計劃奪取滿分，那樣，你可以使用「迷宮直升法」，而毋須這個「八二法則」。（「迷宮直升法」在之後會講。）

2.2 最近五年的題目，則從來不要看，因為你要保留它們，在公開試前的休假月時，來作模擬考試用。

……

2.3 最近五年以前的題目，則可以在課程的那三年間，隨時隨地做。

留意，做試題有兩大主要方法：「分課題」或「分年份」。「分課題」的好處是，可以在你新學一課時，迅速釐清新概念，尤其是用多項選擇題時。「分年份」的好處是，每年的題目包括，幾乎所有課題。那可以避免，你在練習一課時，就忘記了上幾課。所有的章節，你將有永久的記憶。

2.4 而歷屆試題的來源是：最近五年的，可從考試局購買；之前的，共有幾十年的題目，則可從香港公共圖書館的網站，免費觀賞。

3. 星期時間篇

……

4. 分秒時間篇

……

— Me@2024-09-03 09:07:17 AM

Where are you? 2.2

Posted on September 5, 2024 by rpflee

Utopia | 何有之鄉, 2.2

所以，選擇工作職位的原則，並不是找「理想工作」，而是找厭惡得來，仍然可以遷就到的工作。換句話說，排除那些「不可能遷就到」的工作便行。

沒有無刺的玫瑰
但有很多沒有玟瑰的刺

— 叔本華

不可能遷就到，而必須辭職的例子有：

一、薪金低到連正常的，衣食往行也不夠。

二、沒有放工時間。

我以前企圖保留一份職位，心想：「晚一點回家沒所謂。只要每晚回家後到睡覺前，有兩小時的個人時間就可以，我就不會辭職。」但是竟然沒有。有一晚未能回家，在晚上十一時多，仍然在辦工室批改試卷。

三、上司的指令自相矛盾，而自己卻不知道。

我做電腦程式員時，有一次，上司要我為一個程式，加一個功能（甲）。那功能其實完全沒有好處。如果上司堅持的話，加也無妨。但是，功能甲會破壞，那程式的主功能。落實功能甲的話，那個程式就再，不能正常運作了。我花了很多心機，跟上司解釋，他也不明白。我唯有離開。

四、上司的指令構成，任何人的生命安全威脅。

有一次，上司說，可能有一個新項目，叫我研究一下：可不可以透過購買，市面上現成的氣體感應器，來建構一個自動化的，毒氣濃度偵測安全系統。我心想：「不行呀！公司沒有毒氣專家，可以給我諮詢；又沒有電子專家。」

除了微觀短期，可不可遷就外，還要考慮宏觀長期。去或留的大原則是，你要問，究竟該職位，是令你的財富穩步上升，還是穩步下降。

— Me@2024-09-05 11:20:51 AM

Euler problem 22.1

Posted on September 4, 2024 by rpflee

(ql:quickload "str")

(defun score (word)
  (loop :for char :across word
        sum (+ (- (char-code (char-upcase char))
                  (char-code #\A))
               1)))

(defun remove-quote (name)
  (remove #\" name))

(defmacro zipWith (f xs ys)
  `(mapcar ,f ,xs ,ys))

(defmacro range (max &key (min 0) (step 1))
  `(loop :for n :from ,min :below ,max :by ,step
        collect n))

(defun name-scores (filename)
  (with-open-file (stream filename)
    (let* ((names (read-line stream))
           (name-list-q (str:split #\, names))
           (name-list (mapcar #'remove-quote name-list-q))
           (sorted-name-list (sort name-list #'string<))
           (score-list (mapcar #'score sorted-name-list))
           (n-list (range (1+ (length score-list)) :min 1))
           (i-score-list (zipWith #'* n-list score-list)))
      (reduce #'+ i-score-list))))

(name-scores "names.txt")

; SLIME 2.28To load "str":
  Load 1 ASDF system:
    str
; Loading "str"
...

CL-USER> (name-scores "names.txt")
871198282
CL-USER>

— Me@2024-09-04 10:43:19 AM

3 Vector Fields and One-Form Fields, 3.2

Posted on September 3, 2024 by rpflee

Functional Differential Geometry

3.2.1

Instead, the ordinary directional derivative is

$\displaystyle{(Df(x)) \Delta x}$

or

$\displaystyle{\begin{aligned} D_{\mathbf{v}}(f) &= \frac{\left(\delta f\right)_{\mathbf{v}}}{|\mathbf{v}|} &= \left(\nabla f\right) \cdot \hat{\mathbf{v}} \\ \end{aligned}}$

3.2.2

The first generalization of directional derivative is replacing $\displaystyle{\Delta x}$ , a vector independent of $\displaystyle{x}$ , with $\displaystyle{b(x)}$ , a vector function of $\displaystyle{x}$ .

1. Note that

$\displaystyle{(Df(x)) \Delta x \ne D_{\mathbf{v}}(f)}$

Instead,

$\displaystyle{\begin{aligned} (Df(x)) \Delta x &\approx \Delta f \\ (Df(x)) &= D_{\mathbf{v}}(f) \\ \end{aligned}}$

2. The physical meaning of $\displaystyle{b(x)}$ is given by:

p. 25

The function $b$ is the coefficient function for the vector field $v$ . It provides a scale factor for the component in each coordinate direction.

(define (components->vector-field components coordsys)
  (define (v f)
    (compose (* (D (compose f (point coordsys)))
                components)
             (chart coordsys)))
  (procedure->vector-field v))

(define R2->R (-> (UP Real Real) Real))

(define R2-rect-chi-inverse
  (point R2-rect))

(define R2-rect-point
  (R2-rect-chi-inverse (up 'x_0 'y_0)))

(define v
  (components->vector-field
   (up (literal-function 'b^0 R2->R)
       (literal-function 'b^1 R2->R))
   R2-rect))

((v (literal-manifold-function 'f_rect R2-rect))
 R2-rect-point)

(define v
  (literal-vector-field 'b R2-rect))

((v (literal-manifold-function 'f_rect R2-rect))
 R2-rect-point)

4. Eq. (3.7):

$\displaystyle{\text{v}(\chi)(\chi^{-1}(x)) = b_{\chi, \text{v}} (x)}$

p. 25

The vector field $\displaystyle{ \textbf{v} }$ has a coordinate representation $\displaystyle{ v}$ :

$\displaystyle{ \begin{aligned} \textbf{v}(\text{f})(\textbf{m}) &= D( \textbf{f} \circ \chi^{-1})(\chi(\mathbf{m})) b(\chi(\mathbf{m})) \\ &= Df(x) b(x) \\ &= v(f)(x), \end{aligned} }$

with the definitions $\displaystyle{ f = \mathbf{f} \circ \chi^{-1} }$ and $\displaystyle{ x = \chi (\mathbf{m}) }$ .

(define (coordinatize v coordsys)
  (define ((coordinatized-v f) x)
    (let ((b (compose (v (chart coordsys))
                      (point coordsys))))
      (* ((D f) x) (b x))))
  (make-operator coordinatized-v))

(((coordinatize v R2-rect) (literal-function 'f_rect R2->R)) (up 'x_0 'y_0))

— Me@2024-09-02 01:06:17 AM

Physics Town

continues

Monthly Archives: September 2024