3 Vector Fields and One-Form Fields, 1

Posted on September 30, 2023 by rpflee

Chain Rule of Differentiation, 2

Functional Differential Geometry

p. 21

In multiple dimensions the derivative of a function is the multiplier for the best linear approximation of the function at each argument point:

$\displaystyle{f(x + \Delta x) \approx f(x) + (Df(x)) \Delta x}$

In other words:

$\displaystyle{ \begin{aligned} f(x + \Delta x) &= f(x) + a_1 (\Delta x) + a_2 (\Delta x)^2 + \cdots \\ (Df(x)) &= a_1 \\ \end{aligned} }$

The derivative $Df(x)$ is independent of $\Delta x$ .

… the product $\displaystyle{(Df(x))\Delta x}$ is invariant under change of coordinates …

$\displaystyle{(Df(x)) \Delta x}$ is the directional derivative of $\displaystyle{f}$ at $\displaystyle{x}$ with respect to $\displaystyle{\Delta x}$ .

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

Instead, $\displaystyle{(Df(x)) b(x)}$ is a generalization of $\displaystyle{(Df(x)) \Delta x}$ .

— Me@2023-09-29 11:48:36 PM

@dialectphilosophy, 1.2

Posted on September 24, 2023 by rpflee

. . .

The meaning of “velocity is relative” is:

For example, within a car, you cannot know its velocity relative to the ground without seeing outside. In other words,

You cannot know the velocity of A (car) relative to the B (ground) without seeing B (outside).

It is because any two individual objects within the car, if both initially at rest relative to the car, have a constant separation. Also, any individual objects inside the car and any point of the car itself has a constant separation. And here is the proof:

The separation between any 2 objects within the car is

$\displaystyle{ \begin{aligned} &x_2(t) - x_1(t) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &= \Bigl( x_2(0) + v_2 t \Bigr) - \Bigl(x_1(0) + v_1 t \Bigr) \\ \end{aligned}}$ ,

where $\displaystyle{v_1}$ and $\displaystyle{v_2}$ are velocities with respect to the ground of object 1 and object 2 respectively. If the two velocities have the same value,

$\displaystyle{ \begin{aligned} &x_2(t) - x_1(t) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &= x_2(0) - x_1(0) \\ \end{aligned}}$

As long as object 1 and object 2 have the same velocity-relative-to-the-ground as that of the car, $\displaystyle{v}$ , i.e.

$\displaystyle{v_1=v_2=v}$ ,

no matter what value $\displaystyle{v}$ has, the distance between object 1 and object 2 is always constant. In other words, you cannot deduce the value of the $\displaystyle{v}$ by observing the separation changes between any two objects/points within the car.

Even in another case where $\displaystyle{v_1 \ne v_2}$ ,

…

— Me@2023-08-07 05:56:31 AM

Everything 4

Posted on September 24, 2023 by rpflee

It’s only after we’ve lost everything that we’re free to do anything.

— Chuck Palahniuk

— Fight Club

2023.09.24 Sunday ACHK

守時 6.3

Posted on September 22, 2023 by rpflee

朋友同事 6.3

這段改編自 2010 年 4 月 24 日的對話。

所謂的「讀心術」，其實即是「觀人於微」，例如：

1. 對方是否守時；

2. 在公文中，文法正不正確；

3. 說話是否真誠，即有沒有虛假，或含糊其辭；等等。

之前我無留意，然後中招，中得好緊要。

幾年前還會忍受，自己或他人的缺點。現在則視乎，那缺點是否人格問題。人格問題不能忍受，其他可以。

（安：你的「忍受」是什麼？是指「不表達不滿」嗎？）

大概而言，「忍受甲」是指，繼續與甲相處。

詳細來說，以前覺得：

「人總有缺點咖啦……」　　

「遲到沒有所謂啦……」　　

「說謊沒有所謂啦……」　　

意思是，如果介意缺點的話，大概沒有任何朋友。

這個想法，大錯特錯，因為即使「任何人必有缺點」，缺點其實可分成「被動缺點」和「主動缺點」。「被動缺點」可以避開；「主動缺點」則會主動攻擊你。

主動缺點，亦稱「人格問題」。其大鑊之處在於，你想相處，也相處不來，甚至相處不到。例如遲到失約，你想見他也見不到。

— Me@2023-09-18 11:12:31 AM

Euler problem 16.1

Posted on September 17, 2023 by rpflee

(defun digits (n)
  (labels
      ((f-iter (m acc)
         (cond ((= m 0) acc)
               ('t (f-iter (floor m 10)
                           (cons (mod m 10) acc))))))
    (f-iter n NIL)))

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

(sum (digits (expt 2 1000)))

— Me@2023-09-17 01:04:49 AM

Ex 1.30 Driven spherical pendulum, 2

Posted on September 15, 2023 by rpflee

Structure and Interpretation of Classical Mechanics

…

What symmetry(ies) can you have?

Find coordinates that express the symmetry.

What is conserved?

Give [the] analytic expression(s) for the conserved quantity(ies).

~~~

[guess]

…

(define ((F->C F) local)
  (->local (time local)
           (F local)
           (+ (((partial 0) F) local)
              (* (((partial 1) F) local)
                 (velocity local)))))

(define (L-driven m R qs U)
  (compose
   (L-rect m R qs U)
   (F->C (sf->rf qs))))

(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   ((compose L (Gamma q)) 't)))

(+ (* 1/2 m (expt (r t) 2)
            (expt (sin (theta t)) 2)
            (expt ((D phi) t) 2))
   (* -1 m ((D z_s) t)
           (r t)
           (sin (theta t))
           ((D theta) t))
   (* 1/2 m (expt (r t) 2)
            (expt ((D theta) t) 2))
   (* -1 g m (cos (theta t)) (r t))
   (* m ((D r) t)
        ((D z_s) t)
        (cos (theta t)))
   (* (expt R 2) (lambda t))
   (* -1 g m (z_s t))
   (* 1/2 m (expt ((D r) t) 2))
   (* 1/2 m (expt ((D z_s) t) 2))
   (* -1 (lambda t) (expt (r t) 2)))

$\displaystyle{ \begin{aligned} L &= \frac{1}{2} m \left( \dot r^2 + r^2 \dot \theta^2 + r^2 (\sin \theta)^2 \dot \phi^2 + \dot z_s^2 \right) \\ & - m g \left( r \cos \theta + z_s(t) \right) - m \dot z_s \left( r \dot \theta \sin \theta - \dot r \cos \theta \right) \\ & - \lambda(t) (r^2 - R^2) \\ \end{aligned} }$

(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   (((Lagrange-equations L) q) 't)))

(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   ((compose (Lagrangian->energy L) (Gamma q)) 't)))

(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   ((compose ((partial 1) L) (Gamma q)) 't)))

The $\displaystyle{\varphi}$ component of the force is zero because $\displaystyle{\varphi}$ does not appear in the Lagrangian (it is a cyclic variable). The corresponding momentum component is conserved. Compute the momenta:

(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   ((compose ((partial 2) L) (Gamma q)) 't)))

[guess]

— Me@2023-08-20 05:02:09 PM

Pointer

Posted on September 14, 2023 by rpflee

Paradox feeling arises when you try to substitute the representation of the whole sentence by that whole sentence in order to get its meaning, which does not exist. For example, in the sentence “this sentence is wrong“, if you try to substitute “this sentence” by “this sentence is wrong“, you will get

”

“

“

“

… is wrong

” is wrong

” is wrong

” is wrong

“, which is infinitely long.

The sentence “this sentence is wrong” can exist because “this sentence” is just a representation of “this sentence is wrong“, not really the whole sentence itself.

— Me@2015-09-20 09:22:21 AM

Ask HN: Why is nearing completion so demotivating?

Posted on September 13, 2023 by rpflee

mikekchar on May 23, 2018 | next [–]

When your project is finished, the dream is dead and the reality is born. The death of a dream is like the death of a friend. It’s probably been with you for a long time — longer even than the length of the project. A dream is the manifestation of what’s possible. When it is over, the possible diminishes very quickly and you are left with what actually is. Will people respond well to your project — in the dream stage it is possible; everything is possible. In the reality stage, it will only be what it is.

So while it’s common to think of a release as a birth of something new, realise that you also have a significant loss. You will mourn that loss. Give yourself some emotional space to deal with the mourning.

— Hacker News

2023.09.13 Wednesday ACHK

拍拖八年

Posted on September 12, 2023 by rpflee

未來騙局, 3

這段改編自 2023 年 06 月 20 日的對話。

— Dilbert

— Scott Adams

只有本來，沒有未來。

本來無未來。

承諾皆廢話，未來皆騙局。

甲：他們拍了拖八年 …

乙：你說下半句前，我估計，那一定是悲劇，即是結緍不成。

甲：你為何會知道？

乙：拍拖的目的是，透過日常相處，判別對方是否，自己的結緍對象。這個判別的過程，合理時段是數月至兩年之內。須知道，人的「適合生育年齡」，即是所謂的「青春」，就只有那幾年。所以，所謂的「愛情長跑」，實質是「愛情騙局」。

承諾 = 騙局

如果不是結婚對象，為何不立刻分手？

相反情形，既然雙方確認了，是對方的終身，為何不立刻結緍，而要等五年後？

甲：因為現在不夠錢？

乙：為何你會覺得，現在不夠錢的話，五年就會夠錢？

為何結婚需要大量金錢？婚姻登記處的手續費，真的那麼貴嗎？

生兒育女需要錢，結婚本身不需要。

承諾 ~ 行騙

信任 ~ 受騙

— Me@2023-06-23 04:07:36 PM

— Me@2023-08-12 08:22:18 PM

Common Lisp calculator, 1

Posted on September 9, 2023 by rpflee

To run the following code, you have to install the Quicklisp library manager by following the steps in my blog post titled cmu-infix.

(ql:quickload :infix-math)

(asdf:load-system "infix-math/calc")

(infix-math/calc:calc)

This problem of using this calculator is that spaces are required both before and after an operator:

wrong:

1+1

correct:

1 + 1

— Me@2022-12-26 10:19:05 AM

3.6 Analytic continuation for gamma function, 1

Posted on September 8, 2023 by rpflee

A First Course in String Theory

…

$\displaystyle{ \begin{aligned} \Gamma (z) = &\int_{0}^{1}dt~t^{z-1} \left( e^{-t} - \sum_{n=0}^N \frac{(-t)^n}{n!} \right) \\ &+ \sum_{n=0}^N \frac{(-t)^n}{n!} \frac{1}{z+n} + \int_1^\infty dt~e^{-t}t^{z-1} \\ \end{aligned} \\ }$

Explain why the above right-hand side is well defined for $\displaystyle{ \Re (z) > - N - 1 }$ .

…

$\displaystyle{ \begin{aligned} \Gamma (z) &= \int_{0}^{\infty}t^{z-1}e^{-t}dt \\ &= - \int_{0}^{\infty}t^{z-1}de^{-t} \\ &= - \left[ t^{z-1} e^{-t} \right]_{0}^{\infty} + (z-1) \int_{0}^{\infty} t^{z-2}e^{-t} dt \\ &= - \left[ \lim_{t \to \infty} t^{z-1} e^{-t} - 0^{z-1} e^{-0} \right] + (z-1) \int_{0}^{\infty} t^{z-2}e^{-t} dt \\ \end{aligned} \\ }$

Note that this can only prove that for convergence, it may be necessary to have $\displaystyle{ \Re (z) > 1 }$ , but not $\displaystyle{ \Re (z) > 0 }$ .

Also, the condition $\displaystyle{ \Re (z) > 1 }$ may be not necessary at all, because an infinity in one term may be cancelled out by another infinity in a latter term.

$\displaystyle{ \begin{aligned} \Gamma (z) &= \int_{0}^{1}t^{z-1}e^{-t}dt + \int_{1}^{\infty}t^{z-1}e^{-t}dt \\\\ \end{aligned} \\ }$

Prove that $\displaystyle{ \int_{1}^{\infty}t^{z-1}e^{-t}dt }$ converges.

Since $\displaystyle{ \begin{aligned} e^t &= \sum_{s=0}^\infty \frac{t^s}{s!} \\ \end{aligned} \\ }$ , for any integer $r \ge 0$ and any real $t > 0$ :

$\displaystyle{ \begin{aligned} e^t &\ge \frac{t^r}{r!} \\ \end{aligned}~~~}$

…

— Me@2023-09-07 07:38:17 PM

Interstellar 4

Posted on September 7, 2023 by rpflee

Yes. It is Nolan. But it is drama.

Wormhole cannot be passed, physically. The past cannot be influenced by now, logically.

— Me@2015-09-28 03:23:32 PM

iPod

Posted on September 7, 2023 by rpflee

outlook to ipod notes testing

~ 2011

反情感勒索 2.2

Posted on September 7, 2023 by rpflee

這段改編自 2021 年 12 月 16 日的對話。

不要強勢, 不要弱勢, 要平等.

— Me@2009.10.06

You’ve found market price when buyers complain but still pay.

— Paul Graham

如果你知己知彼，即是根據你對自己，和對該職位的理解，你知道你是稱職有餘的話，提出薪金要求時，就千萬不要說「可不可以」、「介不介意」等，例如：

「每月薪金可不可以是三萬元呢？」

那是錯的。

記住，你必須做的，是「要求」，不是「請求」。正確的版本是：　

「我對每月薪金的要求是三萬元。你可以花點時間去考慮。決定同意的話，才再聯絡我。」

如果對方說：　

「二萬八千元可以嗎？」

你則用剛才之話回答：

「你可以花點時間去考慮。決定同意的話，才再聯絡我。」

記住，你一讓步的話，之後在各方面，就會再被迫，不斷讓步。

— Me@2023-07-29 09:15:40 AM

— Me@2023-09-07 09:08:45 AM

… the world only makes sense when you force it to.

— Batman

— The Dark Knight Returns

Standard Model equation

Posted on September 6, 2023 by rpflee

Euler problem 15.2

factorial n = facIter n 1
  where
    facIter m acc =
      if m == 0
      then acc
      else facIter (m - 1) (m * acc)

binomial n r = bIter n 1 `div` factorial r 
  where
    bIter m acc = 
      if m < n - r + 1
      then acc
      else bIter (m - 1) (m * acc)

f n = product [1..n]

b n r = product [(n-r+1)..n] `div` f r

— Me@2023-09-06 08:08:35 PM

Lagrange’s equations Debugged

Posted on September 5, 2023 by rpflee

$\displaystyle{\frac{d}{dt} \left( \frac{\partial L(t, q, \dot q)}{\partial \dot q} \Bigg|_{\begin{aligned} q &= w(t) \\ \dot q &= \frac{d w(t)}{dt} \\ \end{aligned}} \right) - \frac{\partial L}{\partial q}\Bigg|_{\begin{aligned} q &= w(t) \\ \dot q &= \frac{d w(t)}{dt} \\ \end{aligned}} = 0}$

This equation is complete. It has meaning independent of the context and there is nothing left to the imagination. The earlier equations require the reader to fill in lots of detail that is implicit in the context. They do not have a clear meaning independent of the context.

$\displaystyle{\frac{d}{dt} \frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q} = 0}$

— Functional Differential Geometry

2023.09.05 Tuesday ACHK

@dialectphilosophy, 1.1

Posted on September 4, 2023 by rpflee

This post is a debug of dialectphilosophy‘s discussion of relativity.

The word “motion” in physics context should mean only velocity, not displacement, nor acceleration.

Therefore “motion is relative” should only mean “velocity is relative”. Einstein’s goal to prove that “all motion derivatives, including acceleration, are relative” is silly.
Velocity and acceleration are independent variables. “Whether velocity is relative or not” is not relevant to “whether acceleration is relative or not“.

In other words, you cannot use the value of velocity at a moment to derive the acceleration value at that moment. Fundamentally, it is due to fact that $v$ and $\Delta v$ are independent.

…

— Me@2023-08-07 05:56:31 AM

超級撒亞人三型態

Posted on September 1, 2023 by rpflee

A: 明明追到佢之前係善解人意，又剛強獨立，又唔會亂發小姐脾氣，咩都可以攤開黎講，以為自己搵到 the one 。

追到之後反而成日都體虛體弱，成日頭暈唔舒服，借啲易又發脾氣，咩都要人哄，又收收埋埋，性格變得快過天氣，上午可以無咩事，下午就哭住咁話你做咩無做咩咩咩，講嘢大聲啲又話你做咩兇佢…….

我都係想要同一個正常少少既女仔，有一段正常少少既戀愛啫……點解咁難:~(

B: 你以為咁就完?

結埋婚嘅話, 啲女人仲識得二段變身, 好似超級撒亞人咁, 所有嘢都會再升級

生埋小朋友就係超西3

利申: 俾超西3屈埋牆角撳住砌緊:-(:-(:-(

2023.09.01 Friday ACHK

Physics Town

continues

Monthly Archives: September 2023

3 Vector Fields and One-Form Fields, 1

@dialectphilosophy, 1.2

Everything 4

守時 6.3

Euler problem 16.1

Ex 1.30 Driven spherical pendulum, 2

Pointer

Ask HN: Why is nearing completion so demotivating?

拍拖八年

Common Lisp calculator, 1

3.6 Analytic continuation for gamma function, 1

Interstellar 4

iPod

反情感勒索 2.2

Standard Model equation

Lagrange’s equations Debugged

@dialectphilosophy, 1.1

超級撒亞人三型態