Gradient

Assume \((x, y)\) represents the position of an object and \(f(x,y)\) is a scalar field on the \(x\)-\(y\) plane.

Then \(\frac{\partial f}{\partial x}\) represents the change of \(f\) per unit length along the positive \(x\) direction. In other words, it is the spatial rate of change of \(f\) along the \(x\) direction.

Similarly, derivative \(\frac{\partial f}{\partial y}\) represents the spatial rate of change of \(f\) along the \(y\) direction.

For an arbitrary direction, due to the nature of displacement, the change of \(f\) is\[\delta f = \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial x} \delta y\]when the object has finished moving \(\delta x\) in \(x\) direction and then \(\delta y\) in \(y\) direction. 

Then, the spatial rate of change of \(f\) is\[\begin{align}  &\frac{\delta f}{\sqrt{(\delta x)^2 + (\delta y)^2}} \\
&= \frac{\partial f}{\partial x} \frac{\delta x}{\sqrt{(\delta x)^2 + (\delta y)^2}}
+ \frac{\partial f}{\partial x} \frac{\delta y}{\sqrt{(\delta x)^2 + (\delta y)^2}} \end{align}\]

For simplicity, denote the resultant displacement as \(\mathbf{v}\):\[\mathbf{v} = (\delta x, \delta y)\]and define \(\nabla f(x)\) as\[\nabla f(x) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)\]
Then, the change of the \(f\) due to the displacement \(\mathbf{v}\) is\[\begin{align} 
\left(\delta f\right)_{\mathbf{v}}
      &= \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial x} \delta y \\ 
      &= \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x}\right) \cdot (\delta x, \delta y) \\
      &= \left(\nabla f\right) \cdot \mathbf{v} \\
\end{align}\]So the spatial rate of change \(f\) along the direction of the vector \(\mathbf{v}\) is
\[\begin{align}
D_{\mathbf{v}}(f)
&= \frac{\left(\delta f\right)_{\mathbf{v}}}{|\mathbf{v}|} \\
&= \frac{\partial f}{\partial x} \frac{\delta x}{\sqrt{(\delta x)^2 + (\delta y)^2}} 
 + \frac{\partial f}{\partial x} \frac{\delta y}{\sqrt{(\delta x)^2 + (\delta y)^2}} \\
&= \left(\nabla f\right) \cdot \frac{\mathbf{v}}{|\mathbf{v}|} \\
&= \left(\nabla f\right) \cdot \hat{\mathbf{v}} \\
\end{align}\]
\(D_{\mathbf{v}}(f)\) is called directional derivative.

“In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.”

— Wikipedia on Directional derivative

— Me@2016-02-06 09:49:22 PM

2016.02.21 Sunday (c) All rights reserved by ACHK

注定外傳 2.4

Can it be Otherwise? 2.4 | The Beginning of Time, 7

因為沒有指定,追溯到哪一件事,或者哪一刻為止,所以討論會沒完沒了。

(問:不會沒完沒了呀。只會追溯到「時間的起點」。)

我們根本不知道,「時間的起點」(第一因)是怎樣的。那樣,我們又怎能夠,根據「時間的起點」,去判斷某一件事件,是不是注定的呢?

(問:可能可以。所謂「時間的起點」,其實就即是「宇宙的開端」。)

可以這樣說,因為「宇宙」這個詞語,就是指「所有事物」。所以,「時間起點」和「宇宙開端」,是同義詞。

(問:而物理學家知道,「字宙的開端」是「宇宙大爆炸」。所以我們知道,「時間的起點」,就是「宇宙大爆炸」。)

大概而言是。但是,嚴謹一點講:

1. 「宇宙大爆炸」是一件事件,有一個過程,並不是時間上的「一點」,所以不算是「起點」。「宇宙大爆炸這件事的開始那刻」才算是起點。

當然,「宇宙大爆炸這件事的開始那刻」太長太麻煩,可以用同義詞「宇宙開端」代替。但是,「宇宙開端」這四個字,太過空泛,沒有任何詳情。試想想,知道了「時間起點」就是「宇宙開端」,那又怎樣呢?

用「宇宙大爆炸這件事的開始那刻」,起碼可以知道,「宇宙開端」那一刻,開始發生的第一件事,是「宇宙大爆炸」。所以,如果又要細節,又要精簡,把「宇宙大爆炸這件事的開始那刻」,簡稱成「宇宙大爆炸」也無妨,只要上文下理足夠清晰,不會引起誤會就可以。

2. 物理學家根據愛因斯坦的「廣義相對論」推斷,「宇宙開端」那一刻,開始發生的第一件事,是「宇宙大爆炸」。所以,如果「廣義相對論」不正確,「宇宙大爆炸」就未必為真。

3. 即使「廣義相對論」是可信的,普朗克時期(Planck epoch),即是開端後的頭\(10^{−43}\)秒之內,以現時的物理知識,是處理不到的。所以,物理學家推斷不到,那段時間內,發生了什麼事。

如果要知道「普朗克時期」內,宇宙演變的詳情,物理學家就要先把「量子力學」和「廣義相對論」的矛盾化解。這個工序,學名是「把廣義相對論量子化」。

所以,討論任何問題,例如「某一件是否注定」時,即使有「推斷到時間起點」的企圖,也沒有可能做到,除非能夠把「量子力學」和「廣義相對論」合體。

我們至多只能追溯到,「普朗克時間」完結的那一刻,然後講一句:「再之前的,沒有資料」。

— Me@2016-02-15 07:04:56 PM

2016.02.15 Monday (c) All rights reserved by ACHK