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

# 注定外傳 2.4

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

（問：不會沒完沒了呀。只會追溯到「時間的起點」。）

（問：可能可以。所謂「時間的起點」，其實就即是「宇宙的開端」。）

（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」。）

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。

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。

— Me@2016-02-15 07:04:56 PM