28. Coordinate transform and derivative

This article contains additional information about coordinate transformation and fields.¹ The main goal is to understand the relationship between prime numbers, trigonometric functions, differentiation of trigonometric functions, and coordinate transformation. Unit vector differentiation is expressed as follows. Here, the unit vector, $ \hat{e}_r, \hat{e}_\theta$ represents the curvilinear coordinate system.

 

(1)

 

$(\hat{e}_r)'=(\sin^3\theta+\cos^3\theta)\hat{e}_r+(1+\cos^2\theta\sin\theta-\sin^2\theta\cos\theta)\hat{e}_\theta$

 

$(\hat{e}_\theta)'=-(1+\cos^2\theta\sin\theta-\sin^2\theta\cos\theta)\hat{e}_r+(\sin^3\theta+\cos^3\theta)\hat{e}_\theta$

 

 

If $\theta=\pi s/2((s-1)!+1)$, when $s$ is $2, 3$ or large, the equations (1) become like this.

 

$(\hat{e}_r)'=\hat{e}_r+\hat{e}_\theta, (e^{-\theta} \hat{e}_r)'= e^{-\theta}\hat{e}_\theta$

 

$(\hat{e}_\theta)'=-\hat{e}_r+\hat{e}_\theta, (e^{-\theta}\hat{e}_\theta)'=-e^{-\theta}\hat{e}_r$

 

 

$e^{-\theta}\hat{e}_r, e^{-\theta}\hat{e}_\theta$ can be regarded as another unit vectors. The differentiation that is performed at values equal to or less than $1$ unit can be expressed through coordinate transformation between trigonometric functions. From the perspective of number, differentiation is same to coordinate transformation when $s$ is a prime number. Accordingly, the scales of numbers are converted to fit the trigonometric functions. From the results so far, the $\Gamma(s)$ refers to mass, $r$ direction. Unit vectors analogy to $\zeta(s)$. This may be why natural fractal phenomena appear as sequences of complex numbers.

 

 

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^{1. Refer to Coordinate transformation and fields..}