User:Silly rabbit/Sandbox

~~User:Silly rabbit/Sandbox/Spinors~~

~~User:Silly rabbit/Sandbox/Cartan affine connection~~

~~User:Silly rabbit/Sandbox/List of missing differential geometry topics~~

~~User:Silly rabbit/Sandbox/Differentiable manifold~~

~~User:Silly rabbit/Sandbox/Exterior algebra~~

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Proof of (1)

Parallel transport on the sphere.

$\begin{matrix} V & \to & C(q) \\ \downarrow & \swarrow & \\ A && \end{matrix}$

${{a \times n = } \atop {\ }} {{\underbrace{a + \cdots + a}} \atop n}$

Doesn't work, &Ropf;ⁿ &Ropf;ℝn ℝℜℛ

Proof of (1)

$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) \,$	$= a_i \sigma_i b_j \sigma_j \,$
	$= a_i b_j \sigma_i \sigma_j \,$
	$= a_i b_j \left( \delta_{ij} + i \varepsilon_{ijk} \sigma_k \right) \,$
	$= a_i b_j \delta_{ij} + i \sigma_k \varepsilon_{ijk} a_i b_j \,$
	$= \vec{a} \cdot \vec{b} + i \vec{\sigma} \cdot ( \vec{a} \times \vec{b} )\,$

Proof of (2)

First notice that for even powers

$(\hat{n} \cdot \vec{\sigma})^{2n} = I \,$

but for odd powers

$(\hat{n} \cdot \vec{\sigma})^{2n+1} = \hat{n} \cdot \vec{\sigma} \,$

Combine these two facts with the knowledge of the relation of the exponential to sine and cosine:

$e^{ix} \,$	$= \sum_{n=0}^\infty{\frac{i^n x^n}{n!}} \,$
	$= \sum_{n=0}^\infty{\frac{(-1)^n x^{2n}}{2n!}} + i\sum_{n=0}^\infty{\frac{(-1)^n x^{2n+1}}{(2n+1)!}} \,$

Which, when we use $x = a (\hat{n} \cdot \sigma) \,$ gives us

$= \sum_{n=0}^\infty{\frac{(-1)^n (a\hat{n}\cdot \sigma)^{2n}}{2n!}} + i\sum_{n=0}^\infty{\frac{(-1)^n (a\hat{n}\cdot \sigma)^{2n+1}}{(2n+1)!}} \,$

$= \sum_{n=0}^\infty{\frac{(-1)^n a^{2n}}{2n!}} + i (\hat{n}\cdot \sigma) \sum_{n=0}^\infty{\frac{(-1)^n a^{2n+1}}{(2n+1)!}} \,$

The sum on the left is cosine, and the sum on the right is sine so finally,

$e^{i a(\hat{n} \cdot \vec{\sigma})} = \cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} \,$

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A ration of food. (Silly rabbit 15:17, 20 June 2007 (UTC))