Einstein Notation, Levi-Civita Symbol, and Maxwell Equation

by allenlu2007

 

Albert Einstein 有很多重要的物理發現: 相對論, 光電效應, 布朗運動。

除此之外,很多數學物理公式也是以他命名。一個用於 tensor 的是 Einstein notation.

or Einstein summation convention.   主要是簡化 summation over a set of indexed terms in a formula.

 

Einstein Summation Convention (wiki)

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一些 vector 的例子解釋 superscript (column) and subscript (row).

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Inner product

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Cross product (3-dimension)

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Matrix multiplication

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Trace

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Tensor

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Levi-Civita Symbol (wiki)

首先 Levi-Civita symbol 是 anti-symmetric symbol, or alternating symbol.  

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Two dimensional Levi-Civita symbol:

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Three dimensional Levi-Civita symbol:

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Four dimensional Levi-Civita symbol:

N dimensional Levi-Civita symbol:

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Vector Field 

Curl

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Curl of Cross Product

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Gradient, Div, Curl

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Maxwell Equation in Tensor Form

Maxwell equation 本身是二次 differential equation in potential form (𝛗 and 3D-A) and 一次 coupled differential equation in field form (E and B).

如果從 electromagnetic four-potential combined 𝛗 and 3D A into a 4D A, a differential 1-form 出發:

F = dA  (total differentiation)   F is a differential 2-form.

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先定義 F tensor in contravariant matrix form (upper index), and covariant matrix form (lower index)

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η 是 flat space metric tensor (狹義相對論)

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所以原來 4 個 Maxwell differential equations 可以改寫為 2 個 tensor differentiation equations. 

當然重點不是在於從 4 個變成 2 個。而是很容易引入 curved space-time (廣義相對論) into Maxwell equations!

 

Maxwell Equation in Curved Spacetime (wiki)

Maxwell equation can be generalized to curved space-time 只要把 partial derivative 改為 covariant derivatives.  另外 η (flat space metric tensor) 改為 g (curved spacetime metric tensor due to gravity in 廣義相對論).

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Lagrangian Formulation

另外角度是從 Lagrangian formulation –> Euler-Lagrange equation –> 得到 Maxwell Equation

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