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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf SOLUTIONS -- SET 12}\\%
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\noindent
Problem 39: Invariance under Gauge Transformations.
$$ F^{\alpha\beta} = 
   \partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}\,,~~~
   A'^{\,\alpha} = A^{\alpha} + \partial^{\alpha}\Lambda\,.  $$
Then,
$$ F'^{\,\alpha\beta} = 
  \partial^{\alpha}A'^{\,\beta}-\partial^{\beta}A'^{\,\alpha} =
  \partial^{\alpha}A^{\beta}+\partial^{\alpha}\partial^{\beta}\Lambda
 -\partial^{\beta}A^{\alpha}-\partial^{\beta}\partial^{\alpha}\Lambda\,.  
 = \partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha} 
 = F^{\alpha\beta}\,. $$
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