Wait this is illegal
In physics, this is why we call the derivative operator the "generator of translations"pic.twitter.com/iqJoydnylt
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Why's that? This is just the definition of the exponential of a linear operator
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You have it backwards! Translations give us enough to define derivatives...
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I’m going to have to disagree
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Why don't we call it the "operator of translations"?
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Because the derivative is in the Lie algebra and the exponential is the Lie group (which actually does the translating)
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And sometimes (i.e when f is analytic and epsilon is sufficiently small) it's even true.
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It's true in the sense that the differentiation operator is the vector in the Lie algebra of the general linear group that generates the one parameter subgroup of translations.
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Recall the definition of a taylor series about a point
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