Ciro Santilli $$Sponsor Ciro$$ 中国独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件 邓家贵、低端人口、西藏骚乱
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# The derivative is the generator of the translation group

| 🗖 nosplit | ↑ parent "Translation group" | words: 206
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Take the group of all Translations in .
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Let's see how the generator of this group is the derivative operator: $$∂x∂​ (145)$$
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So let's take the exponential map: $$ex0​∂x∂​f(x)=(1+x0​∂x∂​+x02​∂x2∂2​+…)f(x) (146)$$ and we notice that this is exactly the Taylor series of around the identity element of the translation group, which is 0! Therefore, if behaves nicely enough, within some radius of convergence around the origin we have for finite : $$ex0​∂x∂​f(x)=f(x+x0​) (147)$$