Why do I see your tweet like this?pic.twitter.com/lthqFPHI7Q
Any function 𝑓(𝑥) can be split into an even-part and and odd-part. This fact is actually more useful than it may at first appear.pic.twitter.com/nJi2IVMNbX
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oh no:( I use unicode to do boldface and italics.. I guess I should stop that
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Same for tensors als with Glebsch-Gordon coefficients. Like "T_ij=T_[ij]+(T_(ij) -1/n (kronecker delta)_ij T) + 1/n (kronecker delta)_ij T" anti-symmetric [ ], symmetric ( ) and tracefree components (n=3 in 3-dimensions): or: 3×3=3+5+1 or 3×3=3+6. Same for higher order tensors.
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Of course we can extend this to higher rank tensors like: "T_ijk=T_(ijk)+mixed symmetric T_ijk compts +epsilon_[ijk]T" =symmetric+mixed symmetric + antisymmetric irreducible components etc. Where: epsilon is the permutation tensor (Levi-Civita).
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Exp(x) = (Exp(x) + Exp(-x))/2 + (Exp(x) - Exp(-x))/2 i.e Exp(x) = Cosh(x) + Sinh(x)
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I smell hyperbolics
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Moreover, such a decomposition is unique. That is, any vector space of functions is the (internal) direct sum of the subspace of even functions and the subspace of odd functions.
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That also looks a lot like the decomposition of the geometric product of two vectors into its scalar and bivector part.
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A good old-fashioned walk down memory lane with
@MAM1000W
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Woooow!! The applications could be endless. Nice!
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