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Q: Proving Tensor identities using Penrose diagrammatic notation (Page-322, Penrose, Road to Reality)
I want to show that for an anti-symmetric two upper index tensor $S^{ab}$, that:
$$Q^{bcd}=S^{a[b} \nabla_a S^{cd]}=0$$
I'm trying to figure out how to show prove the hint equality is true using the penrose diagrammatic notation. I've tried to make a diagram of it below:
Could someone explain me...
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In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several shapes linked together by lines. The notation has been studied extensively by Predrag Cvitanović, who used it, Feynman's diagrams and other related notations in developing birdtracks (a group-theoretical version of Feynman diagrams) to classify the classical Lie groups. Penrose's notation has also been generalized using representation theory to spin networks...
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I have been reading commutative algebra from lecture notes and I have some questions in a proof of a corollary of Hilbert Nullstellansatz.
Let R be a finitely generated k-algebra. Then for an ideal $I \subseteq R$, $\sqrt{I} =\cap M$, where M are maximal ideals containing I.
Proof: Clearly $\s...
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see Satz) is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz and several other important related theorems named after him (like Hilbert's basis theorem).
== Formulation ==
Let k be a field (such as the rational numbers) and K...
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