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...
Penrose graphical notation seems to be a convenient way to do calculations involving tensors/ multilinear functions. However the wiki page does not actually tell us how to use the notation. The several references, especially ones with Penrose as author, must be good places to start. But it is no...
For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation. Here $\det$ denotes the determinant. Why can the antisymmetrizing bar be inserted in the middle because "there is already antisym...
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...
Consider a continuous time dynamical system where $$\dot x(t) = F(x(t))$$, where $x(t)$ is a coordinate vector of state and the right side of the equation $F$ is a non-linear smooth function. Let the state space be Euclidean. Let $S^t(x_0)$ be the position of the trajectory of $$\dot x(t)= F(x(t)...
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