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I think it's not possible to relax the boundary condition where $\frac{\partial T}{\partial r}|_{r=a} = 0$ , because the temperature function is defined for $a\leq r\leq b$ and $0\leq z\leq L$. If you don't consider the boundary condition, you are eliminating the insulation on the inner surface of the cylinder, whereas the insulation on the inner side is ...

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Full field theories generally take up all of space. However, occasionally it is more convenient to work with sums instead of integrals. This can be done by artificially putting the system in a box. But when putting a system in a box you get edges and just like in any other theory you need to give the boundary conditions (the same is true for example in ...

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The problem is not proving $$n^a n^b n^c =0$$ But in proving $$n^a n^b n^c (\partial_{c}\delta g_{ab} - \partial_{a}\delta g_{cb}) = 0$$ Observe the parenthetic term, we will write it as $$f_{abc} = (\partial_{c}\delta g_{ab} - \partial_{a}\delta g_{cb})$$ Then observe $$f_{abc} = -f_{cba}$$ We get the result immediately, because index gymnastics ...

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