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4

Your problem is that the thing you want to express is only true in the particular basis you're working in. But equations in the summation convention are true in every basis. Your expression involving an explicit summation is fine and is probably what any reasonable mathematician would do. But if you want to show unreasonable devotion to the summation ...


3

The notations $\int f(r)\, dr$ and $\int dr\,f(r)$ are equivalent. The latter is convenient if you have nested integrals, e.g. $$\int_0^\infty dr\int_0^\pi d\theta\,f(r,\theta),$$ so that you can see which integration limit belongs to which integration variable. The notation $d^3r$ is a shorthand for "integrate over 3D space", for example $dx\,dy\,dz$ if ...


3

By convention, vectors are written as column vectors, whereas dual vectors are written as row vectors. This means that in principle, upper indices should index columns and lower indices should index rows. However, in practice, we normally translate rank-2 tensors to matrices by order of the indices, the first one indexing rows, the second one indexing ...


3

$$\Gamma^\mu{}_{\alpha\beta} y^\alpha y^\beta$$ is defined to mean $$\sum_{\alpha = 0}^3 \sum_{\beta = 0}^3 \Gamma^\mu{}_{\alpha\beta} y^\alpha y^\beta $$ that is, each repeated index is summed over independently.


3

This seems to be addition in quadrature of multiple independent uncertainties in a measurement. In particular, if you have a measurement which depends on two quantities $a$ and $b$ whose uncertainties $\delta a$ and $\delta b$ are completely independent and uncorrelated, then their uncertainties will often be combined as $$\delta(a+b)=\sqrt{\delta ...


2

The notation $k^2$ for vector $\boldsymbol{k}$ means the square length/magnitude of the vector; if $\boldsymbol{k} = (k_x,k_y,k_z)$ in Cartesian coordinates then $$ k^2 = \left|\boldsymbol{k}\right|^2 = k_x^2+k_y^2+k_z^2 $$ so that $$ \widetilde{g}(\boldsymbol{k}) = \frac{4\pi}{k_x^2+k_y^2+k_z^2} $$ (using Pythagoras)


2

There are two kinds of derivatives we should differentiate: $$ \frac{\mathrm d\mathcal L}{\mathrm dx}=\lim_{h\to 0}\frac{1}{h}\big[\mathcal L(\phi(x+h),\phi'(x+h),x+h)-\mathcal L(\phi(x),\phi'(x),x)\big]\tag{1} $$ and $$ \frac{\partial\mathcal L}{\partial x}=\lim_{h\to 0}\frac{1}{h}\big[\mathcal L(\phi(x),\phi'(x),x+h)-\mathcal ...


1

As a general note, asking notation questions without providing a reference to the original occurrence (from which we'd be able to infer the context) is an excellent recipe for an unanswerable question. In this particular case, though, it's pretty clear that it refers to the unit basis vector in the $z$ direction, $$\hat e_z=(0,0,1).$$ It arises in this ...



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