## Introduction to tensor notations ### The chain rule, super- and subscript Define a function $F$, the variables are $a,\ b$ and $c$. $F=F(a,b,c)$ And three two-variables functions, the variables are $\mu$ and $\nu$. $A(\mu,\nu),\ B(\mu,\nu),\ C(\mu,\nu)$ These may form a new function $f$, being $f(\mu,\nu)=F(A(\mu,\nu),B(\mu,\nu), C(\mu,\nu))$ Now we examine the derivatives, the first order would be $\frac{\partial f\left(\mu, \nu\right)}{\partial \mu} = \frac{\partial F}{\partial a} \frac{\partial A}{\partial \mu} + \frac{\partial F}{\partial b} \frac{\partial B}{\partial \mu} + \frac{\partial F}{\partial c} \frac{\partial C}{\partial \mu}$ $\frac{\partial f\left(\mu, \nu\right)}{\partial \nu} = \frac{\partial F}{\partial a} \frac{\partial A}{\partial \nu} + \frac{\partial F}{\partial b} \frac{\partial B}{\partial \nu} + \frac{\partial F}{\partial c} \frac{\partial C}{\partial \nu}$ and second order, using the chain rule, $\frac{\partial^2 f(\mu, \nu)}{\partial \mu^2} = \frac{\partial^2 F}{\partial a^2} \left( \frac{\partial A}{\partial \mu} \right)^2 + \frac{\partial F}{\partial a} \frac{\partial^2 A}{\partial \mu^2} + \frac{\partial^2 F}{\partial b^2} \left( \frac{\partial B}{\partial \mu} \right)^2 + \frac{\partial F}{\partial b} \frac{\partial^2 B}{\partial \mu^2} + \frac{\partial^2 F}{\partial c^2} \left( \frac{\partial C}{\partial \mu} \right)^2 + \frac{\partial F}{\partial c} \frac{\partial^2 C}{\partial \mu^2}$ $\frac{\partial^2 f(\mu, \nu)}{\partial \nu^2} = \frac{\partial^2 F}{\partial a^2} \left( \frac{\partial A}{\partial \nu} \right)^2 + \frac{\partial F}{\partial a} \frac{\partial^2 A}{\partial \nu^2} + \frac{\partial^2 F}{\partial b^2} \left( \frac{\partial B}{\partial \nu} \right)^2 + \frac{\partial F}{\partial b} \frac{\partial^2 B}{\partial \nu^2} + \frac{\partial^2 F}{\partial c^2} \left( \frac{\partial C}{\partial \nu} \right)^2 + \frac{\partial F}{\partial c} \frac{\partial^2 C}{\partial \nu^2}$ $\frac{\partial^2 f(\mu, \nu)}{\partial \mu \partial \nu} = \frac{\partial^2 F}{\partial a^2} \frac{\partial A}{\partial \mu} \frac{\partial A}{\partial \nu} + \frac{\partial F}{\partial a} \frac{\partial^2 A}{\partial \mu \partial \nu} + \frac{\partial^2 F}{\partial b^2} \frac{\partial B}{\partial \mu} \frac{\partial B}{\partial \nu} + \frac{\partial F}{\partial b} \frac{\partial^2 B}{\partial \mu \partial \nu} + \frac{\partial^2 F}{\partial c^2} \frac{\partial C}{\partial \mu} \frac{\partial C}{\partial \nu} + \frac{\partial F}{\partial c} \frac{\partial^2 C}{\partial \mu \partial \nu}$ >[!Notice] >Just a reminder, $\frac{\partial^2 A}{\partial \mu \partial \nu}$ and $\frac{\partial A}{\partial \mu} \frac{\partial A}{\partial \nu}$ are entirely different quantities. In similar manner one could calculate the third order expressions. Now we convert above expressions into contracted form. Rename these variables $a,\ b,\ c$ as $a^{1},\ a^{2},\ a^{3}$. These superscript denotes enumeration, not exponentiation. In many cases we write these arguments in a collective manner, by a single letter and suppress the superscript $i$. It is important not to think this single letter a vector with components $a^{i}$. It's just a simplification. In this way, we have $F=F(a^{1}, a^{2}, a^{3})$ or $F=F(a^{i})$ as $F=F(a)$ Similarly, we write $A(\mu,\nu),\ B(\mu,\nu),\ C(\mu,\nu)$ with superscripts, as $A^{1}(\mu,\nu),\ A^{2}(\mu,\nu),\ A^{3}(\mu,\nu)$. This can be simplified as $A^{i}(\mu, \nu)$ or $A(\mu,\nu)$. The $f$ expressed with these compact notation would then be $f(\mu, \nu)=F(A(\mu,\nu))$ The expanded form would be $f(\mu, \nu)=F(A^{1}(\mu,\nu),\ A^{2}(\mu,\nu),\ A^{3}(\mu,\nu))$ Now we can express the partial derivatives above with summation $\sum$. $\begin{aligned} \frac{\partial f(\mu, \nu)}{\partial \mu} =& \frac{\partial F}{\partial a} \frac{\partial A}{\partial \mu} + \frac{\partial F}{\partial b} \frac{\partial B}{\partial \mu} + \frac{\partial F}{\partial c} \frac{\partial C}{\partial \mu}\\=& \frac{\partial F}{\partial a^{1}} \frac{\partial A^{1}}{\partial \mu} + \frac{\partial F}{\partial a^{2}} \frac{\partial A^{2}}{\partial \mu} + \frac{\partial F}{\partial a^{3}} \frac{\partial A^{3}}{\partial \mu}\\ =& \sum_{i=1}^{3} \frac{\partial F}{\partial a^i} \frac{\partial A^i}{\partial \mu}\end{aligned}$ Similarly, $\frac{\partial f(\mu, \nu)}{\partial \nu} = \sum_{i=1}^{3} \frac{\partial F}{\partial a^i} \frac{\partial A^i}{\partial \nu}$ > [!Note]- > The second order partial differential are > $\frac{\partial^2 f(\mu, \nu)}{\partial \mu^2} = \sum_{i=1}^{3} \left( \frac{\partial^2 F}{\partial (a^i)^2} \left( \frac{\partial A^i}{\partial \mu} \right)^2 + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \mu^2} \right)$ > $\frac{\partial^2 f(\mu, \nu)}{\partial \nu^2} = \sum_{i=1}^{3} \left( \frac{\partial^2 F}{\partial (a^i)^2} \left( \frac{\partial A^i}{\partial \nu} \right)^2 + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \nu^2} \right)$ > $\frac{\partial^2 f(\mu, \nu)}{\partial \mu \partial \nu} = \sum_{i=1}^{3} \left( \frac{\partial^2 F}{\partial (a^i)^2} \frac{\partial A^i}{\partial \mu} \frac{\partial A^i}{\partial \nu} + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \mu \partial \nu} \right)$ > We call the $i$ in $\frac{\partial F}{\partial a^i}$ **subscript** (we will be further explained this in [[Covariant and contravariant basis, matric tensors]]), since the index $i$ is in the **denominator**, although it is written in the superscript form $a^i$. But one will see it becomes a "real" subscript if written in other form, like using $\nabla$ sign, $\nabla_i F$, or $F_{|i}$. ### Einstein notation Now we move to the famous **Einstein notation**, or **Einstein summation convention**. The idea is if the same index appears twice, once as a subscript ($\frac{\partial F}{\partial a^i}$) and once as a superscript ($\frac{\partial A^i}{\partial \mu}$), the summation is implied. In this way, we can remove the summation sign and express the partial differential expression before as $\frac{\partial f(\mu, \nu)}{\partial \mu} =\sum_{i=1}^{3} \frac{\partial F}{\partial a^i} \frac{\partial A^i}{\partial \mu}= \frac{\partial F}{\partial a^i} \frac{\partial A^i}{\partial \mu}$ and $\frac{\partial f(\mu, \nu)}{\partial \nu}= \sum_{i=1}^{3} \frac{\partial F}{\partial a^i} \frac{\partial A^i}{\partial \nu} = \frac{\partial F}{\partial a^i} \frac{\partial A^i}{\partial \nu}$ The repeated index $i$ is called a **dummy index**, or a repeated index, or a contracted index. Summation over a dummy index is called a **contraction**. > [!Note]- > The second order partial differential are > $\frac{\partial^2 f(\mu, \nu)}{\partial \mu^2} = \frac{\partial^2 F}{\partial (a^i)^2} \left( \frac{\partial A^i}{\partial \mu} \right)^2 + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \mu^2}$ > $\frac{\partial^2 f(\mu, \nu)}{\partial \nu^2} = \frac{\partial^2 F}{\partial (a^i)^2} \left( \frac{\partial A^i}{\partial \nu} \right)^2 + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \nu^2}$ > $\frac{\partial^2 f(\mu, \nu)}{\partial \mu \partial \nu} = \frac{\partial^2 F}{\partial (a^i)^2} \frac{\partial A^i}{\partial \mu} \frac{\partial A^i}{\partial \nu} + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \mu \partial \nu}$ When we have repeated indices, we have contraction. And the repeated index must appear exactly twice, once at upper and once as lower. Writing something like $\frac{{\partial^{2} F}}{\partial Z^{i} \partial Z^{i}}$ is illegal in tensor calculus and does not denote $\frac{{\partial^{2} F}}{\partial Z^{i} \partial Z^{j}}$, although in some case it is interpretable. It is worth noting that if one index appear more than once and we wanna rename it, then all such index should be renamed, and could not being the same as other indices already exist. For example, for identity $T_{\alpha} = \frac{{\partial F}}{\partial Z^{i} } \frac{\partial Z^{i}}{\partial \mu^{\alpha}}$ if one wanna rename the dummy index $i$ to $j$, or rename the live index $\alpha$ to $\beta$, both index should be replaced at the same time. >[!Notice] >When using Einstein notation, we consider $A_i B^i$ is the result after summation, and the summation performs over entire expression, not a single part. >This is to say, $\sqrt{A_{i}B^{i}}$ stands for $\sqrt{\sum A_{i}B^{i}}$ rather than $\sum \sqrt{A_{i}B^{i}}$. For the latter case, one should write the summation explicitly. >Similarly, for dummy indices $l$ and $m$, $A_l + B_l$ stands for $\sum_l (A_l + B_l)$, while $A_m + B_l$ stands for $\sum _l \sum_{m} (A_m + B_l)$ instead of $\sum_{m} A_m + \sum_l B_l$. ### Contract notation for variables To fully turn above expression into contracted format, we should combine $\frac{\partial f(\mu, \nu)}{\partial \mu}$ and $\frac{\partial f(\mu, \nu)}{\partial \nu}$ into one single expression. This is done by denoting $\mu$ and $\nu$ by $\mu^{\alpha}$ collectively, or sometimes $\mu^{1}$ and $\mu^{2}$. In this way, the function $f(\mu, \nu)$ becomes $f(\mu^{1}, \mu^{2})$, $f(\mu^{\alpha})$ or $f(\mu)$. Now we write the $f$ as if we suppress the index, $f(\mu) = F(A(\mu))$ and the actual form if expanded would be $f(\mu^1, \mu^2) = F\left(A^1(\mu^1, \mu^2), A^2(\mu^1, \mu^2), A^3(\mu^1, \mu^2)\right)$ Write the first order partial differential, $\frac{\partial f}{\partial \mu^\alpha} = \frac{\partial F}{\partial a^i} \frac{\partial A^i}{\partial \mu^\alpha}$ This includes both expressions above, applied Einstein notation and use superscript for simplification. This $\alpha$ could be 1 or 2, and these two values correspond to different expressions. This makes $\alpha$ a **live index**, and it appears at the left hand side. While the right hand side has $i$, the dummy index. >[!Note]- >The second order differential expressions can be simplified with live indices $\alpha$ and $\beta$. We now need two live indices because the left side contains $\mu$ and $\nu$ simultaneously. The original forms are > $\frac{\partial^2 f(\mu, \nu)}{\partial \mu^2} = \frac{\partial^2 F}{\partial (a^i)^2} \left( \frac{\partial A^i}{\partial \mu} \right)^2 + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \mu^2}$ > $\frac{\partial^2 f(\mu, \nu)}{\partial \nu^2} = \frac{\partial^2 F}{\partial (a^i)^2} \left( \frac{\partial A^i}{\partial \nu} \right)^2 + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \nu^2}$ > $\frac{\partial^2 f(\mu, \nu)}{\partial \mu \partial \nu} = \frac{\partial^2 F}{\partial (a^i)^2} \frac{\partial A^i}{\partial \mu} \frac{\partial A^i}{\partial \nu} + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \mu \partial \nu}$ > The simplified form would be > $\frac{\partial^2 f}{\partial \mu^{\alpha}\partial\mu^{\beta}} = \frac{\partial^2 F}{\partial a^i \partial a^{j}} \frac{\partial A^i}{\partial \mu^{\alpha}} \frac{\partial A^j}{\partial \mu^{\beta}} + \frac{\partial F}{\partial a^i} \frac{\partial^2 A^i}{\partial \mu^{\alpha} \partial \mu^{\beta}}$ > In similarly way, we may have > $\frac{\partial^3 f}{\partial \mu^\alpha \partial \mu^\beta \partial \mu^\gamma} = \frac{\partial^3 F}{\partial a^i \partial a^j \partial a^k} \frac{\partial A^i}{\partial \mu^\alpha} \frac{\partial A^j}{\partial \mu^\beta} \frac{\partial A^k}{\partial \mu^\gamma} + \frac{\partial^2 F}{\partial a^i \partial a^j} \left( \frac{\partial^2 A^i}{\partial \mu^\alpha \partial \mu^\beta} \frac{\partial A^j}{\partial \mu^\gamma} + \frac{\partial A^i}{\partial \mu^\alpha} \frac{\partial^2 A^j}{\partial \mu^\beta \partial \mu^\gamma} + \frac{\partial A^i}{\partial \mu^\beta} \frac{\partial^2 A^j}{\partial \mu^\alpha \partial \mu^\gamma} \right) + \frac{\partial F}{\partial a^i} \frac{\partial^3 A^i}{\partial \mu^\alpha \partial \mu^\beta \partial \mu^\gamma}$ > for third order expression. The indices could be freely renamed in most of cases. That is to say we referring to an object $T^{i}$, it does not matter if we call it $T^{k}$ or $T^{j}$ or something else. Because the fact that summation is commute, the indices in contractions also preserve community.