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--- a/seven-wonders.tex
+++ b/seven-wonders.tex
@@ -2,7 +2,7 @@
 \pdfinclusioncopyfonts=1
 %% Author: PGL  Porta Mana
 %% Created: 2015-05-01T20:53:34+0200
-%% Last-Updated: 2024-08-29T07:24:34+0200
+%% Last-Updated: 2024-09-02T21:38:14+0200
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \newif\ifanon
 \anonfalse
@@ -1513,7 +1513,7 @@ \subsubsection{\faIcon{hand-point-right}\enspace Graphical devices}
 \subsubsection{\faIcon{hand-point-right}\enspace Side pictures and quotes}
 
 \marginpar{\vspace{-4\baselineskip}\centering%
-\includegraphics[align=c,width=0.5\linewidth]{images/saitama_image.png}%
+\includegraphics[width=0.5\linewidth]{images/saitama_image.png}%
 \\[\jot]\footnotesize\flushleftright{\color{green}%
 This is an image of Saitama, which actually has nothing to do with the text on the left.}
 }%
@@ -1908,7 +1908,7 @@ \section{Time}
 
 Luckily there's a way to bypass proper-time discrepancies: instead of referring to my proper time or to your proper time, we can agree on assigning a somewhat arbitrary time label to every event: this is called a \emph{coordinate time} and will be discussed in a couple of sections.
 %
-\marginpar{\vspace{-7\baselineskip}\centering\includegraphics[align=t,width=\linewidth]{images/net_rotations.png} \\\footnotesize\color{mpcolor}%
+\marginpar{\vspace{-7\baselineskip}\centering\includegraphics[width=\linewidth]{images/net_rotations.png} \\\footnotesize\color{mpcolor}%
   \url{https://xkcd.com/2882}%
 }%
 When we use coordinate time, some important physics formulae turn out to be the same no matter whether we use General Relativity or an approximate theory such as Newtonian Mechanics. Thanks to this fact, for the most part of these notes we will not need to deal with proper-time details. But I recommend that you keep in mind how time really works, and the small time discrepancies that exist and occur all the time along your \emph{worldline}.
@@ -2399,7 +2399,7 @@ \subsection{Energy and mass are the same}
 \begin{equation*}
   \qty{0.000500000000000000000}{kg}\ .
 \end{equation*}
-Now we stretch the band a little. By doing so we are giving energy to the band, which is said to acquire \enquote*{elastic energy}, say \qty{0.3}{J}.
+Now we stretch the band a little. By doing so we give energy to the band, which is said to acquire \enquote*{elastic energy}. Let's say we have given \qty{0.3}{J} to the band in this way.
 % Refer later to elastic constant
 % 0.5 * 50 N/m * (0.1 m)^2 \approx 0.3 J
 Now we weigh the rubber band again, while stretched. We observe a mass of approximately
@@ -2417,34 +2417,40 @@ \subsection{Energy and mass are the same}
 
 \smallskip
 
-Now set the unstretched band in motion. Owing to the motion, the band is said to have acquired a \enquote*{kinetic energy} of, say, \qty{0.2}{J}. If we could weigh the band while in motion (but without moving the weighing scale), we would actually observe a mass of approximately
-$$\qty{0.000500000000000002225}{kg} \ .$$
+Now set the unstretched band in motion. Owing to the motion, the band is said to have acquired \enquote*{kinetic energy}; let's say an amount \qty{0.3}{J}. If we could weigh the band while in motion (but without moving the weighing scale), we would observe again a mass of approximately
+\begin{equation*}
+  \qty{0.000500000000000003338}{kg} \ .
+%\qty{0.000500000000000002225}{kg}
+\end{equation*}
 The small difference from the initial mass is the additional kinetic energy of the band. Energy has weight; energy is mass.
 
-This case is actually connected with the example of the gas above. If we observed the gas at a molecular level, we would interpret the energy of \qty{60}{J} provided to it as additional kinetic energy of its molecules. The increase in weight was exactly this additional kinetic energy.
+% This case is actually connected with the example of the gas above. If we observed the gas at a molecular level, we would interpret the energy of \qty{60}{J} provided to it as additional kinetic energy of its molecules. The increase in weight was exactly this additional kinetic energy.
 
 \paragraph{Fission and atomic bombs.}
-\marginpar{%
-\includegraphics[align=t,width=\linewidth]{images/atomicbomb.jpg}
-\\[\jot]\color{mpcolor}\footnotesize\emph{Hydrogen Bomb Test, 1954%
+\marginpar{\vspace{-2\baselineskip}\centering\includegraphics[width=\linewidth]{images/atomicbomb.jpg}%
+\\[\jot]\flushleftright\color{mpcolor}\footnotesize\emph{Hydrogen Bomb Test, 1954%
 }\\{(\furl{https://nuclearmuseum.pastperfectonline.com/Archive/716477C1-5E7A-485C-8BE1-857919471563}{National Museum of Nuclear Science \amp\ History})}%
 }
-The \furl{https://www.britannica.com/science/nuclear-fission}{atomic bomb} is a dark example of the fact that mass is energy.
-In the case of nuclear fission, if we weigh the amount of nuclear material, say within a box, before and after fission, we observe that its mass has decreased. But we also observe that a great amount of energy has been released out of the box.
+%
+The \furl{https://www.britannica.com/science/nuclear-fission}{atomic bomb} is a dark example of the fact that mass is energy. In phenomena of nuclear fission, we notice a decrease in the weight, measured at rest, of nuclear material before and after the phenomenon of fission. But we also observe that a great amount of (kinetic) energy is released. This amount is exactly equal to the apparently missing weight.
 
 \paragraph{Electric heater.}
 
 As a final example consider a \qty{1000}{W} electric heater, which is radiating \qty{1000}{J} in one second. The heater is also losing around \qty{0.00000000000001}{kg} of mass every second owing to this heat radiation -- although it's also acquiring the same amount of mass as electromagnetic energy.
 
 
-\medskip
+\subsection{The practical use of the words \enquote*{mass} and \enquote*{energy}}
+\label{sec:dist_mass_energy}
 
+From the examples above it becomes clear that energy and mass are two names for the same thing.
+%
 \marginpar{\vspace{\baselineskip}%
 \footnotesize\color{mpcolor}\enquote{\emph{%
 we are led to the more general conclusion: The mass of a body is a measure of its energy content; if the energy changes by $L$, the mass changes in the same sense by $L/9\!\cdot\!10^{20}$, if the energy is measured in ergs and the mass in grams.
 %\par Perhaps it will prove possible to test this theory using bodies whose energy content is variable to a high degree (e.g., salts of radium).
 }}\sourceatright{\cites{einstein1905d}}%
 }
+%
 The equivalence between energy and mass is given by the famous formula $E=m c^{2}$, where $c$ is the speed of light, \eqn~\eqref{eq:c}. In their respective units this gives
 \begin{equation*}
   \begin{gathered}
@@ -2455,63 +2461,41 @@ \subsection{Energy and mass are the same}
 \end{equation*}
 To grasp these numbers, consider that the mass of the rubber band in the example above, \qty{0.5}{g}, is comparable to the energy released by the \furl{https://www.britannica.com/story/atomic-bombing-of-hiroshima}{atomic bomb over Hiroshima}.
 
-\subsection{The practical use of the words \enquote*{mass} and \enquote*{energy}}
-\label{sec:dist_mass_energy}
-
-From the examples above it becomes clear that energy and mass are two names for the same thing. But it also becomes clear that in our daily experience we deal with energy-mass in two different ways:
+But it also becomes clear that in our daily experience we deal with energy-mass in two different ways:
 
-On the one hand, we deal with huge (atom-bomb-like) amounts of energy packed in very small volumes: the huge amount of energy that goes together with objects like pens, keys, bicycles, cars, houses, and so on. We move, push, pull these huge energy amounts from one place to another, and even put them in our pockets. These energy amounts change a little all the time; see the examples above. But the change is so small as to be often undetectable with ordinary scales, and negligible for practical purposes. We call this energy \enquote*{mass} and measure it with a unit -- \unit{kg} -- that doesn't lead to ridiculously large numbers. And we also agree to neglect the imprecision and fluctuation in its measurement, say any imprecision under \qty{0.000001}{\percent}. % This situation is similar to our measuring the power produced by power plants in megawatts (MW). We can say that another power plant constantly produces \qty{500}{MW}; but with this we don't really mean that it produces \emph{exactly} \qty{500000000.000}{W}: there are fluctuations all the time. And we can say that a power plant produces twice as much power, \qty{1000}{MW}, again with the understanding that the ratio is not \emph{exactly} \num{2.000000000000}.
+On the one hand, we deal with huge (atom-bomb-like) amounts of energy packed in very small volumes: the huge amounts of energy that go together with objects like pens, keys, bicycles, cars, houses, and so on. We move, push, pull these huge energy amounts from one place to another, and even put them in our pockets. These energy amounts change a little all the time, as in the examples with the rubber band above. But these changes are so small as to be often undetectable with ordinary scales, and negligible for practical purposes. We call \enquote*{mass} any such huge amount of energy, and measure it with a unit -- \unit{kg} -- that doesn't lead to ridiculously large numbers. And we also agree to neglect the imprecision and fluctuation in its measurement, say any imprecision under \qty{0.000001}{\percent}. % This situation is similar to our measuring the power produced by power plants in megawatts (MW). We can say that another power plant constantly produces \qty{500}{MW}; but with this we don't really mean that it produces \emph{exactly} \qty{500000000.000}{W}: there are fluctuations all the time. And we can say that a power plant produces twice as much power, \qty{1000}{MW}, again with the understanding that the ratio is not \emph{exactly} \num{2.000000000000}.
 
-On the other hand, we also deal with the small energy changes and exchanges in all these objects. These exchanges that are very important for our daily life: they keep us warm, make our cells work, make our laptops work. In dealing with these energy exchanges, we don't care about the huge energy reservoirs they come from. So we agree to measure them with a unit -- \unit{J} -- that doesn't lead to ridiculously small numbers. And we also agree not to be precise about the total amount in the reservoir from which these energy bits come from.
+On the other hand, we also deal with the small energy changes and exchanges in all these objects. These energy exchanges that are very important for our daily life: they keep us warm, keep our cells active, make our laptops work. In dealing with these energy exchanges, we don't care about the huge energy reservoirs they come from. So we agree to measure them with a unit -- \unit{J} -- that doesn't lead to ridiculously small numbers. And we also agree not to be precise about the total amount in the reservoir from which these energy bits come from.
 
-As an analogy, consider when we speak about the amount of people in different countries. We can say that in Norway there are \num{5} millions, and in India \num{1500} millions, so in India there are \num{300} times more people. By this we don't mean that in Norway there are \emph{exactly} \num{5000000} people and that India has \emph{exactly} \num{300.000000} times more people. These numbers are changing slightly all the time; but we don't care about differences of 10 or even \num{10000} people. At the same time, think of when you have three dear friends or relatives visiting you from abroad: the amount of \num{3} people is now for you very important; and you don't care about how much this amount is compared to the total population of your country.
+As an analogy, think of when we speak about the amount of people in different countries. We can say that in Norway there are \num{5} millions, and in India \num{1500} millions, so in India there are \num{300} times more people. By this we don't mean that in Norway there are \emph{exactly} \num{5000000} people and that India has \emph{exactly} \num{300.000000} times more people. These numbers are changing slightly all the time, but we don't care about differences of 10 or even \num{10000} people. At the same time, if we have three dear friends or relatives visiting us from abroad, then the amount of \num{3} people is now for us very important. And we don't care about how large this amount is, compared to the total population of a country.
 
 \smallskip
 
-The distinction above is of course not clear-cut. In dealing with some physical phenomena, for example with few molecules or in particle physics, the distinction become too blurry and not useful anymore. And indeed in these situations one often uses the terms \enquote*{mass} and \enquote*{energy} interchangeably, as well as a common unit for both (for instance \furl{https://home.cern/tags/13-tev}{\emph{electronvolts}}).
+The distinction above is of course not clear-cut. In dealing with some physical phenomena, for example with few molecules or with subatomic particles, the fictitious but pragmatic distinction between mass and energy becomes too blurry and not useful anymore. In discussing these phenomena, indeed, one often uses the terms \enquote*{mass} and \enquote*{energy} interchangeably, as well as a common unit for both, such as the \furl{https://home.cern/tags/13-tev}{\emph{electronvolt}}.
 
 \medskip
 
 % From the examples above it is clear that the energy changes that we deal with and use in everyday situations are extremely small changes in mass -- so small that we can't measure them within the precision of ordinary weighing scales. Vice versa, the masses that we ordinarily deal with are huge amounts of energy. If we spoke only of \enquote*{energy} or only of \enquote*{mass} in all situations, and used only one unit, either joules or kilograms, we would have to work with very impractical numbers. \enquote*{Mass} can be considered a convenient term for \enquote*{energy} when huge amounts of it are involved, concentrated in small regions of space, and ordinary small energy changes can be neglected.
-In these notes we shall often use the expressions \enquote*{\energym} and \enquote*{\masse} to remember that these two words denote the same thing.
+In these notes we shall often use the expressions \enquote*{\energym} and \enquote*{\masse} to remind ourselves that these two words denote the same physical thing.
 
 \subsection{Different \enquote*{forms} of energy}
 \label{sec:forms_energy}
 
-We often speak of different \emph{forms} of \energym. The most important for us will be \textbf{internal energy}, \textbf{kinetic energy}, \textbf{gravitational potential energy}, to be discussed later; another important one is \emph{electromagnetic energy}.
-
-In \chap\,\ref{cha:bal_energy} we shall see that the differences among these energy forms come from the way they are calculated from other quantities, like matter or magnetic flux and electric charge. For example, if we know that in a volume there's an amount of a particular kind of matter, then we know that there must also be an amount of energy given by a particular formula. And if that matter is moving, then we have to add to the total an extra amount of energy given by another formula. And if in that volume there's a gravitational field (that is, a particular kind of spacetime curvature), then another extra amount must be added, given by yet another formula. Similarly if we know that an electromagnetic field is in that volume.
+We often speak of different \emph{forms} of \energym. The most important forms for us will be \textbf{internal energy}, \textbf{kinetic energy}, \textbf{gravitational potential energy}, to be discussed later. Another important one is \emph{electromagnetic energy}.
 
-We also speak of different forms of flux of energy. The most important for us will be \textbf{heat} and \textbf{mechanical power}. The difference is again in how these fluxes are calculated depending on whether there are also fluxes of matter.
+Later on we shall see that the differences among these energy forms come from the way they are calculated from other quantities, like matter or magnetic flux and electric charge. For example, if we know that in a volume there's an amount of a particular kind of matter, then we know that there must also be an amount of energy, given by a particular formula. And if that matter is moving, then we have to add to the energy an extra amount given by another formula. And if in that volume there's a gravitational field (that is, a particular kind of spacetime curvature), then another extra amount must be added, given by yet another formula. Similarly if we know that an electromagnetic field is in that volume.
 
-\subsection{Amounts and forms of energy are coordinate- and observer-dependent}
-\label{sec:energy_coords}
-
-An aspect of energy that must always be kept in mind is that \textbf{the amount of energy depends on the coordinate system we're using}. If someone points at a specific region of space at a particular instant, and asks \enquote{how much energy is there?}, we \emph{cannot} give an answer until a coordinate system is specified. Once the coordinate system has been chosen, then a precise and unambiguous answer can be given. The same is true for the flow of energy through a surface. This also means that observers using different coordinates will usually assign different amounts of energy to the same regions of spacetime.
-%
-\marginpar{\vspace{-7\baselineskip}\centering\includegraphics[align=t,width=\linewidth]{images/cube_J_coords.jpg}%
-\\[\jot]\footnotesize\flushleftright\color{mpcolor}\enquote{How much energy is there in this volume at this instant?} -- This question cannot be answered until we have specified which coordinate system we're using.%
-}
-
-This is an important difference between energy on one side, and matter and electric charge on the other side. \emph{For matter and electric charge, the questions above can be answered unambiguously independently of any coordinate system}. But not so for energy. The reason of this quirky difference is ultimately connected with the fact that time and space are also observer-dependent.
-
-This coordinate-dependence is not a problem: we must always specify our coordinate system anyway, in order to agree on the time and position of physical events. But it can cause problems when we calculate \emph{changes} in energy. If we first calculate or measure some amount of energy in a coordinate system, then we calculate or measure another amount at some later time in a \emph{different} coordinate system, the difference between the two amounts has no meaning whatsoever. In fact it could happen that there's no energy change at all in one coordinate system or in the other: any change we found was just an artefact of mixing up coordinates.
-
-\begin{warning}[Amounts of \energym\ are coordinate-dependent]
-  Never change coordinate system in the middle of energy calculations!
-\end{warning}
+We also speak of different forms of flux of energy. The most important for us will be \textbf{heat} and \textbf{mechanical power}. The difference is again in how these fluxes are calculated depending on whether there are also fluxes of matter and of other quantities.
 
 \medskip
 
-Also the distinction among different forms of energy is coordinate-dependent. For instance, in one coordinate system we can say that a given volume contains only internal energy; but in another coordinate system that same volume can be said to contain internal and kinetic energy.
-
-The distinction among different forms of energy also depends on the observation scale and on the theory used. For instance, we can observe and model the gas in a container on a scale of metres, seeing it as a uniform flux; in this case we say that there's internal energy in the container. But if we observe and model the gas as a collection of molecules, on a microscopic scale, then we say that there's internal energy (of the single molecules) and kinetic energy in the container (the total energy being the same, unless we have changed coordinate system).
+The distinctions between different forms of energy also depend on the observation scale and the theory used. For instance, we can observe and describe the water in a glass resting on table, on a scale of centimetres. We see it as a still, uniform fluid. In this description we say that the water has internal energy (or that there's internal energy in the glass). But if we observe and model that same water as a collection of molecules, on a microscopic scale, then we say that it has internal energy \emph{and} kinetic energy, because the molecules are in constant motion. The total amount of energy is the same on the centimetre-scale the  microscopic scale, but its partition into different \enquote{forms} depends on the scale.
 %
-\marginpar{\vspace{-3\baselineskip}\centering\includegraphics[width=\linewidth]{images/molecules_20nmb.png}%
+\marginpar{\vspace{-5\baselineskip}\centering\includegraphics[width=\linewidth]{images/molecules_20nmb.png}%
 \\[\jot]\footnotesize\flushleftright\color{mpcolor}\furl{https://doi.org/10.4209/aaqr.2019.04.0177}{Hydrocarbon fuel particles}. The small blobs have size of around \qty{2e-8}{m}.%
 }%
 
-The same is true of energy flux: what we call \enquote*{heat} on one observation scale is \enquote*{work} on a finer scale. % In phenomena that involve matter and electromagnetic fields, any separation between \enquote*{energy of matter} and \enquote*{energy of electromagnetic field} is often arbitrary.
+The same is true of energy flux: what we call \enquote*{heat} on one observation scale, and appears as a flux of energy not associated with the motion of matter, may instead be called \enquote*{work} on a finer scale, and be associated with the motion of matter. % In phenomena that involve matter and electromagnetic fields, any separation between \enquote*{energy of matter} and \enquote*{energy of electromagnetic field} is often arbitrary.
 
 % \begin{warning}[Amounts of \energym\ are coordinate-dependent]
 %   Never change coordinate system in the middle of a calculation of energy change!
@@ -2591,11 +2575,7 @@ \section{Momentum}
 % }
 Given a particular volume at a particular instant in time, and given a coordinate system, we can speak of the total amount of momentum within that volume. This amount is represented by a \emph{vector}. You can imagine a continuous collection of vectors filling the volume, possibly with different directions and small magnitudes; the total momentum is the sum of all these vectors. This visualization obviously comes with many warnings, but it can be very useful if we are careful.
 
-And just as energy, the total amount of momentum is \emph{coordinate-dependent}. So we have the same warning here:
-\begin{warning}[Amounts of momentum are coordinate-dependent]
-  Don't change coordinate system in the middle of calculations of momentum!
-\end{warning}
-
+\medskip
 
 Flux of momentum is what we call \textbf{force}. A static force, like the one you exert when you hold a bag, is often mentally visualized as a static vector. In \chap\,\ref{cha:total_flux} we shall discuss a different visualization, in which force is represented as a sort of flow of vectors.
 
@@ -2647,6 +2627,10 @@ \section{Angular momentum}
 There isn't a clear-cut distinction between translational and rotational motion: usually they involve each other to some degree. A translational motion can be interpreted as a rotation around a point that is very far away; and a rotation of an extended object can be interpreted as small translational motions of its parts.
 
 This is the reason why in many situations we can calculate angular momentum in terms of momentum. % so even if we don't fully grasp it intuitively, we can still calculate it
+%
+\marginpar{\vspace{-\baselineskip}\centering\includegraphics[align=t,width=\linewidth]{images/cube_Nms_coords.jpg}%
+  \\[\jot]\footnotesize\flushleftright\color{mpcolor}The amount of angular momentum within a volume at a given instant is represented by a \textcolor{purple}{3D vector} (curious about the little circulation symbol around the vector? then read on). The magnitude and direction of this vector depend on the coordinate system we're using.%
+}%
 If the momentum in a \emph{small} volume is denoted by the vector $\yP=(P_{x},P_{y},P_{z})$, and the position vector by $\yr=(x,y,z)$, then the angular momentum $\yL=(L_{x}, L_{y}, L_{z})$ \emph{with respect to the origin of coordinates}, in that same volume, is given by the vector product
 \begin{subequations}
   \begin{equation}
@@ -2675,20 +2659,13 @@ \section{Angular momentum}
 \end{subequations}
 Choose whichever you prefer.
 
-%
-\marginpar{\vspace{-5\baselineskip}\centering\includegraphics[align=t,width=\linewidth]{images/cube_Nms_coords.jpg}%
-  \\[\jot]\footnotesize\flushleftright\color{mpcolor}The amount of angular momentum within a volume at a given instant is represented by a \textcolor{purple}{3D vector} (curious about the little circulation symbol around the vector? then read on). The magnitude and direction of this vector depend on the coordinate system we're using.%
-}%
 Angular momentum is something that is associated not only with ordinary bodies (matter), but also with electromagnetic fields. Just like momentum, also angular momentum can be visualized as a \enquote{fluid of vectors}.
 % %
 % \marginpar{%
 % \includegraphics[align=t,width=\linewidth]{images/cubetwistedarrow.pdf}\label{fig:volumetwistedarrow}%
 % \\[\jot]\footnotesize\color{mpcolor}The amount of angular momentum within a volume is represented by a \textcolor{yellow}{(3D) vector}. (Curious about the little circulation symbol around the vector? Then read on.)%
 % }
-Given a particular volume at a particular instant in time, and given a coordinate system, we can speak of the total amount of angular momentum within that volume. This amount is represented by a \emph{vector}, and is \emph{coordinate-dependent}:
-\begin{warning}[Amounts of angular momentum are coordinate-dependent]
-  Don't change coordinate system in the middle of calculations of angular momentum!
-\end{warning}
+Given a particular volume at a particular instant in time, and given a coordinate system, we can speak of the total amount of angular momentum within that volume. This amount is represented by a \emph{vector}.
 
 \smallskip
 
@@ -2718,20 +2695,19 @@ \section{Angular momentum}
 
 You may wonder:
 \enquote{Do we really need angular momentum? after all it just looks like something constructed from momentum}.
+The answer is yes, we really need it, for two reasons. First, angular momentum obeys an important universal law which is independent from those obeyed by energy and by momentum (\cites{truesdell1963b_r1968} tells some of the story of how this was discovered). % \mynotew{rephrase this}
+Second, for some physical phenomena, for example involving liquid \furl{https://www.britannica.com/science/polymer}{polymers},
+% https://www.britannica.com/science/polymer
+elementary particles, or electromagnetic radiation, the angular momentum includes an additional part, called \emph{spin} or \emph{intrinsic angular momentum}, that is \emph{not} related to linear momentum. In the present notes we shall not use this more general kind of angular momentum.
 % \begin{marginfigure}\footnotesize%
 %   \includegraphics[align=b,width=\linewidth]{images/polymer.png}
 % \\\includegraphics[align=c,width=\linewidth]{images/polymer2.png} \\[\jot]%
 % {\color{green}Some liquid polymers (\textbf{top:} Liquid Diethoxymethane Polysulfide) need to be described with a special kind of angular momentum, owing to their molecular structure (\textbf{bottom}).}
 % \end{marginfigure}
-\marginpar{\centering%
-\includegraphics[align=b,width=0.67\linewidth]{images/polymer.png}
+\marginpar{\vspace{-8\baselineskip}\centering\includegraphics[width=0.67\linewidth]{images/polymer.png}
 \\\includegraphics[align=c,width=\linewidth]{images/polymer2.png}
 \\[\jot]\flushleftright\footnotesize\color{mpcolor}Some liquid polymers (\textbf{top:} Liquid Diethoxymethane Polysulfide) need to be described with a special kind of angular momentum, owing to their molecular structure (\textbf{bottom}).%
 }%
-The answer is yes, we really need it, for two reasons. First, angular momentum obeys an important universal law which is independent from those obeyed by energy and by momentum (\cites{truesdell1963b_r1968} tells some of the story of how this was discovered). % \mynotew{rephrase this}
-Second, for some physical phenomena, for example involving liquid \furl{https://www.britannica.com/science/polymer}{polymers},
-% https://www.britannica.com/science/polymer
-elementary particles, or electromagnetic radiation, the angular momentum includes an additional part, called \emph{spin} or \emph{intrinsic angular momentum}, that is \emph{not} related to linear momentum. In the present notes we shall not use this more general kind of angular momentum.
 
 \smallskip
 
@@ -2741,39 +2717,6 @@ \section{Angular momentum}
 
 
 
-\subsection{Angular momentum as a twisted vector}
-\label{sec:twisted_vec}
-
-In order to represent angular momentum we can use a kind of vectors different from the arrow-like ones (called \emph{polar} vectors) with which you are probably familiar. They are called \textbf{twisted vectors}, or also \emph{pseudo}-vectors or \emph{axial} vectors or \emph{outer-oriented} vectors. Twisted vectors represent rotations, and therefore have an orientation, not along them, but \emph{around} them:
-\begin{center}
-\includegraphics[width=0.34\linewidth]{images/io-vector.pdf}%
-\hspace*{0.12\linewidth}\includegraphics[width=0.34\linewidth]{images/oo-vector.pdf}%
-%
-\\\footnotesize
-\makebox[0pt][c]{an ordinary vector}\hspace*{0.46\linewidth}\makebox[0pt][c]{a twisted vector}
-\end{center}
-Their length still represents the magnitude of the vector. They make it immediately clear what is the axis of rotation, and what is the sense of rotation.
-
-The sum of twisted vectors is analogous to the sum of ordinary vectors, with the parallelogram rule:
-\begin{center}
-\includegraphics[width=0.67\linewidth]{images/tvectorsum.pdf}%
-\end{center}
-
-Ordinary vectors and twisted vectors behave very differently if we look at their images through a mirror parallel to their axis: the orientation of ordinary vectors appears unchanged, whereas the orientation of twisted vectors appears \emph{reversed}:
-\begin{center}
-  \includegraphics[width=0.67\linewidth]{images/mirror.pdf}
-\end{center}
-this phenomenon reflects the behaviour of rotations under reflections.
-
-For some mysterious reason many books are afraid of using twisted vectors, and rely on ordinary vectors instead,
-\marginpar{%
-\includegraphics[align=t,width=\linewidth]{images/curlrighthand.png}
-% \\[\jot]%
-% \footnotesize{\color{mpcolor}The amount of momentum within a volume is represented by a \textcolor{cyan}{(3D) vector}}
-}
-introducing the \enquote{right-hand rule} to determine the sense of rotation from the arrow of the ordinary vector. If you've ever asked yourself \enquote{why the right hand, and not the left hand?}, the answer is that it's purely a convention; one could have introduced a left-hand rule instead. Using twisted vectors we don't need these arbitrary conventions and mnemonics: the sense of rotation is unequivocally indicated by the twisted vector.
-
-Use whichever vector representation you prefer! % In these notes we shall use a hybrid notation, as in the side picture of p.~\pageref{fig:volumetwistedarrow}, so as to make everybody (or no one) happy.
 
 \begin{exercise}
   A GPS satellite has, at a given instant, the following position and momentum content:
@@ -2794,6 +2737,28 @@ \subsection{Angular momentum as a twisted vector}
 \end{exercise}
 
 
+
+\subsection{Amounts of energy, momentum, angular momentum are coordinate-dependent}
+\label{sec:energy_momentum_angmomentum_coords}
+
+
+An aspect of energy, momentum, angular momentum that must always be kept in mind is that \textbf{their amounts depends on the coordinate system we're using}. If someone points at a specific region of space at a particular instant, and asks \enquote{how much energy is there?}, we \emph{cannot} give an answer until a coordinate system is specified. Once the coordinate system has been chosen, then a precise and unambiguous answer can be given. The same is true for the flow of energy through a surface, and for the amounts and fluxes of momentum and angular momentum. This also means that observers using different coordinates will usually assign different amounts of energy, momentum, angular momentum to the same regions of spacetime.
+
+This is an important difference between energy, momentum, angular momentum on one side, and matter and electric charge on the other side. \emph{For matter and electric charge, the questions above can be answered unambiguously independently of any spatial coordinate system}.
+%
+\marginpar{\vspace{-5\baselineskip}\centering\includegraphics[align=t,width=\linewidth]{images/cube_J_coords.jpg}%
+\\[\jot]\footnotesize\flushleftright\color{mpcolor}\enquote{How much energy is there in this volume at this instant?} -- This question cannot be answered until we have specified which coordinate system we're using.%
+}
+
+This coordinate-dependence is not a problem: we must always specify our coordinate system anyway, in order to agree on the time and position of physical events. But it can cause problems when we calculate \emph{changes} in energy. If we first calculate or measure some amount of energy in a coordinate system, then we calculate or measure another amount at some later time in a \emph{different} coordinate system, the difference between the two amounts has no meaning whatsoever. In fact it could happen that there's no energy change at all in one coordinate system or in the other: any change we found was just an artefact of mixing up coordinates.
+
+\begin{warning}[{{Amounts of energy, momentum, angular momentum are coordinate-dependent}}]
+  Never change coordinate system in the middle of calculations about energy, momentum, or angular momentum!
+\end{warning}
+
+% Also the distinction among different forms of energy is coordinate-dependent. For instance, in one coordinate system we can say that a given volume contains only internal energy; but in another coordinate system that same volume can be said to contain internal and kinetic energy.
+
+
 \bigskip
 
 \begin{extra}{{What are energy, momentum, angular momentum?}}
@@ -8414,6 +8379,39 @@ \chapter{Balance of angular momentum}
 
 \mynotew{To be written in a later version}
 
+\subsection{Angular momentum as a twisted vector}
+\label{sec:twisted_vec}
+
+In order to represent angular momentum we can use a kind of vectors different from the arrow-like ones (called \emph{polar} vectors) with which you are probably familiar. They are called \textbf{twisted vectors}, or also \emph{pseudo}-vectors or \emph{axial} vectors or \emph{outer-oriented} vectors. Twisted vectors represent rotations, and therefore have an orientation, not along them, but \emph{around} them:
+\begin{center}
+\includegraphics[width=0.34\linewidth]{images/io-vector.pdf}%
+\hspace*{0.12\linewidth}\includegraphics[width=0.34\linewidth]{images/oo-vector.pdf}%
+%
+\\\footnotesize
+\makebox[0pt][c]{an ordinary vector}\hspace*{0.46\linewidth}\makebox[0pt][c]{a twisted vector}
+\end{center}
+Their length still represents the magnitude of the vector. They make it immediately clear what is the axis of rotation, and what is the sense of rotation.
+
+The sum of twisted vectors is analogous to the sum of ordinary vectors, with the parallelogram rule:
+\begin{center}
+\includegraphics[width=0.67\linewidth]{images/tvectorsum.pdf}%
+\end{center}
+
+Ordinary vectors and twisted vectors behave very differently if we look at their images through a mirror parallel to their axis: the orientation of ordinary vectors appears unchanged, whereas the orientation of twisted vectors appears \emph{reversed}:
+\begin{center}
+  \includegraphics[width=0.67\linewidth]{images/mirror.pdf}
+\end{center}
+this phenomenon reflects the behaviour of rotations under reflections.
+
+For some mysterious reason many books are afraid of using twisted vectors, and rely on ordinary vectors instead,
+\marginpar{%
+\includegraphics[align=t,width=\linewidth]{images/curlrighthand.png}
+% \\[\jot]%
+% \footnotesize{\color{mpcolor}The amount of momentum within a volume is represented by a \textcolor{cyan}{(3D) vector}}
+}
+introducing the \enquote{right-hand rule} to determine the sense of rotation from the arrow of the ordinary vector. If you've ever asked yourself \enquote{why the right hand, and not the left hand?}, the answer is that it's purely a convention; one could have introduced a left-hand rule instead. Using twisted vectors we don't need these arbitrary conventions and mnemonics: the sense of rotation is unequivocally indicated by the twisted vector.
+
+Use whichever vector representation you prefer! % In these notes we shall use a hybrid notation, as in the side picture of p.~\pageref{fig:volumetwistedarrow}, so as to make everybody (or no one) happy.
 
 \printpagenotes*
 \clearpage
diff --git a/seven-wonders.toc b/seven-wonders.toc
index bc52803..1f37e26 100644
--- a/seven-wonders.toc
+++ b/seven-wonders.toc
@@ -47,167 +47,167 @@
 \contentsline {paragraph}{Stretched or moving rubber band.}{46}{section*.197}%
 \contentsline {paragraph}{Fission and atomic bombs.}{46}{pagenote.200}%
 \contentsline {paragraph}{Electric heater.}{47}{section*.205}%
-\contentsline {subsection}{\numberline {3.4.2}The practical use of the words \enquote *{mass} and \enquote *{energy}}{47}{subsection.209}%
+\contentsline {subsection}{\numberline {3.4.2}The practical use of the words \enquote *{mass} and \enquote *{energy}}{47}{subsection.206}%
 \contentsline {subsection}{\numberline {3.4.3}Different \enquote *{forms} of energy}{48}{subsection.213}%
-\contentsline {subsection}{\numberline {3.4.4}Amounts and forms of energy are coordinate- and observer-dependent}{49}{subsection.214}%
-\contentsline {section}{\numberline {3.5}Momentum}{50}{section.219}%
-\contentsline {section}{\numberline {3.6}Angular momentum}{52}{section.221}%
-\contentsline {subsection}{\numberline {3.6.1}Angular momentum as a twisted vector}{54}{subsection.229}%
-\contentsline {section}{\numberline {3.7}Entropy}{56}{section.231}%
-\contentsline {section}{\numberline {3.8}Metric}{58}{section.244}%
-\contentsline {section}{\numberline {3.9}Auxiliary quantities}{58}{section.245}%
-\contentsline {section}{URLs for chapter 3}{60}{section*.247}%
-\contentsline {chapter}{\chapternumberline {4}Volume contents and fluxes}{61}{chapter.285}%
-\contentsline {section}{\numberline {4.1}Control volumes and control surfaces}{62}{section.286}%
-\contentsline {section}{\numberline {4.2}Volume content}{64}{section.292}%
-\contentsline {subsection}{\numberline {4.2.1}Scalar quantities}{64}{subsection.293}%
-\contentsline {subsection}{\numberline {4.2.2}Vector quantities}{65}{subsection.295}%
-\contentsline {section}{\numberline {4.3}Flux: scalar quantities}{67}{section.299}%
-\contentsline {subsection}{\numberline {4.3.1}The direction, reckoning, and representation of scalar fluxes}{67}{subsection.300}%
-\contentsline {subsection}{\numberline {4.3.2}How does a scalar flux change, if we change the surface?}{69}{subsection.303}%
-\contentsline {subsection}{\numberline {4.3.3}Flux units: scalar quantities}{70}{subsection.304}%
-\contentsline {subsection}{\numberline {4.3.4}Matter flux}{70}{subsection.305}%
-\contentsline {subsection}{\numberline {4.3.5}Energy flux}{70}{subsection.306}%
-\contentsline {section}{\numberline {4.4}Flux: vector quantities}{71}{section.312}%
-\contentsline {subsection}{\numberline {4.4.1}Representation of vector fluxes}{71}{subsection.313}%
-\contentsline {subsection}{\numberline {4.4.2}Changes of vector flux upon changes of surface}{74}{subsection.324}%
-\contentsline {subsection}{\numberline {4.4.3}Flux units: vector quantities}{75}{subsection.331}%
-\contentsline {section}{\numberline {4.5}Flux of momentum: force}{75}{section.332}%
-\contentsline {subsection}{\numberline {4.5.1}Units}{75}{subsection.333}%
-\contentsline {subsection}{\numberline {4.5.2}Visualizing force as flux of momentum}{75}{subsection.334}%
-\contentsline {subsection}{\numberline {4.5.3}Netwon's Third Law!}{77}{subsection.335}%
-\contentsline {section}{\numberline {4.6}Pressure, tension, shear force}{78}{section.340}%
-\contentsline {subsection}{\numberline {4.6.1}Pressure}{79}{subsection.341}%
-\contentsline {subsection}{\numberline {4.6.2}Tension}{79}{subsection.345}%
-\contentsline {subsection}{\numberline {4.6.3}Shear force}{79}{subsection.349}%
-\contentsline {section}{\numberline {4.7}Closed control surfaces, influxes, effluxes}{80}{section.354}%
-\contentsline {section}{\numberline {4.8}Time-integrated fluxes}{82}{section.364}%
-\contentsline {section}{\numberline {4.9}Fluxes and velocities}{83}{section.370}%
-\contentsline {section}{\numberline {4.10}Symbols for volume contents and fluxes}{85}{section.373}%
-\contentsline {section}{URLs for chapter 4}{86}{section*.374}%
-\contentsline {chapter}{\chapternumberline {5}Physical laws}{87}{chapter.386}%
-\contentsline {section}{\numberline {5.1}Fundamental vs derived laws}{87}{section.387}%
-\contentsline {section}{\numberline {5.2}Universal vs constitutive laws}{89}{section.391}%
-\contentsline {section}{\numberline {5.3}Balance and conservation laws}{91}{section.392}%
-\contentsline {section}{\numberline {5.4}Conservation laws}{91}{section.393}%
-\contentsline {subsection}{\numberline {5.4.1}Differential form of conservation laws}{94}{subsection.398}%
-\contentsline {subsection}{\numberline {5.4.2}Example with a moving control surface}{95}{subsection.400}%
-\contentsline {subsection}{\numberline {5.4.3}Example with a static control surface}{95}{subsection.401}%
-\contentsline {section}{\numberline {5.5}Balance laws}{98}{section.411}%
-\contentsline {subsection}{\numberline {5.5.1}Balance law for vector quantities}{99}{subsection.413}%
-\contentsline {subsection}{\numberline {5.5.2}Differential form of balance laws}{100}{subsection.416}%
-\contentsline {subsection}{\numberline {5.5.3}Example with a moving control surface}{101}{subsection.419}%
-\contentsline {subsection}{\numberline {5.5.4}Example with a static control surface}{102}{subsection.420}%
-\contentsline {subsection}{\numberline {5.5.5}Another example with a moving control surface}{104}{subsection.421}%
-\contentsline {section}{\numberline {5.6}Constitutive relations}{106}{section.423}%
-\contentsline {subsection}{\numberline {5.6.1}Examples}{108}{subsection.427}%
-\contentsline {section}{URLs for chapter 5}{110}{section*.434}%
-\contentsline {chapter}{\chapternumberline {6}The Seven Wonders of the world}{111}{chapter.441}%
-\contentsline {section}{\numberline {6.1}Seven universal balance laws}{111}{section.442}%
-\contentsline {section}{\numberline {6.2}General form of the universal balance laws}{112}{section.449}%
-\contentsline {subsection}{\numberline {6.2.1}Roles of the seven balances}{115}{subsection.454}%
-\contentsline {section}{\numberline {6.3}Numerical time integration and simulations}{116}{section.458}%
-\contentsline {subsection}{\numberline {6.3.1}Prediction and forecast}{116}{subsection.459}%
-\contentsline {subsection}{\numberline {6.3.2}The special role of the universal balance laws, and finite-difference approximations}{117}{subsection.466}%
-\contentsline {subsection}{\numberline {6.3.3}Vector quantities}{120}{subsection.474}%
-\contentsline {subsection}{\numberline {6.3.4}Iterating: numerical time integration and boundary conditions}{121}{subsection.479}%
-\contentsline {subsection}{\numberline {6.3.5}Boundary conditions and constitutive equations}{121}{subsection.487}%
-\contentsline {subsection}{\numberline {6.3.6}Numerical time integration of position}{123}{subsection.492}%
-\contentsline {subsection}{\numberline {6.3.7}Numerical time integration of position: example}{124}{subsection.495}%
-\contentsline {subsection}{\numberline {6.3.8}Applicability of numerical time integration}{125}{subsection.497}%
-\contentsline {subsection}{\numberline {6.3.9}Example script for numerical time integration}{126}{subsection.504}%
-\contentsline {section}{URLs for chapter 6}{129}{section*.592}%
-\contentsline {chapter}{\chapternumberline {7}Conservation \&\ balances of matter}{130}{chapter.606}%
-\contentsline {section}{\numberline {7.1}Formulation and generalities}{130}{section.607}%
-\contentsline {subsection}{\numberline {7.1.1}Balance vs conservation of matter}{131}{subsection.609}%
-\contentsline {section}{\numberline {7.2}Examples of constitutive relations}{131}{section.610}%
-\contentsline {subsection}{\numberline {7.2.1}Relation between matter and mass-energy}{131}{subsection.611}%
-\contentsline {subsection}{\numberline {7.2.2}Conservation of mass: proxy for conservation of matter}{133}{subsection.621}%
-\contentsline {subsection}{\numberline {7.2.3}Radioactive decay}{133}{subsection.628}%
-\contentsline {section}{\numberline {7.3}Examples of applications}{134}{section.635}%
-\contentsline {subsection}{\numberline {7.3.1}Rigid-body and particle mechanics}{134}{subsection.636}%
-\contentsline {subsection}{\numberline {7.3.2}Chemistry}{134}{subsection.637}%
-\contentsline {subsection}{\numberline {7.3.3}Climate}{135}{subsection.650}%
-\contentsline {subsection}{\numberline {7.3.4}Nozzle flow}{136}{subsection.664}%
-\contentsline {section}{URLs for chapter 7}{138}{section*.668}%
-\contentsline {chapter}{\chapternumberline {8}Balance of momentum}{139}{chapter.684}%
-\contentsline {section}{\numberline {8.1}Formulation and generalities}{139}{section.689}%
-\contentsline {section}{\numberline {8.2}Examples of constitutive relations}{141}{section.694}%
-\contentsline {subsection}{\numberline {8.2.1}Newtonian momentum: relation between momentum and matter}{141}{subsection.695}%
-\contentsline {subsection}{\numberline {8.2.2}Hookean spring: relation between momentum flux and distance}{142}{subsection.697}%
-\contentsline {subsection}{\numberline {8.2.3}Non-hookean springs}{143}{subsection.704}%
-\contentsline {subsection}{\numberline {8.2.4}Pairwise forces}{144}{subsection.706}%
-\contentsline {subsection}{\numberline {8.2.5}Gravity and momentum supply near a planet's surface}{145}{subsection.709}%
-\contentsline {subsection}{\numberline {8.2.6}Contact forces}{146}{subsection.715}%
-\contentsline {paragraph}{Normal force}{147}{section*.717}%
-\contentsline {paragraph}{Friction}{148}{section*.718}%
-\contentsline {paragraph}{Static friction}{148}{section*.719}%
-\contentsline {paragraph}{Kinetic friction}{149}{section*.721}%
-\contentsline {section}{\numberline {8.3}Examples of applications}{151}{section.740}%
-\contentsline {subsection}{\numberline {8.3.1}Statics}{151}{subsection.741}%
-\contentsline {subsection}{\numberline {8.3.2}Atmospheric pressure}{153}{subsection.748}%
-\contentsline {subsection}{\numberline {8.3.3}Airborne flight}{155}{subsection.759}%
-\contentsline {subsection}{\numberline {8.3.4}Rockets}{156}{subsection.775}%
-\contentsline {subsection}{\numberline {8.3.5}Statics again: cable cars}{157}{subsection.776}%
-\contentsline {subsection}{\numberline {8.3.6}Momentum fluxes in a gas}{160}{subsection.780}%
-\contentsline {subsection}{\numberline {8.3.7}Hookean spring and harmonic oscillator}{163}{subsection.784}%
-\contentsline {subsection}{\numberline {8.3.8}Many-body systems}{167}{subsection.802}%
-\contentsline {section}{\numberline {8.4}Choice of control surfaces and volumes}{168}{section.812}%
-\contentsline {section}{\numberline {8.5}Numerical time integration: a strategy}{169}{section.813}%
-\contentsline {subsection}{\numberline {8.5.1}Overview of the relevant equations}{170}{subsection.814}%
-\contentsline {subsection}{\numberline {8.5.2}A strategy for writing a numerical time-integration algorithm}{171}{subsection.815}%
-\contentsline {paragraph}{\color {red}0. Find any constants appearing in the equations.}{171}{section*.816}%
-\contentsline {paragraph}{\color {green}1. Find which equations drive the system forward in time.}{172}{section*.817}%
-\contentsline {paragraph}{\color {yellow}2. Choose a \emph {state} for the physical system.}{172}{section*.818}%
-\contentsline {paragraph}{\color {blue}3. From the state, determine the quantities necessary for forward-driving.}{174}{section*.819}%
-\contentsline {paragraph}{\color {cyan}4. Find the new state from the time-updated quantities.}{174}{section*.820}%
-\contentsline {paragraph}{\color {midgrey}5. Decide the condition for stopping the time loop.}{175}{section*.821}%
-\contentsline {subsection}{\numberline {8.5.3}Non-Hookean spring: numerical time integration}{178}{subsection.836}%
-\contentsline {section}{\numberline {8.6}Example script for non-Hookean spring }{180}{section.838}%
-\contentsline {section}{URLs for chapter 8}{183}{section*.957}%
-\contentsline {chapter}{\chapternumberline {9}Balance of energy}{184}{chapter.979}%
-\contentsline {section}{\numberline {9.1}Formulation and generalities}{184}{section.980}%
-\contentsline {subsection}{\numberline {9.1.1}Definitions of \enquote *{total energy}}{185}{subsection.985}%
-\contentsline {subsection}{\numberline {9.1.2}Forms of energy}{187}{subsection.998}%
-\contentsline {subsection}{\numberline {9.1.3}Is energy conserved?}{188}{subsection.1008}%
-\contentsline {subsection}{\numberline {9.1.4}Temperature}{189}{subsection.1021}%
-\contentsline {section}{\numberline {9.2}Constitutive relations for energy content}{190}{section.1029}%
-\contentsline {subsection}{\numberline {9.2.1}Internal, kinetic, gravitational potential energy}{190}{subsection.1030}%
-\contentsline {subsection}{\numberline {9.2.2}The separation between internal and kinetic energy depends on the observation scale}{192}{subsection.1037}%
-\contentsline {subsection}{\numberline {9.2.3}Examples of the exchange between internal, kinetic, gravitational potential energies}{193}{subsection.1038}%
-\contentsline {paragraph}{Bodies in motion.}{193}{section*.1039}%
-\contentsline {paragraph}{Springs and rubber bands.}{194}{section*.1041}%
-\contentsline {paragraph}{Gases.}{194}{section*.1042}%
-\contentsline {paragraph}{Solids and fluids.}{194}{section*.1043}%
-\contentsline {section}{\numberline {9.3}Constitutive relations for energy flux}{195}{section.1044}%
-\contentsline {subsection}{\numberline {9.3.1}Comments about movement of matter at a surface}{195}{subsection.1045}%
-\contentsline {subsection}{\numberline {9.3.2}Heat flux and power}{195}{subsection.1046}%
-\contentsline {subsection}{\numberline {9.3.3}The separation between heat and power depends on the observation scale}{197}{subsection.1048}%
-\contentsline {subsection}{\numberline {9.3.4}Examples of heat and mechanical-power fluxes}{197}{subsection.1049}%
-\contentsline {paragraph}{Holding a cup of hot tea.}{197}{section*.1050}%
-\contentsline {paragraph}{Cooking.}{198}{section*.1054}%
-\contentsline {paragraph}{Spring and body.}{198}{section*.1055}%
-\contentsline {paragraph}{Gases.}{198}{section*.1056}%
-\contentsline {subsection}{\numberline {9.3.5}Summary: energy constitutive relations for matter}{199}{subsection.1064}%
-\contentsline {section}{\numberline {9.4}Rigid bodies}{201}{section.1068}%
-\contentsline {section}{\numberline {9.5}Constitutive relations for ideal gases}{203}{section.1082}%
-\contentsline {subsection}{\numberline {9.5.1}Pressure (momentum flux) of an ideal gas}{204}{subsection.1083}%
-\contentsline {subsection}{\numberline {9.5.2}Internal energy of an ideal gas}{206}{subsection.1089}%
-\contentsline {subsection}{\numberline {9.5.3}Heat flux between sides at different temperatures}{207}{subsection.1091}%
-\contentsline {subsection}{\numberline {9.5.4}Other common assumptions about ideal gases}{208}{subsection.1096}%
-\contentsline {section}{\numberline {9.6}Example applications: ideal gas and piston}{209}{section.1098}%
-\contentsline {subsection}{\numberline {9.6.1}Setup}{209}{subsection.1099}%
-\contentsline {subsection}{\numberline {9.6.2}Control volumes \&\ surfaces}{210}{subsection.1100}%
-\contentsline {subsection}{\numberline {9.6.3}Conservation of matter}{211}{subsection.1103}%
-\contentsline {subsection}{\numberline {9.6.4}Balances of momentum}{211}{subsection.1104}%
-\contentsline {paragraph}{Piston.}{211}{section*.1105}%
-\contentsline {paragraph}{Ideal gas.}{213}{section*.1111}%
-\contentsline {subsection}{\numberline {9.6.5}Balances of energy}{214}{subsection.1113}%
-\contentsline {paragraph}{Piston.}{214}{section*.1114}%
-\contentsline {paragraph}{Ideal gas.}{214}{section*.1115}%
-\contentsline {section}{\numberline {9.7}Surfaces of discontinuity}{219}{section.1231}%
-\contentsline {section}{URLs for chapter 9}{223}{section*.1232}%
-\contentsline {chapter}{\chapternumberline {10}Balance of angular momentum}{224}{chapter.1254}%
+\contentsline {section}{\numberline {3.5}Momentum}{49}{section.218}%
+\contentsline {section}{\numberline {3.6}Angular momentum}{51}{section.220}%
+\contentsline {subsection}{\numberline {3.6.1}Amounts of energy, momentum, angular momentum are coordinate-dependent}{53}{subsection.229}%
+\contentsline {section}{\numberline {3.7}Entropy}{55}{section.230}%
+\contentsline {section}{\numberline {3.8}Metric}{56}{section.243}%
+\contentsline {section}{\numberline {3.9}Auxiliary quantities}{56}{section.244}%
+\contentsline {section}{URLs for chapter 3}{58}{section*.246}%
+\contentsline {chapter}{\chapternumberline {4}Volume contents and fluxes}{59}{chapter.284}%
+\contentsline {section}{\numberline {4.1}Control volumes and control surfaces}{60}{section.285}%
+\contentsline {section}{\numberline {4.2}Volume content}{62}{section.291}%
+\contentsline {subsection}{\numberline {4.2.1}Scalar quantities}{62}{subsection.292}%
+\contentsline {subsection}{\numberline {4.2.2}Vector quantities}{63}{subsection.294}%
+\contentsline {section}{\numberline {4.3}Flux: scalar quantities}{65}{section.298}%
+\contentsline {subsection}{\numberline {4.3.1}The direction, reckoning, and representation of scalar fluxes}{65}{subsection.299}%
+\contentsline {subsection}{\numberline {4.3.2}How does a scalar flux change, if we change the surface?}{67}{subsection.302}%
+\contentsline {subsection}{\numberline {4.3.3}Flux units: scalar quantities}{68}{subsection.303}%
+\contentsline {subsection}{\numberline {4.3.4}Matter flux}{68}{subsection.304}%
+\contentsline {subsection}{\numberline {4.3.5}Energy flux}{68}{subsection.305}%
+\contentsline {section}{\numberline {4.4}Flux: vector quantities}{69}{section.311}%
+\contentsline {subsection}{\numberline {4.4.1}Representation of vector fluxes}{69}{subsection.312}%
+\contentsline {subsection}{\numberline {4.4.2}Changes of vector flux upon changes of surface}{72}{subsection.323}%
+\contentsline {subsection}{\numberline {4.4.3}Flux units: vector quantities}{73}{subsection.330}%
+\contentsline {section}{\numberline {4.5}Flux of momentum: force}{73}{section.331}%
+\contentsline {subsection}{\numberline {4.5.1}Units}{73}{subsection.332}%
+\contentsline {subsection}{\numberline {4.5.2}Visualizing force as flux of momentum}{73}{subsection.333}%
+\contentsline {subsection}{\numberline {4.5.3}Netwon's Third Law!}{75}{subsection.334}%
+\contentsline {section}{\numberline {4.6}Pressure, tension, shear force}{76}{section.339}%
+\contentsline {subsection}{\numberline {4.6.1}Pressure}{77}{subsection.340}%
+\contentsline {subsection}{\numberline {4.6.2}Tension}{77}{subsection.344}%
+\contentsline {subsection}{\numberline {4.6.3}Shear force}{77}{subsection.348}%
+\contentsline {section}{\numberline {4.7}Closed control surfaces, influxes, effluxes}{78}{section.353}%
+\contentsline {section}{\numberline {4.8}Time-integrated fluxes}{80}{section.363}%
+\contentsline {section}{\numberline {4.9}Fluxes and velocities}{81}{section.369}%
+\contentsline {section}{\numberline {4.10}Symbols for volume contents and fluxes}{83}{section.372}%
+\contentsline {section}{URLs for chapter 4}{84}{section*.373}%
+\contentsline {chapter}{\chapternumberline {5}Physical laws}{85}{chapter.385}%
+\contentsline {section}{\numberline {5.1}Fundamental vs derived laws}{85}{section.386}%
+\contentsline {section}{\numberline {5.2}Universal vs constitutive laws}{87}{section.390}%
+\contentsline {section}{\numberline {5.3}Balance and conservation laws}{89}{section.391}%
+\contentsline {section}{\numberline {5.4}Conservation laws}{89}{section.392}%
+\contentsline {subsection}{\numberline {5.4.1}Differential form of conservation laws}{92}{subsection.397}%
+\contentsline {subsection}{\numberline {5.4.2}Example with a moving control surface}{93}{subsection.399}%
+\contentsline {subsection}{\numberline {5.4.3}Example with a static control surface}{93}{subsection.400}%
+\contentsline {section}{\numberline {5.5}Balance laws}{96}{section.410}%
+\contentsline {subsection}{\numberline {5.5.1}Balance law for vector quantities}{97}{subsection.412}%
+\contentsline {subsection}{\numberline {5.5.2}Differential form of balance laws}{98}{subsection.415}%
+\contentsline {subsection}{\numberline {5.5.3}Example with a moving control surface}{99}{subsection.418}%
+\contentsline {subsection}{\numberline {5.5.4}Example with a static control surface}{100}{subsection.419}%
+\contentsline {subsection}{\numberline {5.5.5}Another example with a moving control surface}{102}{subsection.420}%
+\contentsline {section}{\numberline {5.6}Constitutive relations}{104}{section.422}%
+\contentsline {subsection}{\numberline {5.6.1}Examples}{106}{subsection.426}%
+\contentsline {section}{URLs for chapter 5}{108}{section*.433}%
+\contentsline {chapter}{\chapternumberline {6}The Seven Wonders of the world}{109}{chapter.440}%
+\contentsline {section}{\numberline {6.1}Seven universal balance laws}{109}{section.441}%
+\contentsline {section}{\numberline {6.2}General form of the universal balance laws}{110}{section.448}%
+\contentsline {subsection}{\numberline {6.2.1}Roles of the seven balances}{113}{subsection.453}%
+\contentsline {section}{\numberline {6.3}Numerical time integration and simulations}{114}{section.457}%
+\contentsline {subsection}{\numberline {6.3.1}Prediction and forecast}{114}{subsection.458}%
+\contentsline {subsection}{\numberline {6.3.2}The special role of the universal balance laws, and finite-difference approximations}{115}{subsection.465}%
+\contentsline {subsection}{\numberline {6.3.3}Vector quantities}{118}{subsection.473}%
+\contentsline {subsection}{\numberline {6.3.4}Iterating: numerical time integration and boundary conditions}{119}{subsection.478}%
+\contentsline {subsection}{\numberline {6.3.5}Boundary conditions and constitutive equations}{119}{subsection.486}%
+\contentsline {subsection}{\numberline {6.3.6}Numerical time integration of position}{121}{subsection.491}%
+\contentsline {subsection}{\numberline {6.3.7}Numerical time integration of position: example}{122}{subsection.494}%
+\contentsline {subsection}{\numberline {6.3.8}Applicability of numerical time integration}{123}{subsection.496}%
+\contentsline {subsection}{\numberline {6.3.9}Example script for numerical time integration}{124}{subsection.503}%
+\contentsline {section}{URLs for chapter 6}{127}{section*.591}%
+\contentsline {chapter}{\chapternumberline {7}Conservation \&\ balances of matter}{128}{chapter.605}%
+\contentsline {section}{\numberline {7.1}Formulation and generalities}{128}{section.606}%
+\contentsline {subsection}{\numberline {7.1.1}Balance vs conservation of matter}{129}{subsection.608}%
+\contentsline {section}{\numberline {7.2}Examples of constitutive relations}{129}{section.609}%
+\contentsline {subsection}{\numberline {7.2.1}Relation between matter and mass-energy}{129}{subsection.610}%
+\contentsline {subsection}{\numberline {7.2.2}Conservation of mass: proxy for conservation of matter}{131}{subsection.620}%
+\contentsline {subsection}{\numberline {7.2.3}Radioactive decay}{131}{subsection.627}%
+\contentsline {section}{\numberline {7.3}Examples of applications}{132}{section.634}%
+\contentsline {subsection}{\numberline {7.3.1}Rigid-body and particle mechanics}{132}{subsection.635}%
+\contentsline {subsection}{\numberline {7.3.2}Chemistry}{132}{subsection.636}%
+\contentsline {subsection}{\numberline {7.3.3}Climate}{133}{subsection.649}%
+\contentsline {subsection}{\numberline {7.3.4}Nozzle flow}{134}{subsection.663}%
+\contentsline {section}{URLs for chapter 7}{136}{section*.667}%
+\contentsline {chapter}{\chapternumberline {8}Balance of momentum}{137}{chapter.683}%
+\contentsline {section}{\numberline {8.1}Formulation and generalities}{137}{section.688}%
+\contentsline {section}{\numberline {8.2}Examples of constitutive relations}{139}{section.693}%
+\contentsline {subsection}{\numberline {8.2.1}Newtonian momentum: relation between momentum and matter}{139}{subsection.694}%
+\contentsline {subsection}{\numberline {8.2.2}Hookean spring: relation between momentum flux and distance}{140}{subsection.696}%
+\contentsline {subsection}{\numberline {8.2.3}Non-hookean springs}{141}{subsection.703}%
+\contentsline {subsection}{\numberline {8.2.4}Pairwise forces}{142}{subsection.705}%
+\contentsline {subsection}{\numberline {8.2.5}Gravity and momentum supply near a planet's surface}{143}{subsection.708}%
+\contentsline {subsection}{\numberline {8.2.6}Contact forces}{144}{subsection.714}%
+\contentsline {paragraph}{Normal force}{145}{section*.716}%
+\contentsline {paragraph}{Friction}{146}{section*.717}%
+\contentsline {paragraph}{Static friction}{146}{section*.718}%
+\contentsline {paragraph}{Kinetic friction}{147}{section*.720}%
+\contentsline {section}{\numberline {8.3}Examples of applications}{149}{section.739}%
+\contentsline {subsection}{\numberline {8.3.1}Statics}{149}{subsection.740}%
+\contentsline {subsection}{\numberline {8.3.2}Atmospheric pressure}{151}{subsection.747}%
+\contentsline {subsection}{\numberline {8.3.3}Airborne flight}{153}{subsection.758}%
+\contentsline {subsection}{\numberline {8.3.4}Rockets}{154}{subsection.774}%
+\contentsline {subsection}{\numberline {8.3.5}Statics again: cable cars}{155}{subsection.775}%
+\contentsline {subsection}{\numberline {8.3.6}Momentum fluxes in a gas}{158}{subsection.779}%
+\contentsline {subsection}{\numberline {8.3.7}Hookean spring and harmonic oscillator}{161}{subsection.783}%
+\contentsline {subsection}{\numberline {8.3.8}Many-body systems}{165}{subsection.801}%
+\contentsline {section}{\numberline {8.4}Choice of control surfaces and volumes}{166}{section.811}%
+\contentsline {section}{\numberline {8.5}Numerical time integration: a strategy}{167}{section.812}%
+\contentsline {subsection}{\numberline {8.5.1}Overview of the relevant equations}{168}{subsection.813}%
+\contentsline {subsection}{\numberline {8.5.2}A strategy for writing a numerical time-integration algorithm}{169}{subsection.814}%
+\contentsline {paragraph}{\color {red}0. Find any constants appearing in the equations.}{169}{section*.815}%
+\contentsline {paragraph}{\color {green}1. Find which equations drive the system forward in time.}{170}{section*.816}%
+\contentsline {paragraph}{\color {yellow}2. Choose a \emph {state} for the physical system.}{170}{section*.817}%
+\contentsline {paragraph}{\color {blue}3. From the state, determine the quantities necessary for forward-driving.}{172}{section*.818}%
+\contentsline {paragraph}{\color {cyan}4. Find the new state from the time-updated quantities.}{172}{section*.819}%
+\contentsline {paragraph}{\color {midgrey}5. Decide the condition for stopping the time loop.}{173}{section*.820}%
+\contentsline {subsection}{\numberline {8.5.3}Non-Hookean spring: numerical time integration}{176}{subsection.835}%
+\contentsline {section}{\numberline {8.6}Example script for non-Hookean spring }{178}{section.837}%
+\contentsline {section}{URLs for chapter 8}{181}{section*.956}%
+\contentsline {chapter}{\chapternumberline {9}Balance of energy}{182}{chapter.978}%
+\contentsline {section}{\numberline {9.1}Formulation and generalities}{182}{section.979}%
+\contentsline {subsection}{\numberline {9.1.1}Definitions of \enquote *{total energy}}{183}{subsection.984}%
+\contentsline {subsection}{\numberline {9.1.2}Forms of energy}{185}{subsection.997}%
+\contentsline {subsection}{\numberline {9.1.3}Is energy conserved?}{186}{subsection.1007}%
+\contentsline {subsection}{\numberline {9.1.4}Temperature}{187}{subsection.1020}%
+\contentsline {section}{\numberline {9.2}Constitutive relations for energy content}{188}{section.1028}%
+\contentsline {subsection}{\numberline {9.2.1}Internal, kinetic, gravitational potential energy}{188}{subsection.1029}%
+\contentsline {subsection}{\numberline {9.2.2}The separation between internal and kinetic energy depends on the observation scale}{190}{subsection.1036}%
+\contentsline {subsection}{\numberline {9.2.3}Examples of the exchange between internal, kinetic, gravitational potential energies}{191}{subsection.1037}%
+\contentsline {paragraph}{Bodies in motion.}{191}{section*.1038}%
+\contentsline {paragraph}{Springs and rubber bands.}{192}{section*.1040}%
+\contentsline {paragraph}{Gases.}{192}{section*.1041}%
+\contentsline {paragraph}{Solids and fluids.}{192}{section*.1042}%
+\contentsline {section}{\numberline {9.3}Constitutive relations for energy flux}{193}{section.1043}%
+\contentsline {subsection}{\numberline {9.3.1}Comments about movement of matter at a surface}{193}{subsection.1044}%
+\contentsline {subsection}{\numberline {9.3.2}Heat flux and power}{193}{subsection.1045}%
+\contentsline {subsection}{\numberline {9.3.3}The separation between heat and power depends on the observation scale}{195}{subsection.1047}%
+\contentsline {subsection}{\numberline {9.3.4}Examples of heat and mechanical-power fluxes}{195}{subsection.1048}%
+\contentsline {paragraph}{Holding a cup of hot tea.}{195}{section*.1049}%
+\contentsline {paragraph}{Cooking.}{196}{section*.1053}%
+\contentsline {paragraph}{Spring and body.}{196}{section*.1054}%
+\contentsline {paragraph}{Gases.}{196}{section*.1055}%
+\contentsline {subsection}{\numberline {9.3.5}Summary: energy constitutive relations for matter}{197}{subsection.1063}%
+\contentsline {section}{\numberline {9.4}Rigid bodies}{199}{section.1067}%
+\contentsline {section}{\numberline {9.5}Constitutive relations for ideal gases}{201}{section.1081}%
+\contentsline {subsection}{\numberline {9.5.1}Pressure (momentum flux) of an ideal gas}{202}{subsection.1082}%
+\contentsline {subsection}{\numberline {9.5.2}Internal energy of an ideal gas}{204}{subsection.1088}%
+\contentsline {subsection}{\numberline {9.5.3}Heat flux between sides at different temperatures}{205}{subsection.1090}%
+\contentsline {subsection}{\numberline {9.5.4}Other common assumptions about ideal gases}{206}{subsection.1095}%
+\contentsline {section}{\numberline {9.6}Example applications: ideal gas and piston}{207}{section.1097}%
+\contentsline {subsection}{\numberline {9.6.1}Setup}{207}{subsection.1098}%
+\contentsline {subsection}{\numberline {9.6.2}Control volumes \&\ surfaces}{208}{subsection.1099}%
+\contentsline {subsection}{\numberline {9.6.3}Conservation of matter}{209}{subsection.1102}%
+\contentsline {subsection}{\numberline {9.6.4}Balances of momentum}{209}{subsection.1103}%
+\contentsline {paragraph}{Piston.}{209}{section*.1104}%
+\contentsline {paragraph}{Ideal gas.}{211}{section*.1110}%
+\contentsline {subsection}{\numberline {9.6.5}Balances of energy}{212}{subsection.1112}%
+\contentsline {paragraph}{Piston.}{212}{section*.1113}%
+\contentsline {paragraph}{Ideal gas.}{212}{section*.1114}%
+\contentsline {section}{\numberline {9.7}Surfaces of discontinuity}{217}{section.1230}%
+\contentsline {section}{URLs for chapter 9}{221}{section*.1231}%
+\contentsline {chapter}{\chapternumberline {10}Balance of angular momentum}{222}{chapter.1253}%
+\contentsline {subsection}{\numberline {10.0.1}Angular momentum as a twisted vector}{222}{subsection.1255}%
 \contentsline {chapter}{\chapternumberline {11}Balance of entropy}{225}{chapter.1256}%
 \contentsline {section}{\numberline {11.1}Formulation and generalities}{225}{section.1257}%
 \contentsline {subsection}{\numberline {11.1.1}Thermodynamic entropy and statistical entropy}{226}{subsection.1262}%