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18

The symbol $\Delta$ refers to a finite variation or change of a quantity – by finite, I mean one that is not infinitely small. The symbols $d,\delta$ refer to infinitesimal variations or numerators and denominators of derivatives. The difference between $d$ and $\delta$ is that $dX$ is only used if $X$ without the $d$ is an actual quantity that may be ...


13

The wording of the question suggests that the electrons were the first objects or particles whose charge required the people to establish the sign convention. But that's obviously not the case. The electron was discovered by J. J. Thomson in 1897 but for much more than a century before that moment, people had already been studying electric (and magnetic) ...


9

The relative sign is not just a convention. Once you decide that $E$ is represented by $i\hbar \partial/\partial t$, there must be a minus sign in the formula for $p$, namely $p=-i\hbar \partial / \partial x$. Or vice versa. First of all, there has to be $i$ or $-i$ in all the formulae because $\partial/\partial x$ is an anti-Hermitian operator (because of ...


9

First, recall what a partition is. A partition of a set $X$ is a way to write $X$ as a disjoint union of subsets: $X=\coprod_i X_i$, $X_i\cap X_j=\emptyset$ for $i\neq j$. When the elements of the set $X$ are considered undistinguishable, what matters are the cardinals of the set only, and we have a partition of an integer number, $n=n_1+\ldots+n_k$. For ...


9

I had an extensive look around, and I turned up four conventions. This included a short poll of google, other questions on this and other sites, and multiple standards documents. (I make no claim of exhaustiveness or infallibility, by the way.) Using $[q]$ to denote commensurability as an equivalence relation. That is, if $q$ and $p$ have the same ...


8

The kilogram is defined by a prototype (the "International Prototype Kilogram", IPK) -- basically, a kilogram is by definition the mass of a metal cylinder sitting in a vault in Paris. People have made a bunch of other metal blocks with almost exactly the same mass (as near as they could get), called "sister copies". To measure a mass extremely accurately in ...


8

The clockwise direction is normally defined by the right hand grip rule. When your thumb is pointing away from you, your fingers are curled clockwise. So when you look at a clock the axis of rotation is away from you through the clock. I'd guess the downvotes are because people believe your question is not physics related, but in fact this rule is how ...


7

Yes, to some extent. Once you choose which of the electron or positron is to be considered the normal particle, then that fixes your choice for the other leptons, because of neutrino mixing. Similarly, choosing one quark to be the normal particle fixes the choice for the other flavors and colors of quarks. But I can't think of a reason within the standard ...


7

You can absolutely have negative pressure in solids or liquids. Think of an elastic solid being forced to expand to to adhesion to the walls of some chamber. That has negative pressure even if the comparison is a total vacuum. Depending on the bulk modulus of the material being stretched and the strength of the interaction with the walls of the chamber ...


7

Mass isn't always first. For example we write Newton's law for the force between two objects as: $$ F = \frac{Gm_1m_2}{r^2} $$ I don't think there are hard and fast rules. I suspect conventions have arisen over the years and we have all got used to what we learned at school, which was taught by teachers who are used to what they learned at school and so ...


6

There are two separate issues here. (1) Why does it make sense to consider a dipole moment as a vector? (2) Given that it's a vector, why does it make sense to say that it points in this particular direction, rather than the opposite direction. Intuitively, it makes sense to define a dipole as a vector because when we put it in a field, it aligns itself ...


6

This is analogous to the definition of an empty product in mathematics. For a finite non-empty set $S=\{s_1,\ldots,s_n\}$, the product over $S$ can be defined as $$\prod_{s\in S}s=s_1\times \cdots\times s_n.$$ For such a product you'd want disjoint unions to map into products: if $R\cap S=\emptyset$, then you want $\prod_{x\in R\cup S}x=\left(\prod_{s\in ...


6

It is a matter of convention. The sign convention of Clausius and the sign convention of IUPAC are the two prevailing sign conventions. Both of these assign a sign to the work done differently. The former, used primarily in physics assign a positive sign to the work done by the system while the latter assigns positive sign to the work done on the system. ...


6

I would add to John's answer that $a$ is not always constant. It represents the second derivative of motion, and thus is potentially a function of time. So, the overall conventional ordering in equations (in Mathematics as well as Physics) is, $$\mathrm{Constant \times Parameter \times Variable}$$ where I'm distinguishing between, say, $G$ which is ...


5

First off, please don't use units with $c\ne 1$ in GR. It makes everything horribly messy. What we normally think of as a ruler or clock measurement is represented in GR by an upper index quantity like $\Delta x^\mu$. Therefore in a Cartesian coordinate system in the fluid's rest frame, we are guaranteed that $u^\mu=(1,0,0,0)$, not $(-1,0,0,0)$. This is ...


5

Your second equation, $P(\nu,T) = \frac{2 h {\nu}^3}{c^2}$ $\frac{1}{\exp\bigl(\frac{h \nu}{kT}\bigr) - 1}$ is what is commonly referred to as Planck's law for radiation, although a more standard symbol used is $B_\nu(T)$. This is the energy radiated per time, per area, per frequency interval, per steradian. It is a formula for the 'specific intensity' of a ...


5

Your confusion arises because the term (anti)clockwise, when used by itself, is ambiguous, and should always be used with a statement like "as seen from the top" (unless that is absolutely obvious$^1$). The reason for this is that "clockwise" defines a direction of rotation within a plane, but does not specify which side the plane is observed from. (In more ...


5

When radioactive element A decays to produce element B, the (infinitesimal) number of decayed elements A, $dN$, that occurs in a small time interval, $dt$, is proportional to the initial population of A, $N$: $$ -\frac{dN}{dt}\propto N $$ Assuming the proportionality is a constant, then the above becomes $$ -\frac{dN}{dt}=\lambda N $$ which has a known ...


4

I) Consider an arbitrary coordinate transformation $$x^{\mu}\longrightarrow x^{\prime \nu}~=~f^{\nu}(x).$$ Let $$J ~:=~\det(\frac{\partial x^{\prime \nu}}{\partial x^{\mu}})$$ denote the corresponding Jacobian. Traditionally in physics, a scalar $\sigma$ transforms as $$ \sigma ~\longrightarrow~ \sigma^{\prime}~=~\sigma, $$ a pseudo-scalar ...


4

The number $1$ may be linguistically described as "unity". This very number is the original source of various words in the terminology, like the "unit matrix" (a matrix behaving like the number $1$). It is a convention to write down that dimensionless quantities like the Mach number have units $1$ because the multiplication by $1$ changes nothing about the ...


4

It started with conservation of quantum numbers, from baryon number when we did not know about quarks, to lepton number, when we discovered the positron.For the neutrino momentum and energy conservation played a role too, since it is only seen as a missing mass. In time the symmetries in the assignments of the quantum numbers became more and more evident ...


4

You're right, absolute pressure can't be negative. Of course, you can easily have a $20\: \mathrm{PSI}$ pressure differential (although not without pressure above $1\: \mathrm{ATM}$ since that's $14.22\: \mathrm{PSI}$ at sea level). Check out Wikipedia on the zero-reference: Absolute pressure is zero-referenced against a perfect vacuum, so it is ...


4

I suppose you talk about the standard $2\pi$ that appears in the rules for Fourier transform. The factor of $2\pi$ or $1/2\pi$ or two factors of $1/\sqrt{2\pi}$ have to appear "somewhere" in the Fourier transform rules because this is what the mathematics implies. At any rate, if this is your question, it is a mathematical question and you may learn it in ...


4

In the theory of geometry of space-time, it makes much more sense to use $-+++$ because geometric view comes from intuition about 3D space, where we have metric $+++$. Time requires opposite sign, so we end up naturally with $-+++$ (or $+++-$ in some books). In other areas where geometry and space distances are not that important but particles and their ...


4

I endorse Kyle's answer. Just two short comments. The number 36.8% is literally $$ 36.8 \approx 100 \exp(-1) =\frac{100}{2.71828\dots} $$ Moreover, it is right to call this quantity "average lifetime" or just "lifetime" because it is literally the average value of the time for which a nucleus (or something else) from the ensemble lives. If the initial ...


3

To formalize the comments as an answer: The difference between requiring $$(\alpha u,v)=\alpha(u,v)\quad\text{ (mathematician's definition)}$$ and $$\langle u, \alpha v\rangle=\alpha\langle u,v\rangle\qquad\quad\,\,\text{ (physicist's definition)}$$ is purely one of convention, and the two definitions are equivalent as $(u,v)=\langle v,u\rangle$. There's no ...


3

I think you just forgot that the $\int_A^B F\,dl$ is not a scalar expression. Rather it should be written in a form $\int_A^B \vec{F}\cdot d\vec{l}$. Then it comes to the sign of the scalar product: $$\vec{F}\cdot d\vec{l}=F\,dl\,\cos\theta$$ where the angle $\theta$ is taken between the vector $\vec{F}$ and the direction of the tangent to the integration ...


3

In the $({+}{-}{-}{-})$ signature, positive values of $V^\mu V_\mu$ are referred to as "timelike" and negative ones correspond to "spacelike" 4-vectors. The velocities are timelike (positive) vectors because the world lines of massive objects must have a more timelike angle – that's because massless objects can't exceed the speed of light. On the contrary, ...


3

First, if you're going to keep proper track of covariant and contravariant components, you should lower the index on $B$ and make sure the dummy indices are always of opposite types: $B_k = \varepsilon_{kij} \phi^{ij}$. The reason we can be sloppy in Euclidean space is because of how trivial the metric can be. We can always consider our equations in the ...


3

Hmm, what if they thought about it this way: Let's put t for time (latin: tempus) and since time and period are connected (I would think of the period as the smallest time a cyclic phenomena requires to complete one full cycle) let's just denote period with T as t is already taken.



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