Hot answers tagged

36

This is not really an answer to your question, essentially because there isn't (currently) a question in your post, but it is too long for a comment. Your statement that A co-ordinate transformation is linear map from a vector to itself with a change of basis. is muddled and ultimately incorrect. Take some vector space $V$ and two bases $\beta$ and $\...


35

There is a consistent definition, but it involves a couple of arbitrary thresholds, so I doubt you'd consider it rigorous. The construction $X \gg Y$ means that the ratio $\frac{Y}{X}$ is small enough that subleading terms in the series expansion for $f\bigl(\frac{Y}{X}\bigr) - f(0)$ can be neglected, where $f$ is some relevant function involved in the ...


33

Straight from the horse's mouth: Source: Bureau International des Poids et Mesures (Search for "dimensionless" for all guidelines.) The International Bureau of Weights and Measures (French: Bureau international des poids et mesures), is an international standards organisation, one of three such organisations established to maintain the ...


32

Typically: $\rm d$ denotes the total derivative (sometimes called the exact differential):$$\frac{{\rm d}}{{\rm d}t}f(x,t)=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}\frac{{\rm d}x}{{\rm d}t}$$This is also sometimes denoted via $$\frac{Df}{Dt},\,D_tf$$ $\partial$ represents the partial derivative (derivative of $f(x,y)$ with respect to $x$ ...


27

The thing is that $\mathrm{dm}$ is a single symbol, not a combination of two symbols. Yes, it can be understood in terms of a prefix and a base indicator, but it is still a single symbol. An analogy to the concatenation of variable is inappropriate. Reference to an authoritative statement: The grouping formed by a prefix symbol attached to a unit ...


22

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 ...


21

It is an ångström, a unit of length commonly used in chemistry to measure things like atomic radii and bond lengths. Although not an official SI unit, it has a simple relationship to the metric units of length: $$1\:\mathrm{ångström} = 1\:\mathrm{Å} = 10^{−10}\:\mathrm{m} = 0.1\:\mathrm{nm} = 100\:\mathrm{pm}.$$


18

The statement "$A$ happens given that $B$" is equivalent to "If $B$, then $A$", which is symbolically represented as an implication $$ B \Rightarrow A $$ or, if you want to preserve the order of $A$ and $B$ in the original statement $$ A \Leftarrow B $$


16

I agree with Ron Maimon that Large scale structure of space-time by Hawking and Ellis is actually fairly rigorous mathematically already. If you insist on somehow supplementing that: For the purely differential/pseudo-Riemannian geometric aspects, I recommend Semi-Riemannian geometry by B. O'Neill. For the analytic aspects, especially the initial value ...


15

It is, in fact, a double integral! The first notation used $$\varPhi_E = \oint_S \vec{E} \cdot \mathrm{d}\vec{A} = \oint_S \vec{E} \cdot \hat{n} \ \mathrm{d}A$$ is simply a more compact notation. It's much easier to write $\mathrm{d} \vec{A}$ instead of, say, $r \ \mathrm{d}r \ \mathrm{d}\theta$ all the time. Furthermore, it's more general, as $\mathrm{d} \...


15

Surely there is a story behind the print on Feynman's tee, these are the CM-1/CM-2 T-shirts. Quoting Tamiko Thiel: The geometric boxes and their 'hard' connections represent the 12-dimensional 'cube of cubes' that forms the internal hardware network connecting all processor chips with each other in a maximum of 12 steps. Feynman is the one who suggested ...


15

$\lvert A\rangle \langle B \rvert$ is the tensor of a ket and a bra (well, duh). This means it is an element of the tensor product of a Hilbert space $H_1$ (that's where the kets live) and of a dual of a Hilbert space $H_2^\ast$, which is where the bras live. Although for Hilbert spaces their duals are isomorphic to the original space, this distinction ...


14

Bra ket notation has nothing to do with integrals. If I have two vectors in 3D space, $\vec{v}$ and $\vec{w}$ , I can write them as $|v\rangle$ and $|w\rangle$ if I want. In that case, I would write their dot product (a.k.a. inner product) as $\langle v | w \rangle$. This dot product has nothing to do with integration. When your vectors are functions then ...


14

IMHO, the notation $\int_a^b\mathrm{d}x\,f(x)$ is much cleaner than $\int_a^b f(x)\,\mathrm{d}x$, because the integration variable ($x$) and its associated integral range $(\int_a^b$) are kept together. This is particularly important in lengthy and multi-dimensional integrals. Consider $$ \Upsilon_{pq}(k)= \int_0^\infty\mathrm{d}x \int_0^{\beta(x)}\mathrm{d}...


13

I've seen "(1)" used. Radians (and steradians) are also "unitless" but they're clearly not appropriate here.


13

If I saw the word "amp" written as such in a paper in my field (astrophysics) it would strike me as a bit informal. I would expect to see the full "ampere" written. That said, it is rare to actually write out the full name of a unit; usually it follows a number and is given its standard abbreviation. When abbreviated to e.g. "$5\ \mathrm{A}$", I would ...


12

Are those square brackets standard notation in Physics? Yes. See, for example Sean Carroll notes. At least I can tell you from two other classic references using that notation, "General Relativity" by Wald (1984) and "A First Introducion to General Relativity" by Schutz (2009 for the most recent edition) $ $ If I am in a non-curved $\mathbb M$ ...


12

It means that only when m and n are equal the value is 1, otherwise it is zero. Check http://en.wikipedia.org/wiki/Kronecker_delta for more information.


12

It's not just QFT literature. Physicists, especially adult research physicists, find this notation sensible and popular – even though it may be more popular among particle physicists than elsewhere. Formally, $dx\,f(x)$ is a product of two factors and $\int$ is a form of a sum. Because product is commutative, it doesn't hurt when the order is interchanged. ...


12

The notion of tensor product is independent from the Hilbert space structure, it is defined for vector spaces on the field $\mathbb K$ (usually $\mathbb R$ or $\mathbb C$). A formal definition is given below (there are many equivalent approaches). First, if $V$ is a vector space, $V^*$ denotes its algebraic dual space, namely the vector space of the linear ...


11

Here is a video of the film's science advisor explaining what the equation is and how he came up with it: http://www.youtube.com/watch?v=WjfT6MqTCqQ It is based on the Gompertz equation, which is a model of mortality rates, with some added "mathematical glitter."


11

It's c for constant or celeritas, which means speed in Latin. Everyone uses it because it's convention. You could use $\xi$ or $\zeta$ or $\gamma$ or any other symbol you wanted, but then you'd have to explain what it meant, and people would have to go through the trouble to remember this every time they read your papers. Better to go with convention and ...


11

Technically, apparently, your teacher is correct. BIPM and NIST In the official brochure from the Bureau international des poids et mesures (BIPM, the keepers of SI units) in §5.1 Unit symbols we find: It is not permissible to use abbreviations for unit symbols or unit names, such as sec (for either s or second), sq. mm (for either mm2 or ...


11

Capital $\mathrm{C}$, in upright font, is the symbol for the coulomb. Lowercase $c$, italicized, is the speed of light in vacuum. Thanks to Einstein's equation, we can switch between mass and energy ($\mathrm{MeV}$ is a unit of energy) by using factors of $c^2$, and sometimes it's more convenient to know the energy equivalent of a particle's mass rather than ...


11

The notation $\lvert \rangle$ is meant to imply that $\lvert \text{anything here you want to put here} \rangle$ is a vector in a Hilbert space. If you have got some wavefunction $\psi(x)$, then you often denote the abstract vector (instead of the concrete realisation in a basis like $\psi(x)$) it represents by $\lvert \psi \rangle$. If you have got only a ...


10

You can perform $\LaTeX$ search - that is, write formula in LaTeX in an appropriate search engine: http://www.latexsearch.com/ However, as one can type the same expression in different ways and with different symbols, I never used it it practice. (Anyone did?)


10

It's purely notation. Given a real-valued function $f(\mathbf r) = f(x^1, \dots, x^n)$ of $n$ real variables, one defines the derivative with respect to $\mathbf r$ as follows: \begin{align} \frac{\partial f}{\partial \mathbf r}(\mathbf r) = \left(\frac{\partial f}{\partial x^1}(\mathbf r), \dots, \frac{\partial f}{\partial x^n}(\mathbf r)\right) \end{...


10

Quite often everything inside bra or ket is just a label. In this particular case the meaning of $|λ,m_l⟩$ is "a state with the square of the angular momentum being equal to $λ$ (in atomic units, where $\hbar=1$) and with the projection of the angular momentum in some direction ($z$-axis conventionally) being equal to $m_l$". That is, $|λ,m_l⟩$ state is ...


10

The wedge product has its roots in exterior algebra. Exterior algebra lets you talk about objects like planes or volumes as algebraic elements of their own, separate from ordinary vectors, but still obeying the same notions of being "vectors" in their own vector spaces. The wedge product of two vectors is a bivector, and many concepts you may have been ...



Only top voted, non community-wiki answers of a minimum length are eligible