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In your last equation $U$ is a function or $n$ variables. Which of these does your $x$, $y$, and $z$ in that equation represent? To find the contribution to the potential energy due to the action of forces on a particular particle, one has to take the partial of the potential with respect to the variables representing the position of that particle. I ...


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It's not taking partial derivatives with respect to an observed particle's position, but rather the space of all possible positions of that particle. Think of the potential energy as being defined prior to the particle having an actual path. Really, at heart, these things are defined on a phase space not on ordinary physical space.


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I think physical sciences correlate well with mathematics becuase of mechanics. That is to say the laws of physical sciences are written in numbers and geometry. For instance a transmission in a car will have a perfect matematical gear ratio, as well as definite geometry a engine will have exact hp and dimensions etc as well as circular and up/down motion. ...


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Rather than a book, consider the Susskind Physics lectures made available on Youtube as well as Stanford on iTunes via iTunes-University. You want to watch the latest version of the course lectures on Classical Mechanics. After a small bit of introductory material, Susskind gets right into the Lagrangian and Hamiltonian approach to mechanics using a simple ...


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It's not completely clear what you're asking, but I can make one thing clear: quantum mechanics was not developed with the specific aim of correctly describing the precise amounts to which CHSH inequalities can be violated. Quantum mechanics was developed in the 1920s and early 30s. Bell published his first paper on Bell inequalities in 1964. The first ...


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There is a new book called Physics From Symmetry which is written specifically for physicists and includes a long, very illustrative introduction to group theory. I especially liked that here concepts like representation or Lie algebra aren't only defined, but motivated and explained in terms that physicists understand. Plus no concepts are introduced which ...


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Physics from Symmetry is a book that explains all group theoretical concepts needed to understand the foundations of QFT in great detail and is written specifically for physicists. It's not very technical, but it's great if you want to understand quickly what concepts are really important for modern physics and why. For example, it explains why things ...


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As Mark Eichenlaub's Answer states, this is a quote from one of Carrol's admired forerunners. You actually hit nearer the mark with your own words: So, in physics void can mean anything but "void". In modern physics there are only quantum fields, a handful of them: the photon field, electron/ positron field, quark field, gluon field and so forth. In ...


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Look at H, the simplest atom. One neutron and one electron. If your imagination can put H in perspective, it would be the Empire State Building as the nucleus and a grapefruit racing around it at 500 yards. This model is composed mostly of SPACE or VOID!!


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Since, you haven't started Quantum Mechanics, depending on your country, if you country is like mine i.e. Indian Standard, well it will really work well. The reason would be because on your first year, or at least First Semester itself, you'll be taught the old Quantum Mechanics, which is well pretty much still valid for some conditions. So, it won't be a ...


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As a graduate student, I'll give it a shot with some guesses: a) Quantum gravity This is nothing particular of the times we're living on, but since it is a problem that is far from a solution (or even from a complete definition of the problem), I think a lot of people will continue to work on it. b) Quantum information and quantum computing I think this ...


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A vector is a scalar with direction. So Time can be a vector, but what it means depends on the context. In 1D it has only 2 directions, positive and negative with zero being positive. In 2D it can be an angle between ÷/-Pi radians. And so on. Time can be a single dimension attached to the familiar 3 Euclidian spacial dimensions and in this case it is ...


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The difference in the time coordinates for the different observers is entirely a geometrical effect, and is described by a geometrical object called the metric tensor. Entropic arguments tend to be used to try and explain the flow of time i.e. the human perception that time flows from past to future. It is important to understand that the flow of time does ...


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I am trying to go to a bit basic level. The formula work=Force*Displacement works only if the force is constant and not changing its direction or magnitude. When an object moves in circle,the force continuously changes its direction. So to calculate it we have to use integral of F with dl,assuming that force remains constant for a very short displacement dl. ...


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Work is defined as the line integral $\int \mathbf{F} \cdot \mathbf{d\ell}$. The force on an object can be a function of position or time, and could represent external forces placed on the system. Net and total work refer to the same concept, the sum of all work done on an object. For your example, you cannot simply say work is 0 because the object returns ...


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Another way perspective on it is that we have only single a points of observation (i.e our eye only has 1 observation point) for which observe objects. This means that when we observe an object, the visual angle will be smaller for objects which are further (if their sizes are the same). This is similar to mapping a point (our observation point), to any ...


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You could also understand why objects seem smaller from a certain distance and further and why they look bigger when they are to close as this: The retina has in humans a mean distance from the lens of in the eye about 0,02 metres. Thus, from simple trigonometry, we have:$$ {O \over P} = {I \over Q}$$ $$ I=({Q \over P}) O$$, where O is the real size of the ...


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On the very theoretical side of solid state physics is the holographic AdS/CFT correspondence which links strongly coupled condensed matter systems to gravitational theories. Recent work has been done on describing things like phase transitions in this theory. For example models of superconductivity in the gravity dual are promising in describing difficult ...


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There is theory of dispersion in crystals. One can say that the differential geometry is used there. I think it is Group theory + differential geometry.


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Since most constants aren't calculated theoretically, but rather they are measured experimentally, this question is impossible to work out. (Or, if calculated theoretically, one uses measured constants for the calculation) Unless you reach infinite precision (Which is impossible in more than one aspect), there's no difference in measuring a rational or ...


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I believe your professor's trying to tell you that the general solution will be a sum of three terms, and he could have just as easily used $f_{ij}$, $g_{ij}$ and $h_{ij}$ for the three superscript terms. Edit: In other words, you're Matrix $\rho$ can be expressed as a sum of matrices $\rho^{(1)}$, $\rho^{(2)}$, which is proportional to $e^{i\Omega t}$ and ...


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As highlit by Diffeomorphism's answer, one can't really make sense of the notion of "habitable" zone for an individual planet. What I think you mean to ask is "is Venus inhabitable by robots / humans. The habitable zone of a star, in contrast, is the zone where a possible planet might support life (or, by analogy with your question, robots). This is often ...


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It depends on your definition of 'surface'. Yes, on the hard rocky surface, temperature and extreme acidity makes it very hard for any sort of electronics to last more than a few minutes But floating at 50km of altitude in Venus the temperature is barely tropical (around $30$ ºC), and sulphuric acid makes about 2% of the atmosphere. That's definitely on the ...


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My recommendations are: For Mechanics, Mathematical Methods of Classical Mechanics, by Arnold. For Electromagnetism, Modern Electrodynamics, Zangwill. For Quantum Mechanics, Quantum Physics, Le Bellac, or, at an easier level, Introduction to quantum mechanics, Griffiths. For General Relativity, General Relativity, Wald. This should give you a (good) ...


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Objects at a distance appear smaller because the visual angle they subtend becomes more acute with distance. The visual angle may be thought of as a triangle with apex at the eye, and the distant object as its base. The apparent height of an object is directly proportional to its actual height and inversely proportional to its distance from the eye. ...


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Apparent size is not measured as an ordinary size, in meters. It is actually an angle, so it is measured in degrees or radians. See this picture: The left blog is the eye. Look like as the object moves further, the angle becomes smaller. That is what is called perspective. Sometimes people try to compare apparent size and real size, but that makes no ...


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I was taught that the Standard Model was a misnomer; that it ought to be called the Standard Theory. I'm inclined to agree, though theories and models are both indispensable in science. Ultimately, the purpose of a model is provide local understanding of a particular phenomena. A model: Typically considers only fields, objects or quantities relevant to a ...


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A theory is a set of statements that is developed through a process of continued abstractions. A theory is aimed at a generalized statement aimed at explaining a phenomenon. A model, on the other hand, is a purposeful representation of reality. As you can see, both share common elements in their definitions. What differs one from the other (in my opinion) ...


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Short answer His notation is not ambiguous because the expression $$V^{'\mu} \equiv \Lambda^\mu_\nu V^\nu$$ can only mean sum along the $\nu$ component. Since $\Lambda$ is a representation of the Lorentz group, it is a linear operator, hence it can only act on a vector by the usual way that matrices act on vectors. Hence the above is unambiguous. Longer ...


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If you go to Wolfram Alpha and type in You get some very helpful information - the exact date and time of the next new moon (in the local time zone), as well as the moon rise and set times. Doing that right now, I get the following: You can decide if that is the information you need to compute the first day of Ramadan - but you can see here that if ...


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This means that the universe cannot be simulated by a Turing Machine, it's not related to being able to write down th equations of physics. If the laws of physics are non-computable, you then also lift the restrictions any computing device is subject to and expand that to whatever the non-computable laws of physics would allow. E.g. classical physics is ...


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Completeness in mathematics is essentially a metric concept (that means that every Cauchy sequence in the metric space converges to an element of the space). Sometimes (but I think more on a physical standpoint, and I agree is a sort of repetition and not so common) it is used to characterize bases in vector spaces, in the sense that a basis is complete if ...


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You need to be careful with the word span. A mathematician will say that the span of a set of vectors is the set of finite linear combinations, so you can only add linear combinations of finitely many at a time to get something in the span. So there are sets that are mutually orthogonal and all normalized but not enough to span the space with finite linear ...


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As noted, many people use "complete" where perhaps they ought to say "complete and orthogonal and orthonormal" or the like. I'm not sure what I can tell you besides confirming that usage is not always ideal. I'll answer one question you brought up, but I'm worried I may have gotten confused myself by what kind of "complete" you meant: Is it even possible ...


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Imagine a box where each side has an area $A$, and suppose that it contains a continuous flow of rain that falls vertically at a velocity $u$. The density of raindrops is $\rho$ (that's the number of drops per unit volume). The rate at which the drops hit the bottom panel will be $\rho Au$ and the rate they hit the vertical (left) panel will be $\rho Av$, ...


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Well speed would matter. But really there are two things here. When you drive then in your reference frame the rain falls faster and at an angle. Since it falls faster you collect more of it. Since it falls at an angle that is directed at your windshield, the volume of rain it can collect increases. If you are initiated in maths : you just need to project ...


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This is variation on a classic problem - if it is raining, will I get less wet if I run rather than walk home? In this case, however, we're discussing a car driving through rain (and we're not bothered about rain falling on the roof of the car). Let me make a simplification by supposing that your windscreen is vertical and that the rain is coming straight ...



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