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I will explain these two spaces in the context of fluid dynamics. In fluid dynamics, flow velocity components can be expressed as the derivative of scalar stream function. Interestingly, the structure of the stream function is analogous to Hamilton's equation. This similarity was realized in the 80's which gave a new direction to the field. Configuration ...


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In case someone else bumps into the same question, I think I found the answer: The Euler's theorem states that if $f$ is a homogeneous function of degree $n$ in the variables $x_i$, then $$\sum_i x_i\frac{\partial f}{\partial x_i}=nf.$$ So for example, if $f$ is a function of two variables $x_1, x_2$ and it is homogeneous, say to 3rd degree, in ...


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When a photon interacts with an atom, three things can happen: elastic scattering, the photon keeps its energy and changes angle (mirror reflection) https://en.wikipedia.org/wiki/Elastic_scattering inelastic scattering, the photon keeps part of its energy and changes angle (photon transferring vibrational and rotational energies to the molecules, heat up ...


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Angular velocity is a vector quantity, and angular speed is defined as its magnitude (and therefore a scalar quantity). If you are talking about uniform circular motion, the angular velocity after one complete rotation is not zero, but rather a constant (non-zero) vector.


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Angular velocity is a Vector quantity and angular speed is a scalar quantity.angular speed may be can constant but angular velocity not necessarily remaining constant as $\mathbf{\omega}=\mathbf{r}\text{x}\mathbf{v}$.But $\mathbf{v}$ changing with time due to changing direction.


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Homogeneity is a property of composition. If a system is made of the same parts everywhere, then it is considered homogenous. Homogenity has to do with the smallest units that have identical composition or character. The central question here is one of identity. This allows a multi-component system to be described by homogeneity as well. Even a multi-...


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Conventional current_ Current flow from positive to negative termminal of a body due to flow of positive charge known sa conventional current.


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I think there is something wrong with the problem. By convention the direction of the electric field is the direction of the force that a positive charge would experience if placed in the field. In this case no external force is needed to move the charge from A to B as it would naturally be accelerated by the field. Only the field does work. If the charge ...


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What your book is saying, for moving the block from B to A you need to do some work, and this work will be same and opposite of the work done by electrostatic force. Here external force, which provide by you and internal mean electrostatic. You can think about this situation in one more way, if charge to be at rest, at every point on line Ab=force by ...


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When you compare capacitors with discharge tubes, you are comparing those vacuum capacitors sealed in glass tubes with gas discharge tubes. There are many other types of capacitors that do not look like discharge tubes thus are not comparable. Of course the gas pressure and type in the tubes play a big role. In a vacuum capacitor, the residual gas pressure ...


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Exponentials differentiate and integrate better than trig functions, and in general are “easier to combine” than working with trig functions, v.g. complex impedance in a circuit. Taking the real part at the end “brings you back” to the physical fields, voltages, currents etc.


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By "physical values" I assume you mean observables, i.e. quantities that one can measure in real life. Observables are always real number ($\mathbb{R}$) -- at least so far. If you manage to measure a $3\mathrm{i}$ long slab of wood, let me know. Complex numbers enter physical problems in two ways: 1: They are integral part of a theory (e.g. quantum ...


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The shortest answer is that a quantum system is any system that obeys the laws of quantum mechanics. This means that: The system's state at a given time is described by a vector in a complex vector space. This vector is called the system's wave function. The system's wave function evolves over time following the Schrodinger wave equation. Each attribute of ...


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It came up because I was thinking about a state $|\psi\rangle$ which is defined to be the (unique, say) eigenstate of some observable $\hat{o}$ with eigenvalue $\lambda$. Then we see that the time-evolved state $|\psi(t)\rangle = U(t,0) |\psi\rangle$ can be characterized as the eigenstate of the operator $\hat{o}_R(t)$ This property is necessary for the ...


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Yes, they are (for the most part) the same, typically being used interchangeably. And really, both are kind of a misnomer, because things typically not considered "analytic" or "exact" forms are, in fact, very often "exact" in the literal sense. There is nothing "inexact" about, say, the expression $$e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\...


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Free parameters are not predefined but must be estimated by theory or experimentally. Or it can be a parameter used in fitting a dataset with an expression. The free parameters are varied to get a good fit to the data. G in Newton's gravitational equation is a free parameter and has been measured but not with high accuracy.


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I pronounce it “nine point eight zero meters per second per second”, but there isn’t a standard pronunciation for physical quantities. Another pronunciation would be “nine point eight oh meters per second squared”. Only the seconds are being squared. Units are written using the usual mathematical precedence rules. You are probably aware that $ab^2$ means $a(...


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More generally, if the wave equation $\Box y=0$ is satisfied for a scalar field $$\mathbb{R}^{n+1}~\ni~ (\vec{x},t)\quad \stackrel{y}{\mapsto} \quad y(\vec{x},t) ~\in~ \mathbb{R},$$ with spacetime $\mathbb{R}^{n+1}$ as domain, and with a 1-dimensional target space $\mathbb{R}$, we speak of the wave equation in $n+1$-dimensional spacetime, or equivalently, ...


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The 1D wave equation is called like that because it has only one independent space variable, $x$. That's it. The 2D equation has two variables, etc. You are correct that an oscillating rope sweeps out a 2-dimensional plane; in fact, by moving one end in a circle instead of up and down, you can make it occupy a 3D region. But the equation doesn't care about ...


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It is often referred to as the intermediate axis.


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Threads refer to the helical ridge that wraps around a screw—tornillo in Spanish. Each complete rotation of a screw with 10 threads per inch advances it one tenth of an inch. As mentioned in the comments, 126 inches is the circumference of the circle swept out by the handle—the distance you move your hand to put the screw through one full rotation.


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A field configuration that solves the equations of motion of a theory is topologically stable if it cannot be continuously transformed to the vacuum keeping the energy finite. The reason for the name is the fact that topologically stable solutions can be classified by a topological quantum number.


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Particles that exhibit quantum entanglement are called entangled -- that is, "entangled particles". Being entangled is a property, and thus, it is described by an adjective.


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By no first-order changes Feynman means that the first-order functional derivative vanishes, or equivalently, the path is stationary. By the way, no first-order changes is a common talking point of Feynman. Listen e.g. to 46:48-48:48 in the talk The Character of Physical Law, part 4, where he makes similar remarks about the principle of least action.


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Any continuous and differentiable function $f(x)$ can be expressed as a Taylor series: $$ f(x_0+\delta x) = f(x_0) + \frac{\mathrm{d}f}{\mathrm{d}x}\bigg|_{x_0}\delta x + \frac{1}{2}\frac{\mathrm{d} x^2f}{\mathrm{d}^2x}\bigg|_{x_0}\delta x^2 + \dots + \frac{1}{n!}\frac{\mathrm{d}^nf}{\mathrm{d}^nx}\bigg|_{x_0}\delta x^n .$$ Each of these terms are called of ...


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