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Because hydrostatic pressure depends on depth, the problem requires integration. The pressure as a function of $y$ according to Pascal's Law is: $$p(y)=p_0+\rho g y\sin \theta$$ Where $p_0$ is the atmospheric pressure and $\theta=45\:\mathrm{degrees}$. $y\sin \theta$ is the depth. On an infinitesimal piece of door of length $dy$, at position $y$ and ...

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Existence of smooth structures You're asking when does a (topological) manifold $M$ fail to be covered by a smooth atlas. Another way to phrase this is "when does a manifold admit a smooth structure". This is a well-studied problem. It turns out examples of manifolds which do not admit smooth structures only occur in dimensions $\ge 4$ (see e.g. ...

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The phase potrait for this physical system describes the oscillation of the particle described above to be a Homoclinic orbit-that is, the particle oscillating between extreme ends of a double well. No, not even close. I assume that your misunderstanding originates from confusing phase space and geometrical space. Suppose, we describe the dynamical ...

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Yes. That's how the tensor calculus is formulated. For two observers on different frame of reference, all they have to see is the same physical laws. On this basis we formulate our physical laws so that it will be valid for any observers irrespective of their reference frames. Let's take a tensor of rank one (a vector) as example. Suppose two observers on ...

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Yes; he is saying that a tensor encodes geometric information. A vector is a special case, and it's direction and legth are unchanged under a change of basis, though the components do change. A more dramatic example is the determinant, which gives the volume of the paralellopiped whose edges are defined by three coterminous vectors. A change of basis ...

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The precise statement of "self-adjoint operators generate continuous unitary symmetries" is Stone's theorem. It guarantees that there is a bijection between self-adjoint operators $O$ on a Hilbert space and unitary strongly continuous one-parameter groups $U(t)$ that is given by $O\mapsto \mathrm{e}^{\mathrm{i}tO}$. The definition of the exponential for an ...

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Q: 3) There is a final theory explaining everything, and we will find if and only if: i) We are clever enough to find such a theory. ii) We make good and sophisticated enough mathematics. iii) We guess the right axioms/principles/ideas. iv) We interpretate data correctly and test the putative final theory with suitable instruments/experiments. " A: i) ...

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Generically, any square-integrable function is an admissible wave function, and the space of square-integrable complex functions indeed has uncountable dimension as a vector space over $\mathbb{C}$. And it is also true that the eigenstates of the Hamiltonian span the space of states, and that they are countably many. This is the content of the spectral ...

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Assuming the functions are well-behaved (continuous and differentiable), you can change the order of differentiation. $$\left(\frac{\partial T}{\partial V}\right)_S=\frac{\partial}{\partial V}\left(\frac{\partial E}{\partial S}\right) = \frac{\partial}{\partial S}\left(\frac{\partial E}{\partial V} \right) = -\left(\frac{\partial P}{\partial S}\right)_V$$ ...

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Let $H_1$ be the Hamiltonian of a Harmonic oscillator, and let $m=\hbar=\omega=1$, that is, $$H_1=\frac{1}{2}P_1^2+\frac{1}{2}X_1^2-\frac{1}{2}$$ Let $|n_1\rangle$ be the eigenvectors of $H_1$, i.e., $$H_1|n_1\rangle=n_1|n_1\rangle$$ If we define $H=H_1 H_2$ with$^1$ $[H_1,H_2]=0$ we get multiplicative eigenvalues: $$... 0 Your derivation is almost correct, and as I'll show below, one does sometimes make use of it. However, it relies on an unwritten assumption that makes it applicable only in a limited range of situations. That assumption is that the heat capacity c doesn't depend on the temperature. It is this assumption that allowed you to take the factor of mc outside ... 0 Your equation is correct only if:$$\mathrm dQ = mc\,\mathrm dT which is not generally true, indeed, common sense tells you that a change in temperature leads to conclusion that an object being heated up. But we do not encounter gases much in our life, which could be regarded as a general case. In reality your assumption is generally false, a good example ...

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Here is a link that might be useful http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/temper2.html#c2

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You misunderstood the meaning of $H^{\bullet}_{S^1}(U_1) = \mathbb{C}[\Omega]$. This does not mean $H^i_{S^1}(U_1) = \mathbb{C}[\Omega]\forall i$, it means that the cohomology ring (with multiplication given by the cup product) is given by the polynomial algebra in one element that has degree 2. Translated back into the individual degrees $H^i$ this ...

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'Mathematical Physics' by Kusse and Westwig is just the thing you need. The fifth chapter is devoted to the Dirac-delta function. The book is fairly easy to understand and provides the proofs of the theorems that are stated in Arfken-Weber. After having read this, you can read the appendices I and II in Cohen-Tannoudji (Quantum Mechanics) on Fourier ...

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The first thing that must be said is that the question is not really specific enough: Applications to what exactly are you looking for? To me, a book on algebraic geometry and mirror symmetry, and how it relates to mirror symmetry as physicists know it, is very relevant and interesting. However, I have the feeling that this is not exactly what you're looking ...

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