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To avoid to decide if his derivative $\dot u(x)$ is a covariant or a contravariant object (or perhaps to go for the contravariant one). Seriously. Of course not rigorously, nor even formally. Duality will enter scene in the XIXth-XXth centuries. We got used to integrate a density across a path, or to multiply vector and covectors from the tangent and ...


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One can define the (magnitude) of the cross product this way or better $$\mathbf A \times \mathbf B = AB\sin\theta\; \mathbf n $$ where $\mathbf n$ is the (right hand rule) vector normal to the plane containing $\mathbf A$ and $\mathbf B$, Another approach is to start by specifying the cross product on the Cartesian basis vectors: $$\vec e_x \times ...


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Hint : In the Figure below what is the height of your infinitesimal cylinder ??? (a) The red one ($d\mathbf{h}$) ??? (b) The blue one ($d\mathbf{s}$) ???


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If the rigid body is rotating then in general the primed axes will be changing with time. An easier way to see this is to look at the Euler angles themselves as in this diagram. If, for instance, $\alpha$ is changing, then both the line of nodes (the $N$-axis in the diagram) and the $z'$-axis (the $Z$-axis in the diagram) are changing.


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Within the Born-Oppenheimer Approximation the different species have the same structure as all nuclei are considered to be infinitely heavy. In fact, you can determine the structure of simple molecules by changing one of the atoms with one of its isotopes (a so-called isotopologue of your initial molecule) and comparing the rotational spectra. Of course the ...


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Expand $\sin(\omega t + \phi) = \sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi)$. Substituting $x$ into it we get $\frac{y}{\beta} = \frac{x}{\alpha} \cos(\phi) + \sqrt{1-(\frac{x}{\alpha})^2} \sin(\phi)$ $(\frac{y}{\beta} - \frac{x}{\alpha} \cos(\phi))^2 = (1-(\frac{x}{\alpha})^2) \sin(\phi)^2$ After simplification $(\frac{y}{\beta})^2 + ...


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Due to the force of gravity, which goes as the inverse of the square, planets trace out an ellipse in space as they orbit around the sun, which is located at a single focus. The other focus is unphysical. Actually, given two massive bodies, their "difference" vector will trace out an ellipse with the center of mass at the focus. Because the sun is so much ...


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There is no physical object at the location of the second focus. Newton showed that an elliptical path was the consequence of an inverse square radial force from a fixed point. While you can identify the point that is the second focus, nothing associated with that point is required to create the elliptical motion. Deriving Kepler's Laws from ...


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A hyper cone exists in 4-dimensional Minkowski space. You may be confusing the curved geometric properties of a Euclidean solid in flat space with the warped properties of space-time in General Relativity. The equation of a hyper cone is: a^2 + y^2 + z^2 - w^2 = 0 There is no adjustment for relativistic speeds or gravitational fields. You might think of ...


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This calculation assumes that the other screen is very far away, that is, $y \ll D$. So redraw your diagram so that the green line is very nearly horizontal, and you'll get the conclusion.



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