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### Confused about tensor notations of how vector and covectors act on each other

First, you can shorten things considerably by not changing coordinates and then changing back on the second term. $\delta^i_j=v^iv_j=(v^i(w_l)w^l)v_j=v^i(w_lw^l)v_j=v^inv_j=n\delta^i_j$ We can also ...

### Confused about tensor notations of how vector and covectors act on each other

I replaced the repeated $j$ in OP's question with $l$ to make the equations clearer. OP's derivation is as follows. \begin{aligned} \delta^i_j &= v^i v_j \\ &= \left(v^i (w_l) w^l \...
1 vote

The solution occurred to me while I was writing the question, so I answer myself. Let $\vec{R}=n_1\vec{a_1}+n_2\vec{a_2}+n_3\vec{a_3}$ be the position of a point $P$ in the plane $(hkl)$, and let $\... 4 votes ### Clarification on clock synchronization passage from Kleppner and Kolenkow You are not interpreting the described Newtonian procedure correctly. The moon clock is one second ahead of the earth clock so that seen from the earth the two tell the same time. But seen from the ... 5 votes Accepted ### Angle between the two vectors The arcsine function on a calculator or computer will only return values between -90° and 90°. You have to add 180° to your answer of -18.5 to get the correct result of 161.5. Arccosine returns ... 0 votes ### Multiplying non-constant acceleration by a constant on every point - would the total path be multiplied by the same constant? If a(t) is multiplied by c the integral over a(t) and c*a(t) you can easily compare, the same with again taking the integral. So what is the problem? 1 vote Accepted ### Prove that every component of angular momentum commutes with$f$Commuting with$L$is equivalent to being invariant under rotations. The quantities$r^2$,$r\cdot p$and$p^2$are all rotationally invariant, as is any function of them. That is all that is needed. 0 votes Accepted ### Problem trying to graphically represent a 2D vector given angle and intensity The angle of a vector is usually measured from the positive x-axis to the vector, with clockwise angles counting as positive. Thus a vector in the same direction as the positive y-axis has an angle of ... 1 vote ### Understanding conservation of energy in a pulley problem I think you can argue as follows. We have a tension T acting on both masses. The variation of kinetic energy for$m_1$is $$KE^1_{f} - KE^1_{i} = \int_0^h (- m_1 g + T) \ dx,$$ while for$m_2$we ... 1 vote ### Force diagram - Circular motion I don't understand that because if N is the normal contact force, it should be perpendicular to the wall In this case they are just using the variable$N$to represent the total contact force with ... 0 votes ### If there is no net force, but the mass can change, can momentum remain unchanged? In Newtonian mechanics and relativistic mechanics both B and C are true, and A and D are false. If we define, net force as the derivative of momentum for a body we have: $$\frac{\text{d}\boldsymbol{p}}... 1 vote ### Let Q(t,\vec x) solve \partial_t^2 Q = \nabla^2Q. Why \partial_t^2Q = 2 (\partial_r + r^{-1})\partial_{t-r}Q? The impeccable answer of @CWPP identifies the problem and leads to the straightforward correct answer; I'm only writing this as a footnote with an explicit expansion of his point to avoid the ... 0 votes ### Probability of Different States - Canonical Ensemble - Partition Function "Ideal gas" means independent particles. When particles are independent you can study one particle and you know what all other particles do on average. Mathematically, the N-particle ... 3 votes Accepted ### Let Q(t,\vec x) solve \partial_t^2 Q = \nabla^2Q. Why \partial_t^2Q = 2 (\partial_r + r^{-1})\partial_{t-r}Q? You are not transforming the differentials correctly. \partial/\partial t|_r=\partial/\partial u|_r and \partial/\partial r|_t=\partial/\partial r|_u-\partial/\partial u|_r. 1 vote ### Tong QFT Problem set 2, question 6: Normal ordering of angular momentum operator Okay, here is what I get$$\int d^3\vec{x}\ x^jT^{0k}=\int d^3\vec{x}\int \not{d}^3\vec{p}\int \not{d}^3\vec{p}' \omega_p p^{'k}\Big[ a_px^je^{-i(p+p')\cdot x}a_{p'}- a_px^je^{-i(p-p')\cdot x}a_{p'}^{... 1 vote ### Can't figure out my mistake in solving the problem of spring hanged from the ceiling and a weight attached at its end Alternative approach (if you are interested) is to note that in simple harmonic motion the mass will be in equilibrium at the mid-point of the motion, when the extension is$x=2.5$cm. At this point, ... 3 votes Accepted ### Can't figure out my mistake in solving the problem of spring hanged from the ceiling and a weight attached at its end Yes, it is the force$F=mg$that causes the length of the spring to elongate, however, the condition$F=k\Delta x$gives you the elongation$\Delta xof the spring at which the upward force from the ... 2 votes ### Can't figure out my mistake in solving the problem of spring hanged from the ceiling and a weight attached at its end The problem with your approach is that you are assuming the spring to be in equilibrium. The reality is the spring actually undergoes Simple Harmonic Motion. The maximum extension of the spring occurs ... 3 votes ### Why does a stream of falling water get narrower at the bottom? There will be an additional contraction term due to the fact that the water entering the stream sideways carries with it a velocity component in the x and y directions, but I do not know how to ... 8 votes Accepted ### Why does a stream of falling water get narrower at the bottom? Is my line of reasoning correct? Is there any incorrect assumptions that I made? Is this resoult coherent or even correct? Seems reasonable to me, at least to zeroth order. You are basically using: $$... 1 vote Accepted ### Determine velocity vector on sloped surface the components of the velocity vector given in ball system are$$\mathbf v_B= \left[ \begin {array}{c} v\sin \left( \beta \right) \\ v\cos \left( \beta \right) \\ 0 \end {array} \right] $$to ... 0 votes Accepted ### How do I calculate the mass of a moving proton? Relativistic mass = m_{rest}+ \frac{E_{kin}}{c^2} Relativistic mass = \frac{1}{\sqrt{ 1-v^2/c^2 }} * m_{rest} Relativistic mass of proton is good for: Calculating force needed to steer the ... 5 votes ### How do I calculate the mass of a moving proton? Suppose we accelerate a proton to a speed v then crash it into some large mass M so that the proton comes to rest inside the large mass. Then we measure the speed u that the large mass + proton ... 0 votes ### Earth to Low Earth Orbit (LEO), gravity drag and potential energy I suspect what is missing is the free velocity gained by launching in the same direction as the rotation of the Earth. The closer you are to the Equator, the more "free velocity" you can get ... 0 votes Accepted ### Motion in one dimension. One car overtaking a different one Just a small supplement to the solution in the book: With start time t_0 = 0 the truck and car reached a distance s_{truck} = ut \ \ \ (0.1) and s_{car} = 1/2 a t^2 \ \ \ (0.2) w.r.t. the ... 1 vote ### Motion in one dimension. One car overtaking a different one From equation (1) we have t =\frac s u Substituting for t in equation (2) gives s = \frac 1 2 \frac {as^2}{u^2} Rearranging this gives equation (3). 2 votes ### Help with an integral in Peskin & Schroeder - QFT Hint:$$\begin{align} \int_{m}^{\infty}\! dE~\sqrt{E ^2 - m ^2} e ^{-iEt} ~=~~~&\int_{m}^{-i\infty}\! dE~\sqrt{E ^2 - m ^2} e ^{-iEt}\cr ~\stackrel{E=-imz }{=}&-m^2\int_{i}^{\infty}\! dz~\sqrt{... 0 votes Accepted ### Physical interpretation regarding heat equation It implies that the net vertical flow of heat at any elevation y due to the T* part of the solution is zero, and that the net (downward) flow of heat (per unit distance into the paper) is constant at...
P&S do not employ the method of residues to reach the final conclusion you cite. Instead, what they do is simply to make a variable substitution of the form $E=\sqrt{p^2+m^2}$. So, then, if you ...