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  1. K

    Prove Vector Quadruple Product with Levi-Civita/Index Notation

    The given equation is correct: ##\qquad\qquad(\vec A \times \vec B)\times(\vec C \times \vec D) = [ \vec A \cdot (\vec B \times \vec D) ] \vec C - [ \vec A \cdot (\vec B \times \vec C) ] \vec D## The result I got was ##\qquad\qquad(\vec A \times \vec B)\times(\vec C \times \vec D) = [ \vec A...
  2. K

    Prove Vector Quadruple Product with Levi-Civita/Index Notation

    I don't know how and where to start. I'll just think for a minute.
  3. K

    Prove Vector Quadruple Product with Levi-Civita/Index Notation

    I found the mistake in my solution. I don't know if you're interested in knowing where the mistake is.
  4. K

    Vector identity proof using index notation

    The equation to be proven had been improperly written because the vectors in the third and fourth terms had not been properly grouped. Replacing the lower-case-letter vectors with upper case letters, we should have $$\vec \nabla (\vec A\cdot \vec B)=(\vec A \cdot \vec \nabla)\vec B + (\vec B...
  5. K

    Vector calculus identities proof using suffix notation

    Show that $$(1){~~~~~~~~~~~~~~~~~~~~~~~~~}\vec \nabla\cdot(\vec a \times \vec b) = \vec b \cdot (\vec \nabla \times \vec a) - \vec a \cdot (\vec \nabla \times \vec b){~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$$ Using suffix notation, we get for the left hand side of eq. (1)...
  6. K

    Question on special relativity from "Basic Relativity"

    I wonder if this drawing of the moving platform seen fom the top would help solve the problem in this thread.
  7. K

    Show the identity ##\vec{\nabla}(\vec{r} \cdot \vec{u})##

    The OP posted this problem-to-prove question: As I already said in post #11, the correct simplification should lead to$$\vec{\nabla}(\vec{r} \cdot \vec{u}) = \vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + [~(\vec r \times \vec{\nabla}) \times \vec{u}~]$$But I was wondering if the identity to...
  8. K

    Show the identity ##\vec{\nabla}(\vec{r} \cdot \vec{u})##

    I find the use of brute-force method and suffix notations both tedious and cumbersome. Why don't you try using first useful vector identities to simplify your equation. The BAC minus CAB rule is one such useful identity. But I noticed in your OP where you wrote: that you didn't properly group...
  9. K

    Calculation of pressure & volume both isothermally & adiabatically

    I think you're right there. Just two steps away from the answer. Using ln, just doing two computations won't give you the answer yet. You'll need to do at least three; first the value of ln##~y~##, then the value of (ln##~y##)/##n~##, then that of ##exp[{\frac {\ln y}{n}}]~## before you finally...
  10. K

    Calculation of pressure & volume both isothermally & adiabatically

    Isn't ##~x^n = y~\Rightarrow~x = y^{1/n}~##taking the nth root? Why wouldn't a log do, or an ln?
  11. K

    Special Relativity Problem -- Speeding through a red light to make it look green

    @kuruman ... You know fairly well what we were talking about. We were not arguing about the acceptability of settling a $75,000 debt by paying back only $15,000 , which you mentioned in post #26 and that seem to imply, that according to my own point of view, should be acceptable because they...
  12. K

    Special Relativity Problem -- Speeding through a red light to make it look green

    @jbriggs444 ... Of course I wasn't talking about any two numbers having the same order of magnitude in general. I was talking to kuruman about these numbers given by the OP in post #1: and these numbers that I wrote in post #20: I know that they are not equal but for the purposes of my...
  13. K

    Special Relativity Problem -- Speeding through a red light to make it look green

    @kuruman again ... Your post #9 said: There you mentioned nothing about ##|β|~,## that ##β## can be ##\pm~.## Then, suddenly in post #19, only after 10 messages had already been posted, you surprisingly come up with the idea that ##β~,## in the formula you used in #9 , can be ##\pm~.## Why...
  14. K

    Special Relativity Problem -- Speeding through a red light to make it look green

    You said in post #21: I posted in #20 to which post #21 was referring to: I replied in post #22 to answer #21: ##\text {terra} = 10^{12}##
  15. K

    Special Relativity Problem -- Speeding through a red light to make it look green

    @kuruman ... Of course they are, they both have the same order of magnitude There are minimum and maximum values of the frequency for any given color. I used the average value.
  16. K

    Special Relativity Problem -- Speeding through a red light to make it look green

    The correct formula for relativistic Doppler shift can be found in Fowles's book ##\text {Intro. to Modern Optics}~##, 2nd Ed.,©1975. It says there that the observed frequency (frequency received) ##~f'~## is given by $$ f' = f \sqrt { \frac {1 - β } {1 + β } }~\Leftarrow~\begin{cases}...
  17. K

    Special Relativity Problem -- Speeding through a red light to make it look green

    I'm sorry but I disagree with this assesment in post #17 regarding this thread: The solutions given in post #6 and #9 are somewhat arbitrary and has some inconsistencies: I pointed these things out in post #8: It seems that no attention was paid to my misgivings and they only fell on deaf...
  18. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    I only use Insert in MS Word to choose the shape that I want and I paste it on the document. Once you click on it the main menu Shape Format pops out and you can do almost anything the way you want it, to almost any measure that you like. It just needs familiarity and lots of practice. If you...
  19. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    I don't understand this part of your last explanations: Why is the fraction ##~f~## not ##~f\neq a^2\Delta \phi/(πa^2)?## You used one that is half as small that I think it should be. But if one uses the formula for the area of a triangle ½×Base×Height, which I already mentioned in post #23...
  20. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    In the second part of your arguments, the one with (math+physics), I don't get this: Isn't ##~\Delta \phi~## on the plane ⊥ to the ##~z~## axis subtending an ∠ with the ##~x~## and ##~y~## axes? And why do you have the factor 1/2 in the volume enclosed? For an infinitesimal volume, its simply...
  21. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    I'm sorry but I already disagree with you right at the beginning of your arguments. $$\dots~{\rm {assumption~1:}} ~r = 0~ {\rm with} ~[E - ρr/(2ε_0)] \neq0~\Rightarrow~E\neq0~\dots$$ $$\dots~{\rm {assumption~2:}}~rE = 0~\Rightarrow~\begin{cases}\begin{align} & ~(i)~r = 0~{\rm...
  22. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    @kuruman ... I know fully well what open surfaces and closed surfaces are and how they differ from lines. I also happen to know what coaxial cylinders are. I wasn't talking about reducing the ##~r~## of the imaginary Gaussian surface ##~\rm S~## to zero in post #7. I was referring to the range...
  23. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    So, if one knows what happens in the neighborhood of the point in question, then one can make conclusions about the limit at exactly that point? Then, if one wants to talk next about the limit of the magnitude of the electric field ##~E~## at a point, one also has to examine the continuity of...
  24. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    @Delta2 ... Thanks for your comments. But if ##~E=\rho r/2\epsilon_0~## for ##~r\neq 0~##, how did you get ##~\lim_{r\to 0}E=0~## for ##~r = 0~?~## Which expression for ##E~## did you use in evaluating the limit?
  25. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    Could somebody familiar with the way mathematicians argue please take a look at this and make a comment? Thank you. I'm not a mathematician. What follows is about the electric field at the axis of a long uniformly charged right circular cylinder. Following his arguments in post #5, he obtained...
  26. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    Okay, I was wrong. I'm sorry. I accept my mistake. But even without the mathematics, it is clear that ##E = 0## at the center of the uniformly charged cylinder because ##\vec E## run radially away from the axis so that pairs of them moving away opposite one another cancel out. I failed to see...
  27. K

    Help with a Gauss' law problem -- A long cylinder with a uniform charge density throughout

    Given: ... a very long uniformly charged cylinder of radius ##R = 7.2~\rm cm = 0.072~\rm m~##... the volume charge density is ##ρ = 1264~\rm{nC} ⋅ \rm m^{-3} = 1.264~×~10^{-6}~\rm C ⋅ \rm m^{-3}~##... Unknown: ... the values of##~A~##and##~n~##for the electric field with magnitude##~E =...
  28. K

    Problem with differentiation

    ##f(t) = t^2\Rightarrow~##a distance function of time ##\Rightarrow~x(t) = At^2~\Leftarrow~A = 1~\rm m/s^2 = constant## ##\Rightarrow~dx/dt = 2At = v(t)~\dots## ##\Rightarrow~v(2~{\rm s}) = 2A(2~\rm s) = 2(1~{\rm m⋅s^{-2}})(2~\rm s) = 4~\rm m/s~\dots ## ##\Rightarrow~v(3~{\rm s}) = 2A(3~\rm s) =...
  29. K

    Problem with differentiation

    Again, ##f(x) = x^2~\Rightarrow~df/dx = 2x## means that the rate of change of the function ##f(x)## with the independent variable ##x## is given by the function ##g(x) = 2x##. When ##x = 2##, the rate of change of the function ##f(x)## is given by ##g(2) = 2(2) = 4##. When ##x = 3##, the rate...
  30. K

    Problem with differentiation

    Yes, its final value is 4.0 since ##f(x) = x^2~\Rightarrow~df/dx = 2x## and evaluating ##df/dx## at ##x_0 = 2## gives ##f'(x_0) = 2x_0 = 2(2) = 4 = f'(2)##.
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