A friend of mine encountered a problem while studying GR. I'm going to answer it myself if I get some time, but here it is.  The (symmetrized) energy-momentum tensor for the electromagnetic field is $$T^{\mu \nu} = F^{\mu \lambda} F^{\nu}_{\,\, \lambda} - \frac{1}{4} \eta^{\mu \nu} F^{\lambda \sigma} F_{\lambda \sigma} \, .$$  In this calculation, she had to assume conservation of energy-momentum in the form $\partial_\mu T^{\mu \nu} = 0$ to figure how it works out in the presence of field sources. In other words: $$T^{\mu \nu} = T^{\mu \nu}_\mathrm{matter} + T^{\mu \nu}_\mathrm{field}$$ The step that confused her was showing that  $$ \partial_\mu T^{\mu \nu}_\mathrm{matter} = F^{\nu}_{\, \, \mu} j^{\mu} $$  I tried to write it out using the Maxwell field equation $$\partial_\mu F^{\mu \nu} = j^\nu$$ and the Bianchi identity $$\partial_{[ \lambda} F_{\mu \nu ]} = 0$$ but something smelled fishy in my calculations and I didn't manage to get back to it.
In a theoretical physics homework problem, I have to show the following: $$\partial_\nu T^{\mu\nu} = - j_\nu F^{\mu\nu}$$  Where $T$ is the Energy-Momentum-Tensor, $j$ the generalized current and $F$ the Field-Tensor. We use the $g$ for the metric tensor, I think in English the $\eta$ is more common.  I know the following relationships:   Current and magnetic potential with Lorenz gauge condition: $$\mathop\Box A^\mu = \mu_0 j^\mu$$ Energy-Momentum-Tensor: $$T^{\mu\nu} = \frac1{\mu_0} g^{\mu\alpha} F_{\alpha\beta} F^{\beta\nu} + \frac1{4\mu_0} g^{\mu\nu} F_{\kappa\lambda} F^{\kappa\lambda}$$ Field-Tensor: $$F^{\mu\nu} = 2 \partial^{[\mu} A^{\nu]} = \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu}$$ d'Alembert operator: $$\mathop\Box = \partial_\mu \partial^\mu$$ Bianchi identity: $$\partial^{[\mu} F^{\nu\alpha]} = 0$$   So far I have set all the definitions into the formula I have to show, but I only end up a lot of terms from antisymmetrisation and product rule. I also drew all what I have in , but I still cannot see how to tackle this problem.  Could somebody please give me a hint into the right direction?
How do you prove that there are infinitely many primes of the form $5 + 6n$? Possible Duplicate:          So, in my calculus class (one I'm teaching, not taking), the sum $\sum_{n=1}^\infty \frac{\sin(n)}{n}$ has come up a few times.  Unfortunately, as someone not exactly an expert in the convergence of sums, it seems to resist the few techniques I know.  Certain none of the usual first year calculus tricks (integral test, alternating series test, ratio test, etc.) work, and the only more tricky technique, partial summation, I can think of doesn't seem to work either (one would need that $\sum_{n=1}^N\sin(n)$ is bounded, which I believe is false).  It seems like it should converge, since it switches sign quite often, but on the other hand, the harmonic series can mess with your intuition, so I don't have much trust in that.  So, I ask to you:     Does this series converge? Can a textbox be enabled after a certain value from a picklist is selected Find directories and files with permissions other than 775 / 664 Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general? Sharing a Bluetooth Keyboard and Mouse sequentially between two Macs I work out of my house where I have two computers: a mid-2010 Macbook Pro (for my day job) and a mid-2010 iMac (personal). During the workday, my iMac acts as a second display for my Macbook Pro via Mini Displayport.  Currently, I use a USB keyboard and mouse for my work computer and a Bluetooth keyboard and Magic Mouse + Magic Trackpad for use with my personal computer. I'd like to get rid of the wired keyboard and mouse and just use the Bluetooth stuff for both machines (but never simultaneously).   Is it possible to easily share the single set of input devices without having to repeatedly break and reestablish the pairing of the input device to the correct computer? I'm not opposed to buying some type of additional hardware to accomplish this, provided it's not prohibitively costly or cumbersome to use or configure.  It's not clear to me how I would tell the MacBook Pro to leave the devices alone so the iMac can have them for the evening. I want to avoid de-pairing and re-pairing the devices each time I want to shift them back or forth.  Basically is there software or hardware that works as a Bluetooth KVM switch? WPF Get Element(s) under mouse Prove that $g$ is tangent to the graph of $f$ Nautilus bookmarks are missing since upgrade to 13.04 My bookmarks are not shown anymore after upgrading to 13.04 Raring ringtails and I can't add any new bookmarks. The function is not shown and the shortcut Ctrl+D isn't working either. Is this functionality just gone for some reason (as the split screen with F3) or is there something wrong with my upgrade?  Thanks for any hints. How to get a variable name as a string in PHP? Say i have this PHP code:  $FooBar = "a string";   i then need a function like this:  print_var_name($FooBar);   which prints:  FooBar   Any Ideas how to achieve this? Is this even possible in PHP? I know that $\operatorname{Spec} \mathbb{C}[x]$ can be identified with the set $\mathbb{C}\cup *$, where $*$ is a generic point via the correspondence  $$ \prod_{i}(x-a_i) \leftrightarrow \{a_i\}_i , \ \ \ \ (0)\leftrightarrow *. $$ This correspondence holds for any $\operatorname{Spec} k[x]$ with algebraic closed field $k$.   When $k$ is not algebraic closed, how should one understand $\operatorname{Spec} k[x]$? Last activity name still missing from migrated question According to this post  this should be fixed. But it happened again.  See Image below (link to )  There was also another migrated question () in the list where the name was missing, but it was edited on meta. Is every sufficiently large positive integer of the form $ab + ac + bc + 1$? Is every sufficiently large positive integer $A$ of the form $ab + ac + bc + 1$   where $a,b,c$ are some positive integers larger than some given positive integer $d$ ?  How large is sufficiently large , in other words how do we know our number $A$ is sufficiently large ( as a function of $d$ ) ?    Update :  I read the following about positive integers that are not of the form $ ab + ac + bc $:    And I quote about that sequence above : " Note that for n in this sequence, n+1 is either a prime, twice a prime, the square of a prime, 8 or 16 "  This is why I considered the weird +1 in $ab + ac + bc + 1$  So composites larger than 16 not a prime , twice a prime or the square of a prime are of the form $ab + ac + bc + 1$.  And I wonder why !  I guess that relates strongly to the Original posted problem. Maybe I should have mentioned that before , sorry.  Will Jagy's answer gave the same OEIS link which I already knew. This also explains some of the comments.  Im no expert in genus theory or Galois theory but I assume this has a simple answer. How do I change the plymouth bootscreen? With regards to Plymouth --  Are there alternative boot screens available?  What's the easiest way to change the boot screen? I'm attempting to cut a hole in a plane. I added a cylinder object, added the Boolean modifier to the plane, selected "cylinder" and "Difference", but no hole is made. The lower half of the cylinder just highlights, moving/deleting the cylinder reveals a hole-less plane.      How can i cut a hole in the plane? How old is SpongeBob? How old is SpongeBob? I don't mean how old is the series, I mean what is SpongeBob's age if he were a real person. I ask because his age seems to be ambiguous in the show.   He has a full-time job He has his own house He is always referred to as a kid (especially in the movie) He acts childish and immature In one episode, he was pretending to be an adult/grown-up Generate all combinations from multiple lists The square roots of different primes are linearly independent over the field of rationals I need to find a way of proving that the square roots of a finite set    of different primes are linearly independent over the field of    rationals.    I've tried to solve the problem using elementary algebra  and also using the theory of field extensions, without success. To  prove linear independence of two primes is easy but then my problems  arise. I would be very thankful for an answer to this question. By Cauchy's theorem, if a prime number $p$ divides the order of a group $G$, then there is an element $g$ in $G$ whose order is $p$. In addition, if $|G|$ admits a prime decomposition with a factor $p^k$, then Sylow's theorem tells us that a subgroup of order $p^k$ must exist.   My question is the following. If $|G|$ admits a prime decomposition with a factor $p^k$, does this imply that an element of order $p^k$ must exist in $G$?   Update: As explained in Mindlack's answer (among others), the implication does not hold and simple counterexamples exist. In the comments, Mindlack discusses special cases where such an element exists. Cannot create custom config entity fails because ID is blank