I have read that contra-variant and co-variat vectors have different transformation properties , which distinguish them, yet at the same time I have read that a vector can have contra-variant and co-variant components. If a vector can have contra-variant and co-variant components, then isn't this vector both contra-variant and co-variant? But isn't this impossible?
I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear.  One such concept is the notion of contra- and co-variant vectors.  describes a contra-variant vector as varying with the co-ordinates, i.e. the inverse of the reference axes. It gives the example of a change of basis matrix $\mathbf{M}$, and a contra-variant vector $\vec{v}$ and states that in the new set of co-ordinates, we have:  $$\vec{v}'=\mathbf{M}\vec{v}$$  It then states that a covariant vector varies with the co-ordinate axes.     Does this mean that for a covariant vector $\vec{u}$ and change of basis matrix $\mathbf{M}$ we have: $$\vec{u}'=\mathbf{M}^{-1}\vec{u}?$$   I am also having problems understanding the concept of raising and lowering indices. For instance, Wikipedia states that for two four-vectors $\vec{A},\vec{B}$ and using Einstein summation we have:  $$\left\langle \vec{A} \middle|\vec{B} \right\rangle=A^{\mu}B_{\mu}=A_{\nu}B^{\nu}$$  And I am having difficulty understanding how this relates to the other definition of the inner product:  $$\left\langle \vec{A} \middle|\vec{B} \right\rangle =\mathbf{A}^{T}\mathbf{\eta}\mathbf{B}$$  If anyone could help give me a more intuitive understanding of these concepts I would be grateful.
Android and getting a view with id cast as a string Determine the exact value of equations involving more two trig variables $2\cos^2x=1+\sin x$.  Determine the exact values of $x$ such that $0 \leq x \leq 2\pi$.    I am experiencing problems with factoring this question.  First I started by getting everything on to the left side so: $2\cos^2x-\sin x-1=0$.  Then seeing as there are two trig variable here ($\cos$ and $\sin$), I used the pythagorean identity and replaced $2\cos^2x$ with $1-2\sin^2x$ to make it into a one variable question: $1-1-2\sin^2x-\sin x-1=0$.   By collecting like terms it then becomes $-\sin^2x-\sin x=0$.  Now you could factor these with $0$ and $-1$, but then it creates uneven brackets so this musn't work.  I don't think the quadratic formula is necessary, but I am wondering if this could be treated as a difference of squares question.    If you replace the $\sin x$ and make the question into $-x^2-x=0$ it is a little easier to look at, however I am not sure if this is the right approach.  If someone could please clarify my answer, that would be appreciated! Why is it possible to get significant F statistic (p<.001) but non-significant regressor t-tests? What do I need for extreme macro photography? Imagine I have tons of RAM. Let's say 64GB. That's a lot for even gaming PCs. Now the default location of a pagefile in Windows is on the main OS drive, be it HDD or SSD, which are faster in general, but still not as fast as RAM.  Something tells me that disabling the pagefile on the hard drive or creating a virtual RAM drive and letting the pagefile be there could make Windows move all its virtual memory to RAM, and so increase the system's performance, but I'm not very knowledgeable in that area, so that might not be true at all.  I tried both, but I couldn't analyze the results to reach a definite conclusion with my knowledge level in memory things.  Would this work? If not, why? Need polite phrases expressing disagreement with the information/conclusions of another person, especially an educator Show that the set of all polynomials whose constant term is a multiple of 3 is an ideal of the ring $Z[x]$.   Attempt: Suppose then we need to show   if $p(x), q(x) \in I$ then $p(x) + q(x) \in I$  and if $r(x) \in Z[x]$ , then $p(x)r(x) \in I$ and $r(x)p(x) \in I$.  And $0 \in Z$, then $0 \in I$.  Then let   $I =$ {$a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + a_0$ | $a_i \in Z$, $a_0\in 3Z$}  $p(x)  = a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + 3a$  and $q(x) = b_mx^n + b_{m-1}x^{n-1} + ...+b_1x + 3b$  Here we do not assume that $p(x) $ and $q(x)$ have the same degree.  Then $p(x) + q(x)  = [a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + 3a]  +  [b_mx^n + b_{m-1}x^{n-1} + ...+b_1x + 3b] = 3(a + b) + (a_1 + b_1)x + ....+ (a_n + b_m)x^n$.  so $p(x) + q(x) \in I$  And $0 \in Z$ is also in $I$, since for $p(x) = 0x^n + ....+ 3(0) = 0$.  And let $r(x)= c_nx^n + ...+3c  \in Z[x] $.  Then $r(x)p(x) = 3(3ac) + (3b_1 + 3a_1) + ....+ a_nb_mx^{2n}$  So $r(x)p(x) \in I$.  Can someone please help? I am not really sure how to prove this. I am getting confused with the underscrips Any help or better approach would really help. Thanks in advance! How to automatically start programs in specific workspaces on start up I am having Problems converting user rights. I want to assign the following rights to certain files/folders but dont know hat to translate them into a format which chmod will work with like chmod 755.  -rwxrwxrwx    will be?  And  drwxr-xr-x   will be?  And finally  -rw-r--r--   will be? Checkout one file from Subversion Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\subseteq \Omega$. Then by Cauchy Theorem and Morera Theorem, $f$ is holomorphic. Let $f_n$ and $f'$ be the derivatives of $f_n$ and $f$ respectively. Prove that $f_n'\to f'$ uniformly on any compact subset $K\subseteq \Omega$.   How to prove? Can one attack for damage after having grappled an opponent? The grappling rules sure are... light. Supposing one has an Extra Attack:   Assuming one has successfully grappled an opponent with one's first attack, can one then perform a normal attack action with one's Extra Attack? Against the grappled opponent or any opponent within reach? Limitations on weapons (e.g. one-handed only)? Assuming one has successfully grappled an opponent with one's first attack, can one's grappled opponent (now with move 0) perform a normal attack action? Against you or against any one in reach? Do advantage and disadvantage (along the lines of restraint) come into play? Are there grappling-specific attacks for damage (e.g. choke-hold)? HP PSC 1410 all scanner's images are black I have a HP PSC 1410 printer with scanner inside.  I've been scanning documents and images with it for the last couple of years.  Recently my scanner has stopped performing as excepted and i can't scan or copy images\documents.  My problem is that every images\document i'm trying to scan - the result from the printer is a black page.  I've been googling about this issue and couldn't find any helpful stuff beside replacing the whole printer - which is something i'm trying to avoid.  Any ideas ? I've read something about the shipping lock of the scanner but couldn't find any information about how to check it.. How do I set OS X to boot by default? I set Windows 7 to my Boot Camp. Windows 7 starts now by default in the normal boot. However, I want that OSX boots by default and Windows 7 only as I press ALT and then select it.  How can you set OSX to boot by default? Seems like a bug:  The user:   Edits a question Then, he decide to close it. Clicks on the Close button. The popup doesn't appear. Refreshes the page. Click on the close button. The popup appears.   After editing in the link appears a hash: #. But I don't think that this is the problem because if I add the hash without editing the close popup appears.  It's minor, but I think that is good to be reported and solved.  Edit: The close popup doesn't open after we click an edit has been made to this post; click to load. Integral $\int^{ \pi /2}_{0} \ln (\sin x)\ dx$ $$\int^{ \pi /2}_{0} \ln (\sin x)\  dx$$  The answer is $- \frac{\pi}{2} \ln 2$.  I have changed it into $$\frac{1}{2} \int^{1}_{0} t d \ln t^{2}$$  But I didn't get the answer with it. Storing Images in DB - Yea or Nay? Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear? This is an exercise in complex analysis:     Let $\Omega\subset{\Bbb C}$ be open and bounded, and $\varphi:\Omega\to\Omega$ a holomorphic function. Prove that if there exists a point $z_0\in\Omega$ such that   $$ \varphi(z_0)=z_0\qquad\text{and   }\qquad \varphi'(z_0)=1 $$   then $\varphi$ is linear.   I'm trying work out the case $z_0=0$ first, in which $$ \varphi(z)=z+\sum_{n=2}^{\infty}a_nz^2. $$ It suffices to show that $a_n=0$ for all $n\geq 2$. If let $$ \varphi(z)=z+a_2z^2+O(z^3) $$ then  $$ \varphi^k(0)=z+ka_2z^2+O(z^3), $$ and $$ \varphi^k(0)=0,\quad (\varphi^k)'(0)=1. $$ If one can show that $\{ka_2\}_{k=1}^{\infty}$ is uniformly bounded, then one at least has $a_2=0$. But I don't know how to go on. Any idea? I can't think of a decent reason to allow editing audit messages (like  or ). I have abused it for fun before but it really has no purpose other than joking or obfuscating your actions. For that I think that such messages should be untouchable. Please make them so. Stance on repost question instead of editing to invalidate answers?