How does friction allow bodies to rotate on merry go round?
Why does friction play the role of centripetal force during the turning of a car?
Why $SU(3)$ and not $U(3)$? Is there a good reason not to pick $U(3)$ as the colour group? Is there any experiment or intrinsic reason that would ruled out $U(3)$ as colour group instead? Applications of ultrafilters I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters?  I'm at the beginning, so I'd prefer some applications that also a beginner could read. Let $U\subset\mathbb R^3$ be an open set, and $f:U\to \mathbb R$ be a smooth function. Suppose that the level set $S=f^{-1}(\{0\})$ is non-empty, and that at each $p\in S,$ the gradient $\overrightarrow \nabla f(p)$ is not the zero vector. Then $S$ is a smooth two-dimensional surface in $U$, and $p\mapsto \overrightarrow \eta(p)=\frac{1}{||\overrightarrow \nabla f(p)||}\overrightarrow \nabla f(p)$ defines a smooth unit-length normal vector field along $S$. At each $x\in U,$ write $H(f)_{(x)}$ for the $3\times 3$ Hessian matrix specified by $$(H(f)_{(x)})_{ij}=\frac{\partial^2f}{\partial x_i\partial x_j}(x).$$   Show that, at each $p\in S$, the second fundamental form $II_p: T_p(s)\times T_p(s)\to \mathbb R$ is the symmetric bilinear map    $$II_p(\overrightarrow v,\overrightarrow w)=\frac{-1}{||\overrightarrow \nabla f(p)||}\overrightarrow v\cdot H(f)_{(p)}\overrightarrow w,$$for all $\overrightarrow v ,\overrightarrow w \in T_p(s)$.       (Here, we view the tagent space $T_p(S)$ as the two-dimensional subspace $(span\{ {\overrightarrow \eta(p)}\})^{\bot}$ of $\mathbb R^3$.      Edit: Actually my question is why the second fundamental form under the usual definition can be written in this way.      Definition: The quadratic form $II_p$, defined in $T_p(S)$ by $II_p(v)=-&lt;d  N_p(v),v&gt;$ is called the second fundamental form of $S$ at $p$, where $dN_p:T_p(S)\to T_p(S)$ is the differential of the Gauss map.   Hopefully, I express this problem explicitly. I was just wondering how to prove this statement.   I took a diffrential geometry class last semester, and when I organized my notes this morning, I found this statement, but there was no proof...  Looking forward to an understandable explaination. Thanks in advance.     Edit 2:Furthermore, show that, at each point $p\in S$, the expression   $$\phi_p(z)=det\pmatrix{-H(f)_{(p)}-zI_{3\times 3} &amp; \overrightarrow \nabla f(p)\\\ \pm \overrightarrow \nabla f(p)&amp; 0}$$   (the underlying matrix here is $4\times 4$) defines a second-degree polynomial whose roots $\lambda_1$ and $\lambda_2$ are $||\overrightarrow \nabla f(p)||k_1$ and $||\overrightarrow \nabla f(p)||k_2$, where $k_1$ and $k_2$ are the principal curvatures of $S$ at $p$.      Also, if a non-zero vector $\pmatrix {\overrightarrow v \\c}$ lies in the kernel of the $4\times 4$ matrix $$\pmatrix{-H(f)_{(p)}-\lambda_jI_{3\times 3} &amp; \overrightarrow \nabla f(p)\\\ \pm \overrightarrow \nabla f(p)&amp; 0},$$   then $\vec v$ is a non-zero element of $T_p(S)$ and lies in the "principal direction" corresponding to $K_j$. Please add TeX rendering on the Android app MathJax isn't currently working on the Android app, which makes it totally useless for the 'technical' sites like  or .  For example, in the Android app:    Same post in a (mobile) browser:     Please add this functionality! Failed to start TSQL debugger I want to debug in TSQL.When I start the debug the following message box popup. In SSMS v17.2    Then I see this message:    In Microsoft Visual Studio 2015 Server Explorer same Error:    Then I see this message:    Thank You! Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$. I have a LaTeX editing environment that does some assistive stuff such as auto-indenting, closing parentheses, etc.  However, one of the things that it does is unknown to me, and is difficult to search for. If I type an empty set of braces, it replaces them with  {}&lt;++&gt;   If I type two empty sets, for example when I'm setting up a fraction \frac{}{} what I actually get is  \frac\left\{ } \right\}&lt;++&gt;}&lt;++&gt;   Is this a problem? What does &lt;++&gt; do?  Edit: This is using Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics.  I need an clear argument that   involves the equations of motion for just the potentials $A^0$ and $\mathbf{A}$, and not the electric and magnetic fields.   includes sources $\rho$ and $\mathbf{J}$ in the equations of motion.  This is to justify the assertion that there are degrees of freedom that decouple from the rest of system. does not critically rely on quantum field theoretic arguments (although any supplementary remarks are welcome).     To illustrate the level of clarity I expect, I provide an argument in the Coulomb gauge $\nabla\cdot \mathbf{A} = 0$:  Of the four field degrees of freedom, the gauge condition $\nabla\cdot \mathbf{A} = 0$ removes one degree of freedom (the longitudinally polarized EM waves).  To show that among the remaining three degrees of freedom only two are propagating, consider the field equations of motion in the Coulomb gauge:  \begin{align} \nabla^2 A^0 &amp;= -\rho/\epsilon_0,\\ \big[\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\big]\mathbf{A}&amp;= \mu_0 \mathbf{J}-\nabla \frac{1}{c^2}\frac{\partial}{\partial t}A^0. \end{align}  The first equation is NOT a wave equation for $A^0$, and thus does not propagate.  The final equation IS a wave equation, and describes the propagation of two degrees of freedom (Gauss' law in first eq. can be solved, and then inserted into the second equation to show that $\mathbf{A}$ only couples to the Solenoidal part of the current). drawing at least 90% of colors from urn with large populations Why copy constructor not invoked here Java - Splitting a CSV file into an Array I kust realized that some other PC is also getting assigned a IP address (DHCP Lease) from my modem. I am sure this is not one of the PCs in my home.  My modem was using WEP security. I know WEP is far less secure and WPA is better, but this actually is a borrowed modem, and I did not know that WEP was used.  I immediately did the following : On My Modem    Changed the encryption mode from WEP to WPA   Changed the password   After this I do not see the intruder PC on the DHCP list.  On my Mac:     Changed the permissions on the public folder to "everyone - no access"   Turned off "File Sharing" in System Preferences (But I would want to turn it back on sometime)   My mac is the one that is always on, Both the admin user and the user I normally use are password protected.  Is the above sufficient? Should I do something more?  Also, is WPA security sufficient? Should I take other steps? In Ex Machina, can this character survive? Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$.   i) Show that for every $m\in M$ and for every $n\in N$, $mn=nm$.   ii) If $G$ is generated by $M\cup N$ then $G\cong M \times N$.   This is an homework question for abstract algebra. I think I can prove $i)$. But I have no idea how to prove $ii)$. Can anybody give me a hint for this one ?    Here my proof for i): Let $n \in N$ and let $m\in M$. Let $x=mnm^{-1}n^{-1}$. As $N$ is normal subgroup, $mnm^{-1} \in N$. Therefore $x \in N$.  As $M$ is normal subgroup , $nmn^{-1} \in M$. Since $m\in M$, then $x=m(nm^{-1}n^{-1})\in M$. This implies that $c\in M \cap N$, then $c\in \{e\}$, then $c=e$, and we get $mn=nm$. If $A\subseteq\mathbb N$ and $\sum\limits_{a\in A}\frac1a$ converges then $A$ has natural density $0$ In this  to a question about a series, a theorem was stated:     If $A= \{a_i \}$ is a set such that $\sum_{i = 1}^{\infty} \frac{1}{a_i}$ converges, then $d(A) = 0$, where $d(A)$ is the  of the set.   My background in number theory is basically zero and all my attempts to prove this have been utterly unseccessful; would anyone please help me, or at least provide me with a hint to prove it? Exactly half of the elements of $\mathcal{P}(A)$ are odd-sized Let $A$ be a non-empty set and $n$ be the number of elements in $A$, i.e. $n:=|A|$.  I know that the number of elements of the power set of $A$ is $2^n$, i.e.  $|\mathcal{P}(A)|=2^n$.  I came across the fact that exactly half of the elements of $\mathcal{P}(A)$ contain an odd number of elements, and half of them an even number of elements.  Can someone prove this? Or hint at a proof? Why are news headlines in present tense if they refer to past events? In news papers we see headings like this  &quot;India signs a pact with Russia&quot; &quot;Sachin hits another century&quot; &quot;Obama wins presidential election&quot;  These are completed events, aren't they? Then, why are these sentences not mentioned in past tense? How can see call history in iPhone one month back, i also tried with tried party software but that is not perfect , is the any solution from apple? How do I display the nth result of an ls command? I used this command to display the first result of files in my directory.    ls | head -n 1   My simple question is, how can I modify this command to display say the nth result?  Thanks! Is there a difference between "think of something" and "think about something"? I've also met "have heard of/about something".