For an implicit surface $F(x,y,z)=0$ we can take $z=f(x,y)$ and use IFT to calculate its fundamental forms. But they are expressed by $dx^2,dxdy,dy^2$.  I wonder if there are other expressions using $dx^2,dy^2,dz^2,dxdy,dydz,dzdx$, just more symmetric.  Also I want to know how to differentiate entirely without taking coordinate parameters $x,y,z$ to get a direct expression by $dF$ with something.  Any help will be appreciated.
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 $K_j$.
Alice has a secp256k1 key pair that she used some time ago when she communicated with Bob. Bob can no longer user secp256k1 but he only uses ed25519 instead.  Is there a secure algorithm that Bob can use to derive an ed25519 pubkey from original Alice's secp256k1 pub key and also Alice can derive the matching ed25519 private key from her original secp256k1 private key such that they can communicate securely via ed25519 encrypted messages?  Thanks! Mount error: "unknown filesystem type 'exfat'" When trying to mount an exfat filesystem, I get the following error:  Error mounting /dev/sda6 at /media/gkp/Backup: Command-line  `mount -t "exfat" -o "uhelper=udisks2,nodev,nosuid" "/dev/sda6" "/media/gkp/Backup"'  exited with non-zero exit status 32: mount: unknown filesystem type 'exfat'   Exfat is used on some USB sticks and camera sd cards. What can I do to mount this type of filesystem? How to prove positive semidefiniteness of two matrices through Schur Complement? Let matrices $X$ and $Y$ be positive semideﬁnite. Show that $X \succeq Y \succ 0$ is equivalent to $Y^{-1} \succeq X^{-1}$.   The teacher tells me that it is easy to prove via the Schur complement. But I could not get his idea. Could anyone help me and give me some intructions? Thanks in advance! What intuitively is "bias"? I'm struggling to grasp the concept of bias in the context of linear regression analysis.    What is the mathematical definition of bias? What exactly is biased and why/how? Illustrative example? I'm trying to use a mask to hide a part of a render layer, so that when the render layer is shown over a movie, some parts in the movie will appear to be in front of the render layer.  For example, assume I want to add a box to this image, so that it would look like it's behind the person:    I created the following simple object:    If I render the object, it looks great - the object is solid, everything else is transparent (I didn't use any 'sky'):     Now I create a mask to describe the standing person (or at least the part that would otherwise be covered by the box):    What I want now is to create a "hole" in the rendered box, so that when I use Alpha-over, the box look as if it's behind the person, but I can't find an easy way to do that. My first guess was to use 'Set alpha' to apply the mask (inverted) as an alpha for the rendered layer, but this doesn't look good:    The mask did create a "hole" where it should, but now all the other pixels are not transparent. How can I "retain" the transparent pixels while setting more pixels to be transparent based on the mask (use it as a matte)?  Note: I can do this with (I'm ashamed to say) some math work on the alpha channel, but it just doesn't seem right:    Thanks, and sorry for the [oh so] long question. How to increase the signal power of a wi-fi card? I use an atheros card that requires ath9k driver. Are there any tools that can be used to increase the signal power of the wi-fi card? Let us view our deleted comments If moderators can view , can we see comments we've deleted? We can see our deleted answers, but why not our deleted comments? Get dateformat based on Locale Keep "Employee" on the profile of SE employees on meta Currently, SE employees are given diamonds wherever they need them. I agree with  on this to some degree, employees shouldn't be treated any different.  But... on meta, employees are different. They have the power to dole out official responses, and thus we should be able to differentiate between an employee and a normal user. Currently, only some employees have a diamond on MSO (see , ). This makes it confusing, especially when you see official-sounding posts like .  Now, giving diamonds to all employees on meta may not be the best solution to this. I don't think all employees want their precious inboxes spammed with all the suspensions that are doled out on meta, and other mod stuff. They won't really need the diamond powers either (though it wouldn't hurt).  So, I propose this:  Similar to how moderators have "Moderator" clearly on their profile    display "employee" on an employee's profile, along with the diamond if they have one. At least on meta.    It would imho be even better to display it on their usercards in some form, but that has already been shot down.  It would be even even better if non-MSO mods had greyed out diamonds, similar to , but that's just me wanting more shiny stuff ;-) setfacl incorrectly changes group permissions How to interpret the significant interaction of two non-significant main predictors? I have two non-significant main binary variables in my binary logistic regression model, but their interaction is significant. The variables are centered and no multicollinearity is the case (all VIFs about 1.0). The main variables are nonsignificant, but their interaction is. I want to interpret that significant interaction of two non-significant estimates.   It would read something like this: the effect of the variable A is less visible in the level 1 of the variable B (B1) and more visible in the second level of B (B2). Or I can say the same thing about the effect of B being less visible in A1 and more visible in A2...  However, the problem is that neither the effect of A, nor the effect of B are significant! So the above interpretation, although seemingly correct, sounds inconsistent or strange. (How a non-significant effect is supposed to be boosted by the other variable?)...   On a second thought, it seems that it is actually possible. For example, if I exclude cases with B2 from my sample, now the effect of A in the sample would show up as significant (it is B2 cases in the sample that disallow A to appear as significant)... This is getting more clear in my head now, but do you still have something other than the above interpretation in mind? Get two letter country code in Apex method I want to write an Apex method that takes a country name as input and returns the equivalent two letter country code (ISO A2). Is there a way to write it without explicitly hard coding all countries and code in the method? Browsing the internet from the command line How to avoid using sudo when working in /var/www? Can a concentration spell be cast without actually concentrating on it for an "instant" effect? The book says:     Concentration. Some spells require you to maintain concentration in order to keep   their magic active. If you lose concentration, such a spell ends.   So what happens if, for example, you are already concentrating on a spell, and cast Witch Bolt without concentrating. Would you get the initial arc of blue energy, or would it not even go off?  I am asking if you can cast a spell that says concentration without concentrating for an immediate effect. This would be useful to avoid interrupting your existing concentration. In essence, if you are concentrating on thing A (doesn't have to be a spell since other things need concentration), can you cast a concentration spell getting an immediate effect without losing concentration on thing A?  Since the definition of concentration is to keep the spell active (meaning it is already active) I would think that you would get the initial effect (if it had one) without the need to concentrate on it. Load jQuery from Stack's CDN if Google's CDN is blocked (or always) Let $been(x,y)$ denote "person x has been to place y."   Express “Every person has been to 2 or more places” using quantifiers and $been(x, y)$.  This one is completely stumping me. I'm sure that I haven't come across this kind of concept before, and can't wrap my head around how I would approach it. I suspect the quantifiers should be $∀x$, $∀y$, and that it may involve negating the statement "every person has been to one or zero places," but that's as far as I can manage to go. Any help greatly appreciated. XeLaTeX font rendering tend to be slightly bolder compared to PDFlatex How to capture text from the operating system in a practical and simple way? Sometimes we can't select the text to copy &amp; paste, such as dialog boxes.  Sometimes it's just not practical, such as 100 files inside a folder.  Is there any practical and simple way to capture those text from the system into a text file?  I'd wish to know both in OS X and Windows. Maybe it just can't be done today... Upgrade from MaaS 2.6.2 to MaaS 2.7