For my raytracing needs, I'd like to find a way to compute a normal from a given 3d equation. This equation can be either a parametric one or a cartesian one (preferably in canonical form).  From my researches on the internet, it seems to be doable by computing the gradient of the equation but I want to be sure about that.  I manually tried to test against the sphere cartesian equation and it worked but that's not enough to prove that that is the solution.
From this , I now understand the gradient to be the directional derivative that returns the steepest slope at a point. However, reading my textbooks, they all say that the gradient is normal to the tangent plane. In the above SE question, the gradient sounds like it should be parallel to a direction in the $xy$ plane, not actually up on the surface.   My question is therefore do I have a misunderstanding of the tangent plane, gradient or directional derivative?   In addition, I question how the gradient always passes through the origin or the $xy$ plane? If for instance in the graphic, the gradient had been another direction (say horizontal), then it wouldn't pass through the origin! In addition, the gradient of other points in the graphic would satisfy this example. Many thanks!
How do I reset a lost password (using recovery mode requires me to type the password)? Show that  $$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$$ Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I  seem to be having some trouble with the technical stuff. A specific example would be how to find the Galois group of a given polynomial. I know some tricks, and I manage to solve some of those questions, but some not.  For example, one of the tests asks to find the Galois group of $x^{4}-4x+2$.  I can see that it is irreducible over $\mathbb Q$ (Eisenstein), but I have no clue as to how to find its Galois group over $\mathbb Q$. Can someone tell me how to do this? General techniques concerning this sort of problems are also welcome :).  Thanks! Showing that $\mathbb{R}$ is connected So I know that $\mathbb{R}$ is both open and closed. But given a set, $X\subset \mathbb{R}$, $X\ne \emptyset $ that is both open and closed, how does one show that $X=\mathbb{R}$?    Here is my attempt: Using the definitions I was given, $X$ open implies that for every $x\in X$ there is a $\delta \gt 0$, $(x-\delta,~ x+\delta)\subset X$.    Also, since $X$ is closed , $B=\mathbb{R}$\  $X$ is open.    However, I don't know how to proceed. NB: This is not a homework problem. What does $dx$ mean? $dx$ appears in differential equations, such us derivatives and integrals.  For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is. When Frodo put on the One Ring, both the Nazgul and Sauron himself found Frodo. However when Bilbo had the ring and possibly used it many times, Sauron sought Gollum to get a hint on where to look for the ring and send the Nazgul after it.   Sauron sought Gollum before Bilbo wore the ring at his birthday party, slightly after that time, he was very effective at locating Frodo when Frodo put on the ring. So I guess it's not a matter of Sauron gaining any extra powers with time.  Also Bilbo seemed to use the ring without it bothering him at any point, unlike Frodo who seemed to be very troubled when using the ring. How is that possible? How to submit a form when the return key is pressed? Can someone please tell me how to submit an HTML form when the return key is pressed and if there are no buttons in the form? The submit button is not there. I am using a custom div instead of that. Question about the proof of Ramsey's Theorem I was reading up on a proof of Ramsey's Theorem and I can't understand this part of the proof:      Pick a vertex $v$ from the graph, and partition the remaining vertices into two sets $M$ and $N$, such that for every vertex $w$, $w$ is in $M$ if $(v, w)$ is blue, and $w$ is in $N$ if $(v, w)$ is red. Because the graph has $R(r - 1, s) + R(r, s - 1) = |M| + |N| + 1$ vertices, it follows that either $|M| \geq R(r − 1, s)$ or $|N| \geq R(r, s − 1)$.   Why is that $|M| \geq R(r − 1, s)$ or $|N| \geq R(r, s − 1)$? I know from the proof of $R(3,3) = 6$ that given a vertex, at least $3$ are guaranteed to be either red or blue, so what is the pigeonhole argument for the general case? Kill process spawned by ssh when ssh dies Algorithm to return all combinations of k elements from n I want to write a function that takes an array of letters as an argument and a number of those letters to select.   Say you provide an array of 8 letters and want to select 3 letters from that. Then you should get:  8! / ((8 - 3)! * 3!) = 56   Arrays (or words) in return consisting of 3 letters each. I have an object already created in blender, a tree, or a human, or a pipe.. and I need this to be drilled with multiple holes.   The holes are painted in an image, that is, it is an image that has a repeating an images pattern (they can be circles, squares, or any form).  Here are some examples for drilling-pattern: How can I run a program on startup, minimized? I just want Telegram to be run and I have added it to startup apps. The point is that I need it to be minimized. Any commands? What are the solutions to $z^4+1=0$? Scala final vs val for concurrency visibility Are there any official specifications regarding the torque for camera mounts? I've been wondering why I've never seen any official specifications of the maximum allowed torque that can be exerted upon lens mounts. It seems to me like a very useful specification, but I can't seem to find it from any manufacturer.  indicates that there might be no specifications for Nikon mounts. Are there really no manufacturer that publish these specification of their mounts? Solving the diophantine equation $y^{2}=x^{3}-2$ It is known that the diophantine equation $y^{2}=x^{3}-2$ has only one positive integer solution $(x,y)=(3,5)$. The proof of it can be read from the book "About Indeterminate Equation" (in Chinese, by Ke Zhao and Sun Qi). But the method I have known of solving this problem is algebraic number theory.  My question:      Does there exists an elementary method for solving the diophantine equation $y^{2}=x^{3}-2$?   I have gotten an elementary method by the assumption that "$x$" is a prime. That is, we can get a conclusion from the equation that $3\mid x$. Since $x$ is a prime we get the only solution $(x,y)=(3,5)$. But if $x$ is not a prime, the answer seems to be difficult.  Thank you for your help. Evaluating $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\, \mathrm{d}x$ Do I need a transit visa for a layover in the United Arab Emirates? Do I need a transit visa for a layover via airports in the UAE, such as Dubai International Airport or Abu Dhabi International Airport? If so, what are the requirements for such a visa? Can I get the transit visa on arrival? Is it true that an ideal is primary iff its radical is prime?