Why is the magnitude of the gradient equal to the maximum rate of change at that point? I understand the concept of the gradient being a vector of the partials of f with respect to each variable, so essentially the gradient gives you a direction in the input field to travel in order to get the maximum increase in the function f. What I don't understand is why the magnitude of that gradient is the actual maximum rate of change of f - it feels right that it should be but I can't quite join the dots and see a proper reason for it.
Gradient and Swiftest Ascent I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$)  The standard proof is to note that the directional derivative is $$D_vf=v\cdot \nabla f=|\nabla f|\,\cos\theta$$ which is maximized at $\theta=0$. This is a good verification, but it doesn't really help me understand the result.
Do computer viruses affect all user accounts on a computer? Do computer viruses affect all user accounts on a computer?  IF a virus affects a local user will that effect administrator account and vise-versa? Counterexample for the normalizer being a normal subgroup Rooks on a 8 by 8 checker board. Probability problem Two rooks are placed on an 8 by 8 chessboard such that they're in different squares. What is the probability they can't attack each other? Rooks can attack each other if and only if they're in the same row or column. I know that there are questions like this but they give the general numbers not the probability I am not sure how to start the questions and I have looked at this questions for hours. Any help would be welcome. The Data Interoperability Tools .tbx appears in my system toolboxes, however when I double click quick export or import, a red X appears.  I have gone into the administrator settings and the tool doesn't appear to be installed. How do I install it?  I am using ArcGIS 10.2 for desktop trial.  After I attempt to evaluate extensions, I receive a message which states all requested features have already been authorized on this machine. How to redirect users to custom lostpassword page? Mixing tenses in past tense fiction Notepad doesn't open files created by Gedit? Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$ Let $a,b,$ and $c$ be the lengths of the sides of a triangle. Prove that $$\left | \dfrac{a-b}{a+b}+\dfrac{b-c}{b+c}+\dfrac{c-a}{c+a} \right | &lt; \dfrac{1}{8}.$$   The best idea I had was to expand the fractions to get something nicer. So we get $\dfrac{a-b}{a+b}+\dfrac{b-c}{b+c}+\dfrac{c-a}{c+a} = \dfrac{a^2 b-a^2 c-a b^2+a c^2+b^2 c-b c^2}{(a+b) (a+c) (b+c)}.$ Then I would try to relate this to side lengths of a triangle to prove the desired inequality but that is the step where I am unsure. Why the interior of $\mathbb{Q}$ in $\mathbb{R}$ is empty? Find index of nested item in python How does a spinning object "know" that it is spinning? How large is it possible for a PC to get? How large is it possible for a PC to get?  For clarity, here are my parameters:   This is about size categories, not 'feet' or 'inches' or 'meters' or any other precise measurement. This is about Medium to Large to Huge to Gargantuan categories.  This is for an NPC charlatan using this to trick people. (It's an NPC but will have a full PC sheet, so the distinction between PC and NPC doesn't matter). Any means of size category enlargement works for this purpose, including magic spells, magic items, etc. Illusions do not count because it's about actually becoming larger, not just casting an illusion of a gargantuan giant version of yourself.  The character must remain the same physical structure. Polymorphing into a Gargantuan Dragon would not satisfy the need here. Consumable items are fine. This does not need to be permanent. Add something to crontab programmatically (over ssh) I have a deployment script, it must add something to a user crontab (trigger a script that clean logs every XXX days), however this must only be done during the first deployment, or when it needs to be updated.  (I can run xxx.py deploy env or xxx.py update env)  so I have to do this:  Check if my cronJob already exist Put my cronJob if it does not already exist or update my cronjob if one of the parameter of the command is different   I don't see how to add/check/remove something to the crontab without using crontab -e or editing the crontab file (download it, rewrite it, re-upload it)  PS : this is a user specific cronjob, "webadmin" is going to do it and he should not use sudo to do it. Edit:  I foolishly changed my example right before posting this. The following is what I meant to ask.  On the wikipedia page below, it says "a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units."   Then, on the wikipedia page below, it says "an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials."   The page then gives an example of a polynomial that is irreducible "over the rationals" (I assume it is implied that "the integers" is referring to the ring of polynomials over the field of rationals, right?)  $p(x) = x^2 + 2 = (x+\sqrt{2})(x-\sqrt{2})$  On the Wofram page below, it says "the irreducible polynomials are examples of irreducible elements."   If irreducible polynomials are a subset of irreducible elements, how is $x+\sqrt{2}$ a unit?  What can it be multiplied by to equal the multiplicative identity?  I think I have it right this time (sorry. long day)  ====================================================================  The following is the original post (for those who already responded and those looking at the corresponding responses)  On the wikipedia page below, it says "a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units."   Then, on the wikipedia page below, it says "an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials."   The page then gives an example of a polynomial that is reducible "over the integers" (I assume it is implied that "the integers" is referring to some sort of ring, right? due to the claim below)  $p(x) = x^2 + 4x + 4 = (x+2)(x+2)$  On the Wofram page below, it says "the irreducible polynomials are examples of irreducible elements."   If irreducible polynomials are a subset of irreducible elements, how is $x+2$ a unit?  What can it be multiplied by to equal the multiplicative identity? Segmentation fault error occurs for a simple function I have coded How old should you be when you begin Jedi training? How to install Unity 8? I'm using Ubuntu 14.04 and found out that we can install Unity 8 on it. Will Unity 8 replace Unity 7 once I install it? And if it does can I switch back to Unity 7 if I want? How would one go about it? If $ a$, $ b$, $ x$, $ y$ are integers greater than $1$ such that $ a$ and $ b$ have no common factor except $1$ and $ x^a = y^b$ show that $ x = n^b$, $ y = n^a$ for some integer $ n$ greater than $1$.  I started this way: $x^a=y^b$ $=&gt;$ $x=y^{b/a}=(y^{1/a})^b$. Thus $x$ is an integer, and so is $y^{1/a}$, so $y=n^a$ for some integer $n$ greater than $1$. Then from the given relation, we get $x=n^b$.   I am not very convinced with my solution, as I feel I did some mistake. Can anyone point out where the mistake is? Any other solution is also welcome :) 'bridges' on a PCB copper layer I've come across a design where each pad was connected using 4 'bridges' to the GND copper layar. What stands behind these 'bridges'? Why not make a full copper layer with only solder-mask defining the pads? “between” vs. “among”