Why do we ignore squares of infinitesimal quantities in differential calculus , doesn’t that cause error when we require very accurate measurements.?
I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the , right after giving an example of a nonstandard differentiation:     The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity... the typical method from Newton through the 19th century would have been simply to discard the $dx^2$ term.   I've never heard anything like this before, and really find it fascinating that Newton's method was to define the relation $dx^2 = 0$. If we actually formalize the above structure by taking $\mathbb{R}$ and adjoining an element $dx^2 = 0$ to it, we get the "dual numbers," isomorphic to the quotient ring $\mathbb{R}[x]/x^2$. I'd seen some things about how this algebra plays into automated differentiation algorithms for some computer software systems, but I've never heard anything about Newton directly working in this algebra. So I have a few questions:   Does anyone have more historical information on the way that Newton performed differentiation, and its relation to the dual numbers? Does anyone know how effectively real analysis can be formalized with the dual numbers? Does the resulting system play nice enough to develop all of the important modern results? If we start with $\mathbb{C}[x]/x^2$ instead, can we likewise develop complex analysis?   Since this idea is so simple, I'm very curious how powerful it is. I'm also curious if it has any major drawbacks too, since I'm not sure why anyone would mess with the foundational baggage involved in defining the hyperreals if this simple 2-dimensional real algebra could really do the trick.
When to use icons vs. icons with text vs. just text links? Are there any rules on when to use icons vs. icons with text vs. just text links in a web application?  For example:    vs.   save  vs.  save    Extra question: Is  save -or- save  preferable? Book wherein humans were most dangerous/aggressive of numerous species I recall there were two warring alien empires, one of which was led by a race that could telepathically inflict pain on others, but when they tried to do it to humans, it would backfire. something about human's animal instincts being much closer to the surface than the other species.  Humans made the most aggressive and dangerous fighters, so both empires tried to reruit humanity, since the other races involved in the war were fairly monolithic and sided all with one empire or the other. Humanity was all splintered though.  The empire led by the telepathic race claimed that everyone in it was equal but in reality the telepathic race was in charge. If I want to display the IP address that is assigned to eth1, how can I do this in Bash? "He added" vs. "Did he add" (inversion after negative adverb) The following exercise comes from Fulton's Algebraic Curves:     Let $k$ be an infinite field and $V = \{(t,t^2,t^3) \, ; \, t \in k \} \subset \mathbb{A}^3(k)$. Show that $V$ is algebraic and compute $I(V)$ by giving a finite set of generators.   Now, showing $V$ is algebraic is easy by noting that $V = V(X^2-Y, X^3-Z)$, but I can't find a way to describe $I(V)$ explicitly. Obviously $(X^2-Y)(X^3-Z) \in V$, but how can I find the actual generators of this ideal? How to install World of Warcraft? Chromium on Ubuntu 12.04. Youtube says:     This video is currently unavailable   I don't know why but this is the case for some videos, but not all. Well, most videos. And I don't know what's wrong.  Anyone help? We are going to change the IP address of our SharePoint Servers can any one point out the RISK and IMPACT How to integrate $\int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx$ where $a,b > 0$. Can I install the 64 bit version? I want to programmatically create a site using custom web templates in sharepoint 2010.  In 2007 I used to get a   SPWebTemplate template = site.GetCustomWebTemplates(LCID)[number];   and used this SPWebTemplate in  SPWeb newSite = web.Webs.Add(url, title, description, LCID, template, true, false);   This worked quite well for 2007. But in 2010 the site templates are saved into the solution store  SPSite.Solutions   which only returns SPUserSolution objects and not SPWebTemplates.  So my question is: How can I programmatically create sites using the site templates from the solution store.    UPDATE: I tried to use  SPWeb newSite = web.Webs.Add(url, title, desc, LCID, templateName, true, false);   where templateName is a string like '{Template-GUID}#TemplateName', but it still does not work. It throws the following error:  Critical error: File or arguments not valid for site template '{4a2d8952-a0af-49a6-8f55-46a59c8d68a2}#CustomerSiteTemplate'. Parameter name: WebTemplate    Any other ideas how to create a site based on a SPWebTemplate? network connection over thunderbolt bridge between linux and mac biblatex – et al. beginning from second citation? Basically all I need to know is how I can use "et. al" beginning from the second citation of a source. I am using the following  \usepackage[ natbib=true, style=authoryear ]{biblatex}   I need citations to look like this:  First citation of a source     Drobetz, Schillhofer, and Zimmermann (2004)   Second and further citations     Drobetz et al. (2004)   Right now, every citation looks like the first one. With the biblatex option maxcitenames I can only adjust all citations and not all but the first one. Any help? How to unlock device when I forgot my password? I've read  which states that there is an option to disable the lock using my Google account information. However I did not noticed such an option when I entered my password five times wrong. (Although there was also a 20s or 30s delay to complicate brute force attacks.) I use Android 4.0.3 on a Asus Transformer Prime TF201.  Would factory reset work? I am going to install Ubuntu to dual boot with Windows 8 soon, but one of my friends warned me that partitioning my hard drive will slow my computer down. I run on a 500GB HDD and 8GB of ram if that helps at all. Is that true if I do it? Are we inside a black hole? I was surprised to only recently notice that      An object of any density can be large enough to fall within its own   Schwarzschild radius.   Of course! It turns out that supermassive black holes at galactic centers can have an average density of less than water's.  Somehow I always operated under the assumption that black holes of any size had to be superdense objects by everyday standards.  Compare the Earth to collapsing into a mere 9mm marble retaining the same mass, in order for the escape velocity at the surface to finally reach that of light.  Or Mt. Everest packed into one nanometer.  Reading on about this , it increases proportionally with total mass.  Assuming matter is accumulated at a steady density into a spherical volume, the volume's radius will only "grow" at a cube root of the total volume and be quickly outpaced by its own gravitational radius.  Question: For an object the , what would have to be its diameter for it to qualify as a black hole (from an external point of view)?  Would this not imply by definition that:   The Earth, Solar system and Milky Way are conceivably inside this black hole? Black holes can be nested/be contained within larger ones? Whether something is a black hole or not is actually a matter of perspective/where the observer is, inside or outside? Is the axiom of choice used in the proof that every open set is the union of basis elements (Mukres, Lemma 13.1) Is having a skills bar chart on a resume a good idea? How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$ How are answers with the same vote score sorted?