Difference Calculus Finite Difference Form

Standard differential calculus is based on the definition [math]f’(x_o) = \lim_{x-x_0\to0} \frac{f(x)-f(x_0)}{x-x_0} = \lim_{\Delta x \to 0}\frac{\Delta f(x_0)}{\Delta x_0}[/math].The limit in the definition essentially says to measure how the function [math]f[/math] changes with infinitesimal changes in [math]x[/math].Finite difference calculus works not with infinitesimal differences, but with finite differences, as if the limit wasn’t in the definition: [math]\Delta f(x) = f(x+1)-f(x)[/math] (although other similar definitions could be used).You can also take [math]\Delta^2f(x) = \Delta f(x+1)-\Delta f(x) = (f(x+2)-f(x+1)) - ( f(x+1) - f(x)) = f(x+2) - 2f(x+1) + f(x)[/math], and higher-order differences similarly. These are analogous to higher-order derivatives in differential calculus.There are other similarities, too. For instance, if you look at a function like [math]f(x) = x^3[/math], you can generate the following sequences:[math](f(0), f(1), f(2), \dots) = (0,1,8,27,64,127,\dots); f(n) = n^3[/math][math](\Delta f(0), \dots) = (1,7, 19, 37, 61,\dots); \Delta f(n) = 3n^2 +3n + 1[/math][math](\Delta^2 f(0), \dots) = (6, 12, 18, 24, \dots); \Delta^2 f(n) = 6n+6[/math][math](\Delta^3 f(0), \dots) = (6, 6, 6, \dots); \Delta^3 f(n) = 6[/math][math](\Delta^4 f(0), \dots) = (0,0,\dots); \Delta^4 f(n) = 0[/math]This is an example of the fact that the finite difference of a polynomial is a polynomial of one less degree, just like the fact that the derivative of a polynomial is a polynomial of one less degree. Similarly, you have [math]\Delta (f\pm g) = \Delta f \pm \Delta g[/math], just like [math]d(f \pm g) = df \pm dg[/math], and you also have [math]\Delta (cf) = c\Delta f[/math] like [math]d(cf) = c df[/math].There’s also a product rule that’s similar to the differential case: [math]\Delta (f(n)g(n)) = g(n)\Delta f(n) + f(n+1)\Delta g(n) = g(n)\Delta f(n) + f(n)\Delta n + \Delta g(n)\Delta f(n)[/math], which you can see has the same basic concepts as the differential product rule [math]d(uv) = vdu + udv[/math]. You could say [math]d(uv) = vdu + udv + dvdu[/math] and it would be identical, but because of the limit in the definition of the differential, [math]dvdu \to 0[/math] and vanishes from the product rule.You can see the product rule in a case like [math]\Delta n^2 = \Delta (nn) = n\Delta n + (n+1)\Delta n = n((n+1)-n) + (n+1)((n+1)-n) = n(1) + (n+1)(1) = 2n+1[/math].Since derivatives of exponentials are exponentials, you probably wouldn’t be surprised to find the finite differences of exponentials are also exponentials. Finite differences of trig functions are not as clean: if [math]f(x)[/math] is a sine or cosine, then [math]\Delta f(n) = \sin 1 f’(n) + (\cos 1 - 1)f(n)[/math], where [math]f’(n)[/math] is the standard derivative. The idea is similar, but a bit more complicated.A chain rule exists, but it is extremely complicated (one source lists it as [math]\Delta(f\circ g)(x) = \sum_0^{\Delta g(n)}\Delta f(g(n) + i)\delta i[/math], which itself is a somewhat non-standard notation). In general, it isn’t used.There are a few generalizations one can do with finite differences: One is to use a different step size:[math]\Delta_s f(x) = f(x+s) - f(x)[/math]One is to divide by the step size:[math]\Delta_h f(x) = \frac{f(x+h)-f(h)}{h}[/math]Or both, separately:[math]\Delta_{s,h} = \frac{f(x+s) - f(x)}{h}[/math]Another is to distinguish between forward, backward, and symmetric differences:[math]\Delta_+ f(n) = f(n+1)-f(n), \Delta_- f(n) = f(n)-f(n-1), \Delta f(n) = \frac{f(n+1) - f(n-1)}{2} = \frac{\Delta_+ f(n) + \Delta_-f(n)}{2}[/math]These generalizations can change the exact values of the differences, but the general results (polynomials are reduced in order, products follow a product rule, etc) hold.One useful application of finite differences is that it can make it easy to tabulate the values of polynomials. For instance, if you had a polynomial approximation to some function (like, say, a Taylor polynomial of order [math]n[/math]), you can take [math]n[/math] finite differences to get a [math]f(0), \Delta f(0), \Delta^2 f(0), \dots, \Delta^n f(0)[/math], the last function, [math]\Delta^n f(n) = c[/math] for some constant. Then you can simply do [math]n+1[/math] additions for each next value of [math]f(n)[/math] you want. Before the advent of computers, this is essentially how tables of mathematical functions were generated.

For just about everyone, “finite” means what has a limit. For theologians, “infinite” means what has NO LIMITS WHATSOEVER; and so, for theologians, “infinite” means God and only God. For scientists and mathematicians (though many of them will deny it most adamantly and, if you disagree with them, accuse you of lacking a good background in high school math) “infinite” means what’s merely indeterminate and thus has no end (i.e.: a particular kind of limit) we can establish. Effectively, it means for them no more than a blank regarding how much further to go. By no means does “infinite” signify for them what has NO limits WHATSOEVER. Understandably, for Theology buffs like myself, “infinite”, as used by scientists and mathematicians, is something of a comedy indicating a grave lack of common sense. See page 49 of his book “A Brief History Of Time” (Bantam Books 1998 paperback), where the late Stephen Hawking is honest enough to admit that math can’t really deal with infinite numbers. See also the 1996 “Webster’s New Universal Unabridged Dictionary” which, to define “infinite” uses terms like “immeasurable” and “indefinite”.

Finite and infinite are vague in study of calculus. If we study calculus at beginning when we have taught derivative of single valued function and study of Rolle's theorm exist for a particular function at infinity. Even existence of derivative we notice many functions. In analytical continuation of complex variable exist at infinite extend. In case of Maclaurin series expansion we observe for x€R, f(x)=f(0)+ h /1! f'(0)+ h²/2! f”(0)+ h³/3! f'''(0)+ …+hⁿ/n! fⁿ(0) which is finite of degree n if n->∞ then extend infinite i.e. f(x)=f(0) + h/1! f'0) + h²/2! f”(0) + h³/3! f'''(0) + …. up to ∞. So in 1st case we treat finite and 2nd case we treat it infinite.In case of finding asymptomatic behavior of a curve,we find one branch of this curve is finite and another branch is infinite. But there is no segmentwise study of finite and infinite calculus.

NOOOOOOO. You are talking to a military romance scammer. I received an email from the US Army that directly answers your question that is pasted below please keep reading.I believe you are the victim of a military Romance Scam whereas the person you are talking to is a foreign national posing as an American Soldier claiming to be stationed overseas on a peacekeeping mission. That's the key to the scam they always claim to be on a peacekeeping mission.Part of their scam is saying that they have no access to their money that their mission is highly dangerous.If your boyfriend girlfriend/future husband/wife is asking you to do the following or has exhibited this behavior, it is a most likely a scam:Moves to private messaging site immediately after meeting you on Facebook or SnapChat or Instagram or some dating or social media site. Often times they delete the site you met them on right after they asked you to move to a more private messaging siteProfesses love to you very quickly & seems to quote poems and song lyrics along with using their own sort of broken language, as they profess their love and devotion quickly. They also showed concern for your health and love for your family.Promises marriage as soon as he/she gets to state for leave that they asked you to pay for.They Requests money (wire transfers) and Amazon, iTune ,Verizon, etc gift cards, for medicine, religious practices, and leaves to come home, internet access, complete job assignments, help sick friend, get him out of trouble, or anything that sounds fishy.The military does provide all the soldier needs including food medical Care and transportation for leave. Trust me, I lived it, you are probably being scammed. I am just trying to show you examples that you are most likely being connned.Below is an email response I received after I sent an inquiry to the US government when I discovered I was scammed. I received this wonderful response back with lots of useful links on how to find and report your scammer. And how to learn more about Romance Scams.Right now you can also copy the picture he gave you and do a google image search and you will hopefully see the pictures of the real person he is impersonating. this doesn't always work and take some digging. if you find the real person you can direct message them and alert them that their image is being used for scamming.Good Luck to you and I'm sorry this may be happening to you. please continue reading the government response I received below it's very informative.   You have contacted an email that is monitored by the U.S. Army Criminal Investigation Command. Unfortunately, this is a common concern. We assure you there is never any reason to send money to anyone claiming to be a Soldier online. If you have only spoken with this person online, it is likely they are not a U.S. Soldier at all. If this is a suspected imposter social media profile, we urge you to report it to that platform as soon as possible. Please continue reading for more resources and answers to other frequently asked questions:  How to report an imposter Facebook profile: Caution-https://www.facebook.com/help/16... < Caution-https://www.facebook.com/help/16... >   Answers to frequently asked questions:  - Soldiers and their loved ones are not charged money so that the Soldier can go on leave.  - Soldiers are not charged money for secure communications or leave.  - Soldiers do not need permission to get married.  - Soldiers emails are in this format: [email protected] < Caution-mailto: [email protected] > anything ending in .us or .com is not an official email account.  - Soldiers have medical insurance, which pays for their medical costs when treated at civilian health care facilities worldwide – family and friends do not need to pay their medical expenses.  - Military aircraft are not used to transport Privately Owned Vehicles.  - Army financial offices are not used to help Soldiers buy or sell items of any kind.  - Soldiers deployed to Combat Zones do not need to solicit money from the public to feed or house themselves or their troops.  - Deployed Soldiers do not find large unclaimed sums of money and need your help to get that money out of the country.  Anyone who tells you one of the above-listed conditions/circumstances is true is likely posing as a Soldier and trying to steal money from you.  We would urge you to immediately cease all contact with this individual.  For more information on avoiding online scams and to report this crime, please see the following sites and articles:   This article may help clarify some of the tricks social media scammers try to use to take advantage of people: Caution-https://www.army.mil/article/61432/< Caution-https://www.army.mil/article/61432/>   CID advises vigilance against 'romance scams,' scammers impersonating Soldiers  Caution-https://www.army.mil/article/180749 < Caution-https://www.army.mil/article/180749 >   FBI Internet Crime Complaint Center: Caution-http://www.ic3.gov/default.aspx< Caution-http://www.ic3.gov/default.aspx>   U.S. Army investigators warn public against romance scams: Caution-https://www.army.mil/article/130...< Caution-https://www.army.mil/article/130...>   DOD warns troops, families to be cybercrime smart -Caution-http://www.army.mil/article/1450...< Caution-http://www.army.mil/article/1450...>   Use caution with social networking  Caution-https://www.army.mil/article/146...< Caution-https://www.army.mil/article/146...>    Please see our frequently asked questions section under scams and legal issues. Caution-http://www.army.mil/faq/ < Caution-http://www.army.mil/faq/ > or visit Caution-http://www.cid.army.mil/ < Caution-http://www.cid.army.mil/ >.  The challenge with most scams is determining if an individual is a legitimate member of the US Army. Based on the Privacy Act of 1974, we cannot provide this information. If concerned about a scam you may contact the Better Business Bureau (if it involves a solicitation for money), or local law enforcement. If you're involved in a Facebook or dating site scam, you are free to contact us direct; (571) 305-4056.   If you have a social security number, you can find information about Soldiers online at Caution-https://www.dmdc.osd.mil/appj/sc... < Caution-https://www.dmdc.osd.mil/appj/sc... > . While this is a free search, it does not help you locate a retiree, but it can tell you if the Soldier is active duty or not.  If more information is needed such as current duty station or location, you can contact the Commander Soldier's Records Data Center (SRDC) by phone or mail and they will help you locate individuals on active duty only, not retirees. There is a fee of $3.50 for businesses to use this service. The check or money order must be made out to the U.S. Treasury. It is not refundable. The address is:  Commander Soldier's Records Data Center (SRDC) 8899 East 56th Street Indianapolis, IN 46249-5301 Phone: 1-866-771-6357  In addition, it is not possible to remove social networking site profiles without legitimate proof of identity theft or a scam. If you suspect fraud on this site, take a screenshot of any advances for money or impersonations and report the account on the social networking platform immediately.  Please submit all information you have on this incident to Caution-www.ic3.gov < Caution-http://www.ic3.gov > (FBI website, Internet Criminal Complaint Center), immediately stop contact with the scammer (you are potentially providing them more information which can be used to scam you), and learn how to protect yourself against these scams at Caution-http://www.ftc.gov < Caution-http://www.ftc.gov > (Federal Trade Commission's website)

You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if a question is answerable by one of a few powerful general purpose "machines"  (e.g., continuity arguments, the correspondences between geometric and algebraic objects, linear algebra, ways to reduce the infinite to the finite through various forms of compactness) combined with specific facts you have learned about your area. The number of fundamental ideas and techniques that people use to solve problems is, perhaps surprisingly, pretty small -- see http://www.tricki.org/tricki/map for a partial list, maintained by Timothy Gowers.You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It's not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion. More on this in the next few bullets.Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a "fixed point" which does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don't know that they shouldn't be straining to visualize things for which they don't seem to have the visualizing machinery.) Instead...When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights. For example, you might imagine two- and three-dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don't have the desired property. Then you think about what was important to the examples and try to distill those ideas into symbols. Often, you see that the key idea in the symbolic manipulations doesn't depend on anything about two or three dimensions, and you know how to answer your hard question. As you get more mathematically advanced, the examples you consider easy are actually complex insights built up from many easier examples; the "simple case" you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical illumination about something intractable; you work to reduce it to the things that feel friendly.To me, the biggest misconception that non-mathematicians have about how mathematicians work is that there is some mysterious mental faculty that is used to crack a research problem all at once. It's true that sometimes you can solve a problem by pattern-matching, where you see the standard tool that will work; the first bullet above is about that phenomenon. This is nice, but not fundamentally more impressive than other confluences of memory and intuition that occur in normal life, as when you remember a trick to use for hanging a picture frame or notice that you once saw a painting of the street you're now looking at.In any case, by the time a problem gets to be a research problem, it's almost guaranteed that simple pattern matching won't finish it. So in one's professional work, the process is piecemeal: you think a few moves ahead, trying out possible attacks from your arsenal on simple examples relating to the problem, trying to establish partial results, or looking to make analogies with other ideas you understand. This is the same way that you solve difficult problems in your first real maths courses in university and in competitions. What happens as you get more advanced is simply that the arsenal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try. Sometimes, during this process, a sudden insight comes, but it would not be possible without the painstaking groundwork [ http://terrytao.wordpress.com/ca... ].Indeed, most of the bullet points here summarize feelings familiar to many serious students of mathematics who are in the middle of their undergraduate careers; as you learn more mathematics, these experiences apply to "bigger" things but have the same fundamental flavor.You go up in abstraction, "higher and higher". The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves -- that is, you "zoom out" so that every function is just a point in a space, surrounded by many other "nearby" functions. Using this kind of zooming out technique, you can say very complex things in short sentences -- things that, if unpacked and said at the zoomed-in level, would take up pages. Abstracting and compressing in this way makes it possible to consider extremely complicated issues with one's limited memory and processing power.The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques. A theoretical physicist friend likes to say, only partly in jest, that there should be books titled "______ for Mathematicians", where _____ is something generally believed to be difficult (quantum chemistry, general relativity, securities pricing, formal epistemology). Those books would be short and pithy, because many key concepts in those subjects are ones that mathematicians are well equipped to understand. Often, those parts can be explained more briefly and elegantly than they usually are if the explanation can assume a knowledge of maths and a facility with abstraction. Learning the domain-specific elements of a different field can still be hard -- for instance, physical intuition and economic intuition seem to rely on tricks of the brain that are not learned through mathematical training alone. But the quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts that make learning other fields easier, as long as you are willing to be humble and modify those mathematical habits that are not useful in the new field.You move easily among multiple seemingly very different ways of representing a problem. For example, most problems and concepts have more algebraic representations (closer in spirit to an algorithm) and more geometric ones (closer in spirit to a picture). You go back and forth between them naturally, using whichever one is more helpful at the moment. Indeed, some of the most powerful ideas in mathematics (e.g., duality, Galois theory, algebraic geometry) provide "dictionaries" for moving between "worlds" in ways that, ex ante, are very surprising. For example, Galois theory allows us to use our understanding of symmetries of shapes (e.g., rigid motions of an octagon) to understand why you can solve any fourth-degree polynomial equation in closed form, but not any fifth-degree polynomial equation. Once you know these threads between different parts of the universe, you can use them like wormholes to extricate yourself from a place where you would otherwise be stuck. The next two bullets expand on this.Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days figuring out why a result follows easily from some very deep and general pattern that is already well-understood, rather than from a string of calculations. Indeed, you tend to choose problems motivated by how likely it is that there will be some "clean" insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities. (Nevertheless, detailed calculation of an example is often a crucial part of beginning to see what is really going on in a problem; and, depending on the field, some calculation often plays an essential role even in the best proof of a result.)In A Mathematician's Apology [http://www.math.ualberta.ca/~mss..., the most poetic book I know on what it is "like" to be a mathematician], G.H. Hardy wrote:"In both [these example] theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and  economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with  the far-signNowing results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way."[...]"[A solution to a difficult chess problem] is quite genuine mathematics, and has its merits; but it is just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly) which a real mathematician tends to despise."You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles. Mathematicians don't really care about "the answer" to any particular question; even the most sought-after theorems, like Fermat's Last Theorem, are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing. The accomplishment a mathematician seeks is finding a new dictionary or wormhole between different parts of the conceptual universe. As a result, many mathematicians do not focus on deriving the practical or computational implications of their studies (which can be a drawback of the hyper-abstract approach!); instead, they simply want to find the most powerful and general connections. Timothy Gowers has some interesting comments on this issue, and disagreements within the mathematical community about it [ http://www.dpmms.cam.ac.uk/~wtg1... ].Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: "First I will lay this foundation, then I will build this framework using these familiar pieces, but leave the walls to fill in later, then I will test the beams..." All these steps have mathematical analogues, and structuring things in a modular way allows you to spend several days thinking about something you do not understand without feeling lost or frustrated. (I should say, "without feeling unbearably lost and frustrated"; some amount of these feelings is inevitable, but the key is to reduce them to a tolearable degree.)Andrew Wiles, who proved Fermat's Last Theorem, used an "exploring" metaphor:"Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them." [ http://www.pbs.org/wgbh/nova/phy... ]In listening to a seminar or while reading a paper, you don't get stuck as much as you used to in youth because you are good at modularizing a conceptual space, taking certain calculations or arguments you don't understand as "black boxes", and considering their implications anyway. You can sometimes make statements you know are true and have good intuition for, without understanding all the details. You can often detect where the delicate or interesting part of something is based on only a very high-level explanation. (I first saw these phenomena highlighted by Ravi Vakil, who offers insightful advice on being a mathematics student: http://math.stanford.edu/~vakil/... .)You are good at generating your own definitions and your own questions in thinking about some new kind of abstraction. One of the things one learns fairly late in a typical mathematical education (often only at the stage of starting to do research) is how to make good, useful definitions. Something I've reliably heard from people who know parts of mathematics well but never went on to be professional mathematicians (i.e., write articles about new mathematics for a living) is that they were good at proving difficult propositions that were stated in a textbook exercise, but would be lost if presented with a mathematical structure and asked to find and prove some interesting facts about it. Concretely, the ability to do this amounts to being good at making definitions and, using the newly defined concepts, formulating precise results that other mathematicians find intriguing or enlightening. This kind of challenge is like being given a world and asked to find events in it that come together to form a good detective story. You have to figure out who the characters should be (the concepts and objects you define) and what the interesting mystery might be. To do these things, you use analogies with other detective stories (mathematical theories) that you know and a taste for what is surprising or deep. How this process works is perhaps the most difficult aspect of mathematical work to describe precisely but also the thing that I would guess is the strongest thing that mathematicians have in common.You are easily annoyed by imprecision in talking about the quantitative or logical. This is mostly because you are trained to quickly think about counterexamples that make an imprecise claim seem obviously false.On the other hand, you are very comfortable with intentional imprecision or "hand-waving" in areas  you know, because you know how to fill in the details. Terence Tao is very eloquent about this here [ http://terrytao.wordpress.com/ca... ]: "[After learning to think rigorously, comes the] 'post-rigorous' stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the 'big picture'. This stage usually occupies the late graduate years and beyond."In particular, an idea that took hours to understand correctly the first time ("for any arbitrarily small epsilon I can find a small delta so that this statement is true") becomes such a basic element of your later thinking that you don't give it conscious thought.Before wrapping up, it is worth mentioning that mathematicians are not immune to the limitations faced by most others. They are not typically intellectual superheroes. For instance, they often become resistant to new ideas and uncomfortable with ways of thinking (even about mathematics) that are not their own. They can be defensive about intellectual turf, dismissive of others, or petty in their disputes. Above, I have tried to summarize how the mathematical way of thinking feels and works at its best, without focusing on personality flaws of mathematicians or on the politics of various mathematical fields. These issues are worthy of their own long answers!You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer. (The theoretical computer scientist Richard Lipton lists some examples of potentially "deep" ignorance here: http://rjlipton.wordpress.com/20...) This makes it more comfortable to be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being very intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.

I’m sure someone already wrote the answer, but here it goes:Nationality = the country from where you are from regardless of race/ethnicity.Ethnicity = usually inherited; people from the same ancestry sharing the same culture, traditions, language, rituals, etc…Race = a social construct that states humans that share the same physical qualities (these qualities may differ)Example One: Both Celia Cruz and Desi Arnaz were Cuban entertainers. They shared the same nationality (both were from Cuba) yet were different races as Celia was a black woman and Desi a mestizo? (mixed Spaniard with indigenous ancestry?). It is possible that Celia had Spanish ancestry although her physical appearance had dark skin and it is possible that Desi had a tiny bit of African ancestry (only he knows and DNA tests weren’t available then).Example Two: Two Hypothetical Nigerians.Two men are both Nigerians (same nationality), but are from different ethnic groups (one is Yoruba, the other is Ibo), they are both of the black race (not using an example of a mixed-race Nigerian).Example Two Hypothetical African-AmericansA man and a woman are born in the USA and they are the fifth generation. They are both American (nationality), they are both black (race), and they are of varying ethnic groups (took DNA tests) of all the Africans who were transported during the Atlantic Slave Trade including the European slave holders who produced children with their slaves.

I was selected for a summer internship 2016.I tried to be very open while filling the preference form: I choose many products as my favorite products and I said I'm open about the team I want to join.I even was very open in the  location and start date to get host matching interviews (I negotiated the start date in the interview until both me and my host were happy.) You could ask your recruiter to review your form (there are very cool and could help you a lot since they have a bigger experience).Do a search on the potential team.Before the interviews,  try to find smart question that you are going to ask for the potential host (do a search on the team  to find nice and deep questions to impress your host). Prepare well your resume.You are very likely not going to get algorithm/data structure questions like in the first round. It's going to be just some friendly chat if you are lucky. If your potential team is working on something like machine learning, expect that  they are going to ask you questions about machine learning, courses related to machine learning you have and relevant experience (projects, internship). Of course you have to study that before the interview. Take as long time as you need if you feel rusty. It takes some time to get ready for the host matching (it's less than the technical interview)  but it's worth it of course.

The differential in basic calculus is an example of the differential form in differential geometry.Vector fields X act on functions f, by forming the expression f(p + tX(p)) and taking the derivative with respect to t for t=0. In other words, Xf(p) is how fast f is changing near p in the direction X(p). If f is a smooth function, then Xf is another smooth function. X(f+g) = Xf + Xg, X(ag) = a X(g), and X(fg) = (Xf)g + f(Xg). The smooth functions form a ring, and such a transformation on a ring is called a derivation. A linear form is a linear map from vector fields to functions, which is not just linear for real numbers, but even for functions: w(fX) = fw(X).On a manifold, you either repeat this discussion in a coordinate system, or you replace p + tX(p) with any curve s for which s(0) = p, and whose tangent vector at 0 is X(p).The differential of calculus is the 1-form df which is the evaluation map: df(X) = Xf. All the usual rules for derivatives follow from the fact that X is a derivation. Any differential form is a sum of smooth functions multiplied by the differentials of smooth functions.