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Intuitively, the fundamental theorem of calculus states that "the total change is the sum of all the little changes". $f'(x) \, dx$ is a tiny change in the value of $f$. You add up all these tiny changes to get the total change $f(b) - f(a)$. In more detail, chop up the interval $[a,b]$ into tiny pieces: a = x_0 < x_1 < \cdots < ...
To combine asymptotic analysis with nonstandard analysis. By the definition of derivative, $$f'(x) = \frac{f(x + \epsilon) - f(x)}{\epsilon} + o(1)$$ ($o(1)$ means the error is infintiesimal) If $H$ is a positive, infinite, nonstandard integer, then by the left endpoint rule, using the shorthand $\xi_i = a + i (b-a)/H$, \begin{align}\int_a^b f'(x) ... 6 Others have said that the total change is the sum of the infinitely many infinitely small changes, and I agree. I will add another way of looking at it. Think of \displaystyle A = \int_a^x f(t) \, dt, and imagine x moving. Draw the picture, showing the t-axis, the graph of f, the vertical line at t=a that forms the left boundary of the region ... 5 Expanding on my comment, let me inspire further: Retraction Suppose you have a topological space X and A\subset X is a subspace. Hence there is a natural injection map (inclusion) \imath:A\hookrightarrow X. We say the continuous map r:X\to A is a retraction map if r|_{A}=\mathrm{id} and r(X)=A. In other words although r doesn't touch A, it ... 5 For one, in studying an abelian group you can utilize a lot of your intuition from doing addition. But more to the point about normal subgroups - One should note that in an abelian group G, every subgroup H is normal, so you can always take the quotient G/H. In a non-abelian group, you need special conditions on H. The definition of a normal ... 5 This reminds me at the Zeno's paradox, in which Zeno said that dividing infinitely many times the distance between Achille and the turtle of a half he will not be able to move, because of the infinitude of the process (I'm not writing for now the details of the paradox). The matter is that not all the processes which are infinite are automatically not ... 4 As mentioned above, countably infinite formally means a bijection (one to one and onto mapping) to the naturals exists, but I wanted to offer a personal response to "Why distinguish between countably infinite and uncountably infinite?" Well, we don't always distinguish between the two. For example if you were answering a question in elementary linear ... 3 I think one surprising consequence is that we tend to think of transcendental numbers like \pi and e as being incredibly rare. (After all, how many can you name?) However, all but countably-many real numbers are transcendental. It is the algebraic numbers that are rare. In fact, the algebraic numbers account for precisely 0\% of the real numbers. You ... 3 One especially useful application is to give very easy proofs of existence of objects that can otherwise be much harder to construct. To take your example, here is the shortest and easiest proof that transcendental numbers exist: the reals are uncountably many, but the algebraic numbers are countable, so there must be reals that are not algebraic. Before ... 3 The question in its current form is too general. However, the case \pi_n(S^{n-1}) is relatively easy and was discovered first. In Milnor's Topology from differential viewpoint, you can find a quite intuitive explanation of the fact that elements of \pi_3(S^2) correspond to framed cobordism classes of 1-dimensional smooth submanifolds of S^3, that ... 3 You are rewriting the old Zeno paradoxes. They come from the confusion between a well-defined finite value and an infinite process that defines it. By the way, this phenomenon is not at all specific to irrational numbers, you could rephrase it for any unlimited decimal number, and even integers! Taking your example, \lim_{n\to\infty}\frac1n=0, the ... 3 As A.S.'s comment indicates, both distributions relate to the same kind of process (a Poisson process), but they govern different aspects: The Poisson distribution governs how many events happen in a given period of time, and the exponential distribution governs how much time elapses between consecutive events. By way of analogy, suppose that we have a ... 2 Exponential is a continuous distribution while poisson is a discrete one. 2 One of my teachers always told me "don't know definitions, don't know math." At the time I was pretty annoyed, but he was completely right. The only way to learn math is to have the fundamentals down cold. This involves both a rigorous side, (memorizing them is a good start) and an intuitive side. So at an entry level, I strongly recommend spending a long ... 2 It turns out that there is a relationship between analytic torsion and torsion in (co)homology. The idea is that analytic torsion equals Reidemeister torsion by the Cheeger-Muller theorem, and Reidemeister torsion is equal to the alternating product of sizes of torsion subgroups of integer homology (modulo some normalizing factors called regulators). This ... 2 This is not a probability specific concept. Rather, it has to do with the concept of densities and their transformation laws. First, let us write a 1d probability integral in the following equivalent form:P_\alpha(b) - P_\alpha(a) = \int_a^b p_\alpha(\alpha) e^\alpha \cdot e_\alpha \, d\alpha$$where e^\alpha \cdot e_\alpha = 1. From a geometric ... 2 There are really two FTCs. One is what you have written. The other is$$\frac{d}{dx} \int_a^x f(y) dy = f(x)$$for continuous f. The latter is easier to understand. If you replace x by x+\Delta x for small positive \Delta x, then you add area which is "well-approximated" by a rectangle of height f(x) and width \Delta x. You can intuitively ... 2 Take your favourite example of a universal arrow, say \{a, b\} → U\{a, b\}^* (the free monoid generated by letters a, b). Now \{a, b\}^*  is certainly big enough to factor every arrow \{a, b\} → UM, and it will remain so even if you add some clutter, making it into eg. \{a, b, c\}^*. But now the factorization is not unique, because you can map c to ... 2 Part 2 is easier to answer: no, the way monotone operators are defined in functional analysis, -F is not in general monotone when F is. (This is unlike the concept of monotonicity in real analysis). Monotone operators correspond to (non-strictly) increasing functions. The reverse inequality defines dissipative operators. Part 1, geometric ... 1 This is not really an answer, but it was getting too long to be a comment. Mathematics draws much of its power from deep, sometimes mysterious dualities between geometry and algebra, so I do not think there is any way to understand the relationship between geometric intuition and symbol manipulation in general. Diophantine equations are one of the least ... 1 any integer number c can be written uniquely in any base b representation. For simplicity let's look at non-negative numbers (to extend on negative n, just look at the corresponding -n positive number and put a minus sign up front): c = \sum a_n b^n,  where 0 \leq a_n< b for n \in \mathbb{Z}+. It is easy to see that there is no alternative ... 1 Formally, a set S is countably infinite if there is a bijection between S and the natural numbers \mathbb N. What this means is that one can assign each element of S to an element of \mathbb N, allowing one (in theory) to list all of the numbers. Uncountable sets (for example, \mathbb R) cannot be listed. If a subset of \mathbb R is countable, ... 1 The motivation for destinguishing between countable and uncountable sets, I think, is telling whether infinite sets are 'of the same size'. A countable set has an injection with the natural number and is therefore in a sense 'of the same size' as the natural numbers. The uncountable sets are the sets not having this property and are therefore 'bigger'. Some ... 1 Infinite has caused lots of troubles in the history of mathematics. This concept often leds to paradoxes, like Hilbert's paradox of the Grand Hotel. This is even used to define infitine sets as those that can be put into a one-to-one correspondence with a strict subset (Dedekind infinite). \mathbb{N} is the first most natural infinite set. It defines ... 1 From a compuatability point of view, a set being countably infinite means that searching for an element in that set is now semi-decidable. That is, if the element is in that set, you can search in a way that you will eventually find it. (You may search forever if that element is not in that set). Basically, because a countably infinite set has a bijection ... 1 Old question here, but it is often asked by students of measure theory. In our text Real Analysis (Bruckner{}^2*Thomson) Example 2.28 in Section 2.7 there is a motivation for this that Andy used in his classes. You would have seen that the inner/outer measure idea was successful for Lebesgue measure on an interval. Certainly if you are hoping for ... 1 As mentioned by Yuval in the comments, this question has previously been discussed on MathOverflow. I have replicated the accepted answer by Mark below. Here is an argument that may give some intuition: Assume that m^{*} is an outer measure on X, and let us assume furthermore that this outer measure is finite: m^* (X) < \infty ... 1 The problem with using five (1+x+x^2+x^3+x^4+x^5)^5 is that you get duplicated partitions in that way. For instance 4+3+5=12 and 3+5+4=12 gives in fact the same partition. The intuition behind the solution is that, z_1 represent the number of 1s in the partition, then z_2 represent the number of 2s in the partition and so on, in this way the ... 1 I don't know how much this helps, but in computer science the concept of parametric polymorphic function is quite useful. What the polymorphic function principle basically says is this: Every function f:F(a)\rightarrow G(a) that is definable and polymorphic in a is a natural transformation from F to G. The base language wrt which these ... 1 Intuition and logic are not the same thing. Take, for example, the idea that$$\lim_{x\to\infty} \frac{1}{x}=0 What does this mean? Intuitively, you can imagine a graph of the function and see that it gets closer and closer to $0$, but who's to say that the limit isn't actually $0.0001$? To show that this isn't the case, you need a formal definition of ...