# Tagged Questions

For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also [nonstandard-analysis]).

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### Is non-standard analysis worth learning?

As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides ...
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### What is $dx$ in integration?

When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board. $$\int f(x)\, dx$$ When he came to explain the meaning of the $dx$, he told us ...
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### What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
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### Justification of algebraic manipulation of infinitesimals

As an engineering student, I regularly see people making arguments like this: Consider a rectangle of dimensions $x\times 4x$. If we make $x$ bigger by a small quantity $dx$ then this will make $4x$ ...
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### Newton's “Famous Blunder”?

On page $225$ of Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini (see here for a link), I read In the following demonstration... Newton made a famous blunder... He wrote, ...
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### Transcendental a infinitely close to rationals?

Apologies that this question is rather vague, but I do not know how to state it more precisely. Is, say pi, infinitely "close" to some rational number? More importantly, are all transcendental numbers ...
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### What does limit actually mean?

I have been in a deep confusion for about a month over the topic of limits! According to our book the limit at $a$ is the value being approached by a function $f(x)$ as $x$ approaches $a$ I have a ...
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### How is the concept of the limit the foundation of calculus?

My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton ...
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### What is this limit called? Is it a different kind of derivative?

(first I should notice you this is not something I can look up in a textbook, because I'm learning partial derivatives, alike I do with most Maths, as a hobby. If something below is wrong, blame the ...
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### Infinitesimal calculus

I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ...
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### Definition of tangent

What is the formal definition of a tangent to a curve? The only one I can find is that it is a straight line drawn between two infinitely close points on the curve.
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### Using infinitesimals to find the volume of a sphere/surface area of a sphere

I've always thought of $dx$ at the end of an integral as a "full stop" or something to tell me what variable I'm integrating with respect to. I looked up the derivation of the formula for volume of a ...
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### Are there concepts in nonstandard analysis that are useful for an introductory calculus student to know?

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense. Can someone elaborate on this ...
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### Products of Infinitesimals

In my physics class my professor was abusing the derivative, as per so many physics classes I've been in. This time, he took the quantity $(x+dx)(y+dy)$ and argued that the $dxdy$ term should ...
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### What is the topology of the hyperreal line?

Denote by $\Bbb R$ the real line and by $\Bbb R^*$ the hyperreal line. For any real numbers $x < y < z$ and infinitesimal $\epsilon$ the following holds: \forall a,b,c \in \Bbb ...
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### First year calculus student: why isn't the derivative the slope of a secant line with an infinitesimally small distance separating the points?

I'm having trouble with the limit approach to calculus ever since I heard about the infinitesimal definition. Maybe you can help me settle what's been bothering me this year. Looking at the limit ...
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### What problems arise when using infinitesimals in calculus?

In contemporary real analysis we use a limit definition in terms of deltas and epsilons. Before that, people used infinitesimals to calculate limits. Is there a specific non-philosophical reason why ...
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If $\mathrm{d}x$ is treated as a hyperreal infinitesimal we can easily do derivations. How do we interpret and perform integrals using infinitesimals? What is the $\mathrm{d}x$ in $\int x^3\,... 1answer 75 views ### Can this sequence be a hyperreal number? What would be its real part? Consider the sequence$\{a_n\} = \{\sin(n) \mid n\in \mathbb N \}$. Can this sequence be viewed as a hyperreal number? What could be its real part? Any intuition would be highly appreciated :) ... 2answers 165 views ### How can this result in Thermodynamics be rigorously proved? In Fermi's "Thermodynamics" there's a proof of the formula: $$W=\int _{V_1} ^{V_2} p\,\text dV,$$that is, the work done by the pressure of a gas that expands from a volume$V_1$to a volume$V_2$on ... 1answer 77 views ### Interpretation of$d\phi(z)$in differential geometry In "Exercises and Solutions in Mathematics", Ta-Tsien, 2nd Edition, exercise 3343. Statement of the exercise Let$(\mathbb{H}, g)$be the two-dimensional hyperbolic space, where \... 3answers 146 views ### infinitesimal intervals in physics The density of states of a system in an interval$[E, E+dE]$is given implicity by$dV = D(E)dE$(Or I suppose explicitly, by$D(E) = \frac {dV}{dE}$, but we'll be integrating it anyway, so it doesn't ... 2answers 276 views ### Non-standard analysis way of proving that derivative of$e^x$is$e^x$What is the non-standard (infinitesimal) analysis way of proving that the derivative of$e^x$is$e^x$? I tried to prove it myself, but I am having a hard time proving this without recourse to ... 2answers 83 views ### How to show$\chi_{{}^{*}P} ={}^{*}\chi_{P}$by transfer principle? Let$\mathfrak{R}$be the real number system,$(\mathbb{R},+,\cdot,<)$and${}^{*}\mathfrak{R}$be the hyperreal number system$({}^{*}\mathbb{R},{}^{*}+,{}^{*}\cdot,{}^{*}<)$. Transfer ... 1answer 84 views ### Determining hyperreal class for$\frac{\epsilon + \delta}{\sqrt{\epsilon^2 + \delta^2}}$I'm solidifying my calculus by going through Keisler's book that uses a hyperreal/infinitesimal approach. I'm stuck on this problem. Given infinitesimals$\epsilon,\delta > 0$, deterimine whether ... 2answers 278 views ### How is an infinitesimal$dx$different from$\Delta x\,$? [duplicate] When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard$dx$is called an infinitesimal and I don't know what this means.... 1answer 118 views ### Integrating over a power of the infinitesimal I don't know if the title makes sense (or if the question makes sense at all for that matter) but here I go. Suppose I have a piecewise constant function$y=f(x)$with$x,y\in\mathbb{R}^+$, described ... 1answer 61 views ### Rigorous Justification of Infinitesimal Techniques As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ... 1answer 94 views ### Perturbation in characteristic p, or Why, really, does Lie's theorem fail? While recalling some basics of Lie theory, I found a funny proof of the main lemma in Lie's theorem on triangularity of representations of solvable Lie algebras. It turns out that this proof has a ... 0answers 193 views ### Nontrivial trivial integrals Consider two propositions in geometry: Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ... 3answers 670 views ### Why don't infinite sums make any sense? Using the infinite sum series, an infinite sum of (1/5)to the nth power, where n goes from zero to infinity, the general summation equation tells us that the answer is 5/4. However, how is this ... 3answers 155 views ### If$dx$is an infinitesimal, can't you list all real numbers as the sequence each whole number times dx? I'm taking calculus right now. If the difference between each real number and the next is an infintesimal, then wouldn't the following sequence$\{0\,dx, 1\,dx, -1\,dx, 2\,dx, -2\,dx, \ldots\}$be a ... 2answers 143 views ### How do mathematics define a point? I have a serious doubt. How do mathematicians define a 'point' in a space or a plot? If we have a clear explanation for a 'point' , I think my doubt on infinitesimals and infinity will be clarified. 3answers 246 views ### How$\frac{dx}{dy}=f(x)g(y) \Leftrightarrow \int \frac{dx}{f(x)} = \int g(y)dy\$?

In my intro differential equations class we have often used the "equivalence" stated in title. It seems to me that somehow, the intermediate step $$\frac{dx}{f(x)} = g(y)dy$$ is being used, in which ...