Questions tagged [infinitesimals]

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

Intuitively, an infinitesimal is an infinitely small number. They played a significant role in both Leibniz's and Newton developments of calculus, as well as in the work of Archimedes.

For a more rigorous treatment of infinitesimals, also see .

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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 "think of it as a full stop". For whatever reason…
Sachin Kainth
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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 some formalism to this type of calculus. So, do you…
user13255
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Why can't the second fundamental theorem of calculus be proved in just two lines?

The second fundamental theorem of calculus states that if $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$ on the same interval, then: $$\int_a^b f(x) dx= F(b)-F(a).$$ The proof of this theorem in both my textbook and Wikipedia is…
Newton
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Category-theoretic description of the real numbers

The familiar number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ all have "natural constructions", which indicate, why they are mathematically interesting. For example, equipping $\mathbb{N}$ with the usual sucessor function and the constant $0$,…
user158047
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Problem with basic definition of a tangent line.

I have just started studying calculus for the first time, and here I see something called a tangent. They say, a tangent is a line that cuts a curve at exactly one point. But there are a lot of lines that can cut the same point just like shown in…
Aaryan Dewan
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Is it a new type of induction? (Infinitesimal induction) Is this even true?

Suppose we want to prove Euler's Formula with induction for all positive real numbers. At first this seems baffling, but an idea struck my mind today. Prove: $$e^{ix}=\cos x+i\sin x \ \ \ \forall \ \ x\geq0$$ For $x=0$, we have $$1=1$$ So the…
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What is the use of hyperreal numbers?

For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes of calculus on a sound footing. From what I have…
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Are infinitesimals equal to zero?

I am trying to understand the difference between a sizeless point and an infinitely short line segment. When I arrive to the notion coming from different perspectives I find in the mathematical community, I arrive to conflicting conclusions, meaning…
Carvo Loco
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How small is an infinitesimal quantity?

When speaking of infinitesimals, I see some mathematicians say that it represents an "extremely small" element, such as an infinitesimal area on a manifold. What bothers me about this naive definition is how small is, an infinitesimal area, for…
Steven
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Why is it considered that $(\mathrm d x)^2=0$?

Why is it okay to consider that $(\mathrm d x)^n=0$ for any n greater than $1$? I can understand that $\mathrm d x$ is infinitesimally small ( but greater than $0$ ) and hence its square or cube should be approximately equal to $0$ not exactly $0$ .…
user369582
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Has anybody ever considered "full derivative"?

When differentiating we usually take a limit and drop the infinitesimal terms. But what if not to drop anything? First, we extend the real numbers with an infinitesimal element $\varepsilon$ which has its own inverse $1/\varepsilon=\omega$. And…
Anixx
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Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are these dangerous? (2) The ban on infinitesimals and…
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Calculus over $\mathbb{Q}$

The mismatch between the sensitivity of 'mathematical calculus' and the flexibility of 'real world calculus' has been bothering me a bit recently. What I mean is this: in the real world, I can trust that calculus will work whether $\mathbb{R}$ is…
Robert Mastragostino
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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 cannot be a stationary value because if so then…
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Rigorous definition of "differential"

When it comes to definitions, I will be very strict. Most textbooks tend to define differential of a function/variable like this: Let $f(x)$ be a differentiable function. By assuming that changes in $x$ are small enough, we can say: $$\Delta…
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