Articles of infinitesimals

Why Cauchy's definition of infinitesimal is not widely used?

Cauchy defined infinitesimal as a variable or a function tending to zero, or as a null sequence. While I found the definition is not so popular and nearly discarded in math according to the following statement. (1). Infinitesimal entry in Wikipedia: Some older textbooks use the term “infinitesimal” to refer to a variable or a […]

Learning differential calculus through infinitesimals

In class, we’ve studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and then integrals. But the teacher really did everything to avoid talking about infinitessimals. For instance when we talked about variable changes we had to swallow that for a […]

Basic question about nonstandard derivative

I’m trying to understand how the nonstandard derivative works. For instance, consider the function $f(x) = \frac{1}{2} x^2$ The derivative is $f'(x) = st \left( \frac{\frac{1}{2}(x + \epsilon)^2 – \frac{1}{2}x^2}{\epsilon} \right)$ for some infinitesimal $\epsilon$. This works out to $f'(x) = st(x + \frac{\epsilon}{2}) = st(x)$ So the derivative of $\frac{1}{2} x^2$ is the standard […]

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 read about hyperreal numbers I understand that they are an extension of […]

Non-Standard analysis and infinitesimal

Can someone please explain how Non Standard Analysis is used to justify infinitesimals? I am not very clear about this but apparently it has something to do with hyperreals.

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: \begin{equation} \forall a,b,c \in \Bbb R:~~~x + a\cdot \epsilon<y+b\cdot \epsilon<z +c\cdot \epsilon \end{equation} This, together with the ordering of $\Bbb R$ being a subset of the […]

What is the point of making dx an infinitesimal hyperreal?

It seems fairly common to describe $\mathrm{d}x$ in nonstandard analysis as an infinitesimal. But after thinking it through (and then skimming Keisler’s text), I can’t see the point and actually think it’s misleading! First, let me clearly point out that $\mathrm{d}y$ is not being used here as to express a “difference in $y$”; this post […]

Is $0$ an Infinitesimal?

For the definition of Infinitesimal, wikipedia says In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size; or, so small that it cannot be distinguished from zero by any available means. MathWorld says An infinitesimal is some quantity that is explicitly nonzero and yet […]

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 the picture- WHY aren’t we […]

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 and Leibniz. The concept of limits (and their epsilon-delta proofs) was what allowed calculus to get past the shaky logic of infinitesimals. (The […]