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- What is the meaning of infinitesimal? - Mathematics Stack Exchange
Likewise, Infinitesimal is a concept; its value is smaller than any value you can imagine Check out this video and you will appreciate why Infinity and Infinitesimal cannot be "explained" to someone seeking to find "applications" "methodology"
- Definition of an Infinitesimal - Mathematics Stack Exchange
That chapter defines: A number $\epsilon$ is said to be infinitely small, infinitesimal, if: $-a < \epsilon < a$ And goes on to an introduction to the hyperreal line However, this definition seems to imply an infinitely small number ($\epsilon$) is one which is between $\pm a$, which seems to be a very large range if you choose, for example
- ordinary differential equations - What is the Lie group infinitesimal . . .
Our prof gave us the definition of the Lie group infinitesimal generator and it's kth-coordinate, he also explained why the kth-coordinate is called "the kth-coordinate", but nonetheless he didn't want to explain the Lie group infinitesimal generator because "we will understand it later as the course will continue"
- How do you understand Infinitesimals? - Mathematics Stack Exchange
While it is possible to create number systems with infinitesimals, that does not validate the intuition that you've stated, that somehow $0 \overline{9}$ and $1$ differ by an infinitesimal You lose a lot of nice properties of the real numbers when you add infinitesimals
- Whats an example of an infinitesimal? - Mathematics Stack Exchange
Cauchy said that a sequence converging to zero becomes an infinitesimal rather than is an infinitesimal The idea that he meant a null sequence to generate an infinitesimal is confirmed by the fact that he was interested in rates of growth of sequences and even tried to classify those
- Are infinitesimals equal to zero? - Mathematics Stack Exchange
Hence, zero is also an infinitesimal But not necessarily exactly like other infinitesimals, because it seems you cannot add zero to itself any number of times and arrive to anything other than zero, while you can add other infinitesimals to themselves and arrive to real values
- infinity times infinitesimal - Mathematics Stack Exchange
and define an infinitesimal number as the difference between a convergent geometric series and its sum: $ x+1 -\displaystyle\sum_{i=0}^{n\rightarrow\infty} \left(\frac{x}{x+1}\right)^i$ If the x is the same in both the infinity and the infinitesimal their product will converge to the finite number x(x+1) as n increases without bound
- Precisely how is infinitesimal calculus meaningfully different from . . .
Infinitesimal calculus is really how it was thought of when it was first created We know the area under a curve because we just add up all of these infinitesimally small slices However, and really think about this, how small is an infinitesimal? This is why I said that this is how calculus was thought of when it was first created We don't
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