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1d
comment Why can we treat infinitesimals as real numbers in integration by substitution?
So, my question is if anyone can elaborate on exactly why it is logically rigorous to treat infinitesimals as real numbers There is nothing necessarily non-rigorous about infinitesimals. The most common way of making them rigorous is through non-standard analysis (NSA). In NSA, infinitesimals obey all the same elementary axioms of arithmetic that the real numbers obey.
1d
comment Why can we treat infinitesimals as real numbers in integration by substitution?
possible duplicate of What is $dx$ in integration?
Jun
30
comment What is it called when you have two systems of measurement and each scale has two different numbers that can represent the same thing?
This isn't about history. It would be more appropriate for math.SE.
Apr
16
awarded  Nice Answer
Mar
31
awarded  Good Question
Dec
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awarded  Popular Question
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awarded  Autobiographer
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awarded  Popular Question
Aug
4
comment In calculus, which questions can the naive ask that the learned cannot answer?
+1, but I don't think a naive first-year calc student would be at all likely to come up with this series. It takes a lot of insight to understand why its convergence is difficult to establish, and why the exponents 3 and 2 work.
Aug
3
revised Intuitive explanation of the difference between waves in odd and even dimensions
Kevin Brown is the mathpages guy, not John Baez
Jul
26
comment Could we assign a numerical value to an infinitesimal?
I don't understand the point of the question. When I first read it, I thought maybe you were trying to reinvent the wheel, and you just needed to be told about the existence of systems such as non-standard analysis (NSA) and smooth infinitesimal analysis (SIA). But then I saw in one of your comments that you had already heard about SIA. So what is this question asking? Are you proposing a third system and asking whether it's useful or consistent?
Jul
26
comment What are some conceptualizations that work in mathematics but are not strictly true?
@EricLippert: Symbols like $dy$ and $dx$ can be defined as infinitesimal numbers. That's historically what they originally meant, and there's nothing wrong with it. See math.stackexchange.com/questions/21199/…
Jul
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awarded  Yearling
Jul
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comment Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?
@jpmc26: The foundational aspects of calculus have gone through a series of changes over the centuries, and therefore different people have different ideas about the meaning of a symbol such as $d\Phi$. The original meaning was that it was an infinitesimal number. That fell out of favor ca. 1850-1960, when the foundations of calculus were rebuilt in terms of limits. Then infinitesimals, which scientists and engineers had never stopped using, were rehabilitated. A nice book on this topic at the undergraduate level is Keisler, math.wisc.edu/~keisler/calc.html
Jul
14
awarded  Popular Question
Jul
13
comment Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials
Related: math.stackexchange.com/questions/865559/… . I think the basic issue you're running into is that Leibniz notation predates the notion of a function by a couple of hundred years. Leibniz and his contemporaries thought in terms of expressions, not functions, and the notation implements that attitude.
Jul
12
asked Modern notational alternatives for the indefinite integral?
Jul
10
comment Best applications-oriented introductory calculus textbooks?
Agnew may actually be public domain now. Its copyright was in 1962, and it doesn't appear to have been renewed.
Jul
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awarded  Curious
Jun
24
revised Standard terminology for infinite limits with opposite sign on the two sides?
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