In ordinary calculus,if we don't make any absurd notation,the use of infinitesimals is extremely useful and intuitive which ended up giving correct results. For example $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ are obvious if we treat $dy,dx,dt$ as infinitesimals. Can we do the same in case of multivariable calculus? For example if $V$ is a function of $x$ and $y$ and $x,y$ are both functions of $r,\theta$,can we still do stuffs like $\frac{\delta V}{\delta x}=\frac{\frac{\delta V}{\delta \theta}}{\frac{\delta x}{\delta \theta}}$? Or are there scenarios where such use of infinitesimals lead to incorrect results?
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2Google queries with "infinitesimals" and "pitfalls" will give you answers like this one. – Jean Marie Nov 09 '23 at 11:51
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- If you're asking it it's possible to use infinitesimals for multivariable, just combine "Elemetary Calculus: An Infinitesimal Approach" with a multivariable text; I'd be happy to expand a bit on that in an answer. 2. If you're asking when a partial derivative can be treated like a fraction, that's close to this question about reciprocals. 3. If you're asking some third thing (e.g. "how to think about the multivlariable chain rule intuitively through the lens of infinitesimals"), could you rephrase your question to clarify?
– Mark S. Nov 09 '23 at 12:51 -
@MarkS.,the link you provided answered my previous question and that led to a bit more questions of mine. Would you be available to discuss those in this chatroom? https://chat.stackexchange.com/rooms/149603/calc-problems – madness Nov 09 '23 at 13:09
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1Not a question on infinitesimals again! Take $V(x,y)$ and apply the chain rule. Then treat whatever infinitesimal pops up as a fraction. Further hints: non standard analysis or differential forms. – Kurt G. Nov 09 '23 at 13:16
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@MarkS.,the question I have is if $z$ is a function of $x,y$,would $\frac{\partial x}{\partial z}=0$? We can discuss further in chat room. – madness Nov 09 '23 at 13:16
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3@JeanMarie, the pitfalls are common to the non-infinitesimal approach and the infinitesimal approach. One certainly has to be careful with notation when working with partial derivatives. – Mikhail Katz Nov 09 '23 at 14:22
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There are pitfalls with using notation that is too abbreviated when working in multivariate calculus, whether or not one uses infinitesimals. Partial derivatives are treated carefully in Keisler's textbook Elementary calculus with infinitesimals. While in single-variable calculus, it is obviously an advantage to be able to express chain rule as $\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}$ as literally involving fractions, there are some subtleties here, as well. Thus, $\frac{dy}{dx}$ is a bit more complicated than "the y-increment over the x-increment". The precise justification is in terms of the standard part. This is also explained very well in Keisler.
Mikhail Katz
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