Wikipedia:Reference desk/Archives/Mathematics/2014 August 29

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August 29

notation of differentiation versus notation of integration

We have an article notation for differentiation but no article of notation for integration which is telling. Why? — Preceding unsigned comment added by 174.3.125.23 (talk) 17:14, 29 August 2014 (UTC)

Because one is a big enough subject for its own article and the other is a small topic best dealt as a subsection in Integral. Dmcq (talk) 17:58, 29 August 2014 (UTC)

I mostly agree with that, but it doesn't really answer the underlying motivation: why are there several conventions for notation of differentiation still in modern use, but only one for integration (at least restricting to functions of a single real variable)? Put another way, why do we still use sometimes use Newton's notation for derivatives, but not for integrals? I suspect the answer is that the various options for differentiation have different strengths and weaknesses, while in contrast, the integral notation doesn't have any real downsides. Of course, there are a few different notations for different types of integrals, e.g. path integral, double integral, surface integral, Ito integral etc. In that light, it wouldn't be so strange to have an article that mentions each of these briefly. Checking the articles, the notation is fairly consistent, but sometimes in text books the integral symbol gets adorned in different ways, depending on context. SemanticMantis (talk) 21:40, 29 August 2014 (UTC)

Let's look at the question in another way: $1/1=1^{1}=1^{-1}$. $1^{-1}$ is the inverse of $1/1$ but there is only one way to write this. Differentiation on the other hand has many different ways, but the inverse, integration, has one way. Why?174.3.125.23 (talk) 22:17, 29 August 2014 (UTC)

Actually there's an article Integral symbol. I just remembered about that as it describes how the Germans and Russians use much more upright versions. Dmcq (talk) 22:18, 29 August 2014 (UTC)

Don't forget the physicists habit of writing the d-whatever right after the integral sign as opposed to after the integrand. YohanN7 (talk) 22:19, 29 August 2014 (UTC)

I'm quite liable to leave it out altogether sometimes ;-) Dmcq (talk) 14:16, 30 August 2014 (UTC)

This mention makes me think of differential geometry, where the integral does not form a notational pairing with a formal variable of integration (as in Exterior derivative#Stokes' theorem on manifolds); it is only over a region of a manifold. This might be relevant in that while it looks similar, it is a distinct notation. —Quondum 20:54, 30 August 2014 (UTC)

Ok, let's ask another question. We know that "n" is any number, "b" is any number. Newtonian notation uses "dt". Is "dt" = "dx"? Why?174.3.125.23 (talk) 16:07, 30 August 2014 (UTC)

I can't quite make out what you are saying but Newtonian notation does not use dx or dt. It assumes a single independent variable, t normally but something else can be assumed instead. For instance ${\ddot {x}}=-cx$ describes simple harmonic motion with time as the independent variable but ${\dot {y}}=y$ might describe the exponential function with x as the independent variable - but in mechanics it would just be time again. Dmcq (talk) 17:50, 30 August 2014 (UTC)

Ok, my situtationsituation is at a Math 31 level, which is a grade 12 calculus course in Alberta. I am stuck on the quotient rule. I need a proof. I believe where I was stuck uses Leibniz notation. I think the quotient rule is one multiplied by another, but I don't understand why.174.3.125.23 (talk) 20:13, 31 August 2014 (UTC)

This is quite different from your original question. Try reading quotient rule and product rule. —Quondum 01:20, 1 September 2014 (UTC)

That is a poor explanation of my question. Here's another question, why is d over dx?174.3.125.23 (talk) 01:28, 1 September 2014 (UTC)

The purpose of the reference desk is not to act as a tutoring service, but is primarily to provide references such as I gave you; in particular, you need to be prepared to take the information and links given and extract the information that is relevant to your question. If you cannot frame your questions so that it is clear what information you seek, and especially if you are so dismissive, you can't expect much of a response. You are not demonstrating that you are trying to synthesize the information that you have been given. —Quondum 01:51, 1 September 2014 (UTC)