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6h
comment Notation for “partial” derivative
If $f(x,c,y) = x + c + y$ and $c = c(x)$ is a function of $x$ then you could set $g(x,y) = f(x,c(x),y)$ and $\partial g/\partial x = 1 + dc/dx$. But it's natural to think it is a bad way to take care of the notational problem by changing the name of the function from $f$ to $g$.
6h
comment Notation for “partial” derivative
There is no conventional notation if you want to keep writing the function by the same letter throughout and write derivatives in Leibniz notation to make the chain rule look like "canceled fractions". The notation developed historically and does not hold up well to strict logic. It's similar to the situation with the Euler-Lagrange equation where you see expressions like $\partial f/\partial y'$. Ultimately you simply need to understand what is going on. You can always use words to remind the reader at some point where you think there may be a misunderstanding.
7h
comment Notation for “partial” derivative
In your two examples the context is different. For $\partial f/\partial t$ you have a function $f(t,x,y)$, so the partial derivative is with respect to its first variable (the other two being fixed). But for $\partial z/\partial u$ you're given a function $z(x,y)$ with $x$ and $y$ both functions of two new variables $u$ and $v$. So the context of $\partial f/\partial t$ and $\partial z/\partial u$ is not the same. This is what I meant about looking at the notation for the multivariable chain rule: the notation is inherently ambiguous if you insist it have a unique possible meaning.
7h
comment Notation for “partial” derivative
If you're dealing with the Euler-Lagrange equations then you have to work with a far worse abuse of notation than what you're asking about, namely $\partial f/\partial y'$.
8h
comment Notation for “partial” derivative
Just explain what all your notation means (e.g., that $c$ is a function of $x$) and then there won't be confusion when you write $\partial f/\partial x$. For comparison, look at the notation used for the multivariable chain rule.
18h
comment Are there any relevance between coefficients of simple continued fraction of quadradic algebraic number and algebraic number with degree $2^n$
@GH from MO: I think the OP meant to ask about the relations between these continued fractions rather the relevance (which doesn't make grammatical sense).
1d
comment Ideal classes fixed by the Galois group
@MatthiasWendt: the Galois group of a quadratic field $K$ can act trivially on the ideal class group of $K$ even if the class group is nontrivial. For example, let $K = {\mathbf Q}(\sqrt{-5})$, with class group of order 2 generated by the ideal class of the prime $\mathfrak p = (2,1+\sqrt{-5})$. This prime ideal is fixed by the Galois group, so its ideal class is fixed. Therefore the Galois action on the ideal class group is trivial. More generally, if all elements of the class group have order 1 or 2 then the Galois action on ideal classes is trivial since ideal classes of primes are fixed.
Aug
27
comment What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$
Are you expecting there to be a simple rule? There isn't.
Aug
23
awarded  Nice Answer
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
You didn't show how your Fourier method leads to an antiderivative for $1/x$ (what is it?). Concerning the role of differential equations, your question asks for a reason that "antiderivatives are given with arbitrary constants rather than these distinguished ones" and differential equations provide a context that answers that question. Have you solved differential equations and used the undetermined constants to find the unique solution fitting some initial conditions, or used the general constants to find a formula for the general solution of the equation?
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
@Vectornaut: in more concrete terms, what you're saying is that the only constant function in $L^1({\mathbf R})$ is $0$. Of course there are many reasons people might be interested in functions defined on closed and bounded intervals, where the result you describe breaks down.
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
I don't want to remove that paragraph since then some of the comments on my answer will no longer make sense. I have edited the start of that paragraph to make it clearer that it is my interpretation of what you are essentially doing. In any case, I do wish you had at least once addressed the part of my answer dealing with the application of undetermined integration constants in the solution to differential equations rather than ignore it so completely.
Aug
23
revised Why calculus textbooks do not include the natural integration constants in the tables of integrals?
added 33 characters in body
Aug
23
revised Separation of lattice points on the Mordell elliptic curve
added 54 characters in body
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
@Anixx: Your question is not research-level, so it's not appropriate here. Please post a question on math.stackexchange, and I suggest simply asking there why undetermined constants of integration are important in math, without offering your proposal by Fourier transforms as part of the question.
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
@OlegEroshkin: why would "no one need" integrals of functions defined only a finite interval? Studying differential or integral operators on intervals $[a,b]$ is a basic topic in functional analysis, for instance. Or does the term "finite interval" mean something to you other than closed bounded intervals?
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
Кстати, по-английски фраза «в точке» = at a point, не in a point (напр., "at x = 0" вместо "in x = 0").
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
I asked you to address the role of the undetermined integration constants in solving differential equations if you want to make further comments on my answer, since you are ignoring it. Does that application not provide an explanation to you for why we want the flexibility of the undetermined constants in integration? Differential equations are the first topic in analysis where those undetermined constants are fundamentally important (they certainly are not important for computing areas and volumes).
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
I've already answered your original equation by pointing out the very important role of undetermined constants in differential equations. If you want to make further comments on my answer, please address that part of it.
Aug
23
comment Why calculus textbooks do not include the natural integration constants in the tables of integrals?
Let $f(x) = 2x+3$. The antiderivative vanishing at $x = 0$ is $x^2 + 3x$. The antiderivative vanishing at $x = 1$ is $x^2 + 3x - 4$. The antiderivative vanishing at $x = a$ changes as you change $a$. If you know an antiderivative takes a particular value at a particular point then that antiderivative is determined elsewhere if it is defined on an interval, but changing the point of evaluation or the preferred value will change the antiderivative by a constant, which is the entire issue you are trying to avoid in the first place, so it isn't going away after all. Your strategy is ill-defined.