Questions tagged [differential-calculus]

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103 votes
13 answers
36k views

How misleading is it to regard $\frac{dy}{dx}$ as a fraction?

I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as ...
58 votes
22 answers
12k views

Which high-degree derivatives play an essential role?

Q. Which high-degree derivatives play an essential role in applications, or in theorems? Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration), and the ...
43 votes
1 answer
5k views

Did Leibniz really get the Leibniz rule wrong?

A couple of posts ([1], [2]) on matheducators.SE seem to suggest that Leibniz originally got the wrong form for the product rule, perhaps thinking that $(fg)'=f'g'$. Is there any actual historical ...
user avatar
34 votes
2 answers
2k views

Is it always possible to calculate the limit of an elementary function?

I already asked this question on https://math.stackexchange.com/questions/2691331/is-it-always-possible-to-calculate-the-limit-of-an-elementary-function but as I received no answer; maybe it is not as ...
Olivier Esser's user avatar
33 votes
1 answer
2k views

How quickly can the derivative of an everywhere differentiable function change sign?

Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$. By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...
Siméon's user avatar
  • 635
29 votes
1 answer
3k views

Is there an explicit formula for the hessian of "Determinant"?

Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function. Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$,...
Ali Taghavi's user avatar
25 votes
2 answers
1k views

Is there a convenient differential calculus for cojets?

I understand the basics of exterior differential geometry and how to do calculus with exterior differential forms. I know how to use this to justify the notation dy/dx as a literal ratio of the ...
Toby Bartels's user avatar
  • 2,644
23 votes
1 answer
3k views

Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix?

In his 1841 article De determinantibus, Jacobi remarked that the notation $\frac{\partial z}{\partial x}$ for partial derivatives is ambiguous. He observed that when $z$ is a function of $x,y$ as well ...
Michael Bächtold's user avatar
19 votes
4 answers
12k views

How did Bernoulli prove L'Hôpital's rule?

To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to ...
John Palmieri's user avatar
18 votes
4 answers
1k views

Variable-centric logical foundation of calculus

Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For ...
Jason Howald's user avatar
18 votes
0 answers
420 views

An integral in Gradshteyn and Ryzhik

Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...
Victor Moll's user avatar
15 votes
3 answers
1k views

Derivative formula

While trying to understand a paper of Cayley, he left something unexplained, I managed to show that it is equivalent to the following formula, which I got stuck at: $$k \cdot (f^k)^{(k-1)} = \sum_{j=0}...
gurtonn's user avatar
  • 173
13 votes
3 answers
3k views

Differentiability of Eigenvalues - Perturbation Theory

first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
xmonetx's user avatar
  • 138
9 votes
1 answer
2k views

Differential Calculus and the De Rham Homotopy Operator

Suppose $M$ is a smooth manifold, $\Omega^k(M)$ the vector space (or $C^\infty(M)$-module in case this is better suited) of $k$-differentialforms on $M$ and $$I: \Omega^k(M\times \mathbb{R}) \to \...
Mark.Neuhaus's user avatar
  • 2,004
8 votes
1 answer
598 views

Example of a function with a curious property

Denote by $L^1(0,1)$ the space of Lebesgue integrable functions on the interval $(0,1)$. $\textbf{Question:}$ Does there exist a function $F:(0,1)\rightarrow\mathbb{R}$ such that: $\frac{F(x)}{x}\in ...
Tony419's user avatar
  • 401
8 votes
0 answers
301 views

Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them

I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I have in ...
Giovanni Moreno's user avatar
7 votes
3 answers
880 views

A definition of the fractional derivative

I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it? $$\frac{d^n}{dx^n}f(x) = \lim_{h \...
Halbort's user avatar
  • 1,129
7 votes
1 answer
6k views

The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial. In particular, it's the case ...
7 votes
2 answers
452 views

Interpretation of second order term in Fokker-Planck equation

Let $G:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix-valued smooth function. Let us define a quantity by $$ \begin{align*} \nabla^2\cdot G(x) &=\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{d}\...
Peter's user avatar
  • 131
7 votes
1 answer
337 views

A property of $C^2$ functions

Let $f\in C^2(\Bbb R^m), f\geq 0$, Hessian matrix of $f$ is upper bounded by some constant $C$. Do we have $|\nabla f|\leq \alpha \sqrt{f}$ for some $\alpha$, even if the Hessian matrix is degenerate?
zhangwei's user avatar
7 votes
2 answers
1k views

Common roots of polynomial and its derivative

Suppose $f$ is a uni-variate polynomial of degree at most $2k-1$ for some integer $k\geq1$. Let $f^{(m)}$ denote the $m$-th derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots then,...
Gorav Jindal's user avatar
6 votes
3 answers
826 views

"Universal" differential identities

(This is a cross-post from MSE). Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth. The mixed derivatives commute: $f_{xy}=f_{yx}$. This identity is "universal" in the sense that it holds for any smooth ...
Asaf Shachar's user avatar
  • 6,611
6 votes
2 answers
376 views

Existence and uniqueness of an Euler-type ODE with varying parameters

Consider this ODE on $[1, \infty)$ $(r^2 - 2ar)f''(r) + 2(r-a) f'(r) - ({4a} + m(m+1))f(r) = -4af(1) $ with initial conditions $\frac{a}{1-2a} f(1) + f'(1) = C, \qquad \lim_{r\to \infty} f(r) = 0$ ...
Laithy's user avatar
  • 865
6 votes
1 answer
339 views

Law of unconsious statistician: application in characteristic function

Let $g(x)=(x-a)\mathbf 1_{x\ge a}$ for some $a>0$ and let $X$ be a non-negative random variable with cdf $F$ and $E[X]<+\infty$. I want to calculate $$\frac{d}{da}E[g(X)]$$ To do that I thought ...
Jimmy R.'s user avatar
6 votes
0 answers
133 views

Injectiveness of a monotonic surjective mapping $\mathbb R^n \to \mathbb R^n$ with $\det J \neq 0$

Consider a surjective mapping $F \colon \mathbb R^n \to \mathbb R^n$, $F\in C^1$, $\dfrac{\partial F_i}{\partial x_j} > 0$, and $\det \left(\!\left( \dfrac{\partial F_i}{\partial x_j} \right)\!\...
jokersobak's user avatar
5 votes
3 answers
1k views

Solving a limit about sum of series

what's the limit of $\sqrt{1-t}\sum _{n=0}^{\infty}t^{n^2}$ as $t$ goes to the left of $1$? i.e. $t\to 1^{-}$? I tried several times but failed. Here is my thought: This is a $0\cdot\infty$ problem, ...
Xu Shan's user avatar
  • 95
5 votes
3 answers
536 views

Osculating circle

(This question may be too elementary for this site — I'm fine if it needs to be moved to math.stackexchange.) If I approximate a nice planar curve by a straight line, the tangent, then the second ...
Stopple's user avatar
  • 10.8k
5 votes
3 answers
767 views

Mathematical Techniques to Reduce the Width of a Gaussian Peak

In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this ...
AChem's user avatar
  • 803
5 votes
1 answer
627 views

An inequality inspired by the isoperimetric inequality

Let us consider the simplest isoperimetric inequality. Consider a smooth simple closed curve given by $r=\rho(\theta)$ in polar coordinates, where $\rho(\theta)>0$ can be regarded as a smooth ...
user50396's user avatar
  • 141
5 votes
1 answer
325 views

A differential inequality involving gradient and laplacian

Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$. What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...
tituf's user avatar
  • 311
5 votes
1 answer
594 views

Signed distance function and level set

For $\phi\in C^1(\mathbb{R}^N)$ with $$\omega_{\phi}=\{x\in\mathbb{R}^N\ |\ \phi(x)>0\}$$ being a bounded set with $\nabla\phi (x)\neq 0,\ \forall\ x\in\phi^{-1}(0)=\partial\omega_{\phi}\neq \...
Bogdan's user avatar
  • 1,330
5 votes
0 answers
255 views

Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero. By the product ...
Arrow's user avatar
  • 10.3k
5 votes
0 answers
961 views

Boundary of an open, bounded and convex set in $\mathbb{R} ^n$

Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
The Convex Man's user avatar
4 votes
1 answer
478 views

Higher-order derivatives of $(e^x + e^{-x})^{-1}$

I am currently trying to build the derivatives of $$f(x) = \frac{1}{e^x+e^{-x}}.$$ It is fairly straightforward to obtain $$ \frac{d^n f}{dx^n} = \frac{P_n(e^x)}{e^{(n-1)\cdot x} (e^x+e^{-x})^{n+1}}, $...
tobias's user avatar
  • 739
4 votes
1 answer
857 views

Derivative is Zero on a dense G_delta set

I have the following question: I have a function $f: \mathbb R \to \mathbb R$ which is differentiable everywhere. I also have a set $G\subset\mathbb R$ which is dense in $\mathbb R$ and a $G_\delta$-...
Neslihan's user avatar
  • 495
4 votes
1 answer
233 views

Does the homeomorphism have a non-negative or non-positive determinant?

Let $ \Omega_1 $ and $ \Omega_2 $ be domains (open and connected) in $ \mathbb{R}^2 $. $ \psi:\Omega_1\to\mathbb{R} $ and $ \phi:\Omega_1\to\mathbb{R} $ are $ C^1 $ functions with two variables. ...
Luis Yanka Annalisc's user avatar
4 votes
1 answer
292 views

A Conjecture about the integral related to Chebyshev polynomial

I am interested in the following integral related to the Chebyshev polynomials $$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$ where $n,m\in \mathbb{Z}^+.$ It is easy to see ...
Jacob.Z.Lee's user avatar
4 votes
1 answer
525 views

Hegel's disproof of Newton [closed]

I know it's not a very comprehensive question but I've nowhere else to ask. A friend relayed to me a portion of a book from Hegel where he seemingly disproves Newton's way of finding a differential. I ...
AdivonSlav's user avatar
4 votes
1 answer
236 views

Inequality involving sigmoid function

Let $\sigma$ denote the sigmoid function $\sigma(x) = \frac{1}{1+e^{-x}}$, let $x,y \in \mathbb{R}$. Given the following two conditions: $|\sigma(-x) - \sigma(y)| < \epsilon$ and $x - y > c > ...
luw's user avatar
  • 317
4 votes
1 answer
528 views

Is the space of tangents actually the tangent space?

This is a crosspost of this MSE question. Given a locally Euclidean (locally homeomorphic to some Euclidean space) topological subspace $X\subset\mathbb R^n$ and $p\in X$, let $\mathrm{T}_pX$ denote ...
Arrow's user avatar
  • 10.3k
4 votes
1 answer
225 views

Ratio of the first squared and the second moment

Let $G(t)$ be a probability generating function of some integer and non-negative r. v. $X$. Suppose that $$\lim_{t\to1}G'(t)=+\infty.$$ That is $$ \mathbb{E}X=+\infty. $$ Can you show that $$ \lim_{t\...
Fancier of Mathematica's user avatar
4 votes
1 answer
879 views

Laplace-Beltrami of the mean curvature

For a surface $S$ defined in 3D space, denote its mean curvature as $H$, and the Laplace-Beltrami operator as $\Delta_S$. I know that there is a result for the Laplace-Beltrami of coordinate functions:...
Lightmann's user avatar
  • 141
4 votes
1 answer
106 views

$AC^p$ curves and pointwise metric speed in abstract metric spaces?

For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if $$ d(\gamma(s)...
leo monsaingeon's user avatar
4 votes
2 answers
237 views

Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
Programmer1's user avatar
4 votes
0 answers
132 views

Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root

Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...
baronbrixius's user avatar
4 votes
0 answers
109 views

Properness of real analytic maps?

Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...
Arrow's user avatar
  • 10.3k
4 votes
0 answers
145 views

Basic calculus on topological fields

Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $ \mathbb Q_p$). 1) Let $f: K^n \to K$ be a ...
Antongiulio Fornasiero's user avatar
3 votes
4 answers
1k views

Intrinsic definition of arc length [closed]

Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?
Felix Goldberg's user avatar
3 votes
2 answers
478 views

Curl as a divergence... Is it possible? [closed]

I want to know if it is possible to express the operation $$ \nabla \phi \times (\nabla \times \mathbf A) $$ as the divergence of second order tensor field $T$. Here $ \phi$ is a scalar field and $\...
nodarkside's user avatar
3 votes
1 answer
281 views

Are all Helmholtz decompositions related?

Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...
MrPie 's user avatar
  • 185