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-1
votes
0answers
38 views

Improper integral calculation - limit at infinity [on hold]

Will you please help me prove the following limit is zero ? $ lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} cos(x\cdot u) du $ Thanks in advance
-3
votes
0answers
20 views

multi-variate closed form solution [closed]

I have a loss function with two paramters. I found that when I fix one I can obtain the closed form solution of the other. In that case how can I optimize to learn those two parameters. Lets my ...
-1
votes
0answers
50 views

How do I solve this differential equation by differentiation? [closed]

How do I solve this equation $$ f(x) = \int_{y=0}^{1-x}f(x+y)\, dy + \int_{y=1-x}^1 y\, dy $$ by differentiation?
1
vote
0answers
94 views

Universal property of (k-fold) differential $d^k$

does the differential $d^k:C^\infty(U,\mathbb{R}) \to C^\infty(U,S^k(X)^*)$ for $U \subseteq X$ fulfill some kind of (strict) universal property under all '$k$-fold derivations' (you know what i ...
2
votes
1answer
105 views

Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$

Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states ...
-1
votes
1answer
69 views

How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable? In this context PSD means symmetric PSD.
0
votes
0answers
62 views

Existence and Uniqueness of Volterra integral equations of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
2
votes
0answers
102 views

Slice a compact C1 surface in R3 by a moving transverse plane. Does the length of the slice depend C1 on the plane?

To be more precise I am interested in questions similar to the one below (I asked the question below on math.stackexchange last week but got not answer.) I have a $C^1$ function $f:[0,1]^2 \to ...
0
votes
0answers
62 views

Sufficient Conditions For Monotonic Decreasing of Multivariate Function

I found the following theorem on sufficient conditions for decreasing monotonicity of an absolutely continuous function $\phi : \mathbb{R} \rightarrow \mathbb{R}$, I would like to know if it is ...
2
votes
1answer
75 views

Is first term of my cost function convex?

I have an optimization problem in the form of [\begin{array}{l} \mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...
3
votes
1answer
154 views

A functional equality

I don't know if this is known, but I was fiddling around with this equality : $$f:(-1,1)\to (-1,1)\quad \text{satisfies}\quad(f(z)+1)^s=\sum_{j=0}^{\infty}\dbinom{s}{j}f(z^k) \quad \forall z\in ...
1
vote
1answer
211 views

Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: ...
3
votes
0answers
168 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 ...
2
votes
0answers
161 views

Looking for author of calculus quote

When I was a lowly calculus student many many years ago, my calculus teacher quoted some famous mathemtician: "Calculus is the last course in arithmetic and the first course in mathematics that one ...
3
votes
1answer
72 views

Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$. I would like to know how many extrema $p$ has on the standard simplex ...
1
vote
1answer
210 views

Request for help with two integrals

It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n ...
0
votes
0answers
147 views

Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...
1
vote
1answer
1k 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 ...
8
votes
1answer
838 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 ...
4
votes
0answers
157 views

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

Hello! 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 ...
23
votes
2answers
828 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 ...
17
votes
4answers
5k 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 ...
2
votes
0answers
2k views

What is the geometric meaning of the third derivative of a function at a point? [closed]

What is the geometric meaning of the third derivative of a function at a point? This question is now asked on the sister site: ...
2
votes
2answers
351 views

Finding the Universal Ideal of a (Covariant) Differential Calculus

Let $(\Omega,d)$ be a differential calculus over an algebra $A$. It is easy to show that $\Omega$ is always equal to a quotient of $\Omega_u(A)$, the universal calculus over $A$, by some ideal $N$ of ...