The differential-calculus tag has no wiki summary.

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### 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 ...

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### 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 ...

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### What does mean the exact value of derivative [migrated]

i'm starting my calculus's journey and i have a question.
What does mean the exact value of a derivative
Take an easy example we have a derivative of $f(x)=x^2$, that is $f'(x)=2x$.
Someone would ...

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### 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 ...

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### 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{ - ...

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### 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 ...

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### 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:
...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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: ...

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### 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 ...