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        Questions tagged [differential-calculus]

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        integral of the product of an arbitrary function and a linear function [on hold]

        In engineering mechanics, a classic approach for the calculation of displacements (virtual work method) requires the evaluation of the definite integral of the product of two continuous functions in $[...
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        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 ...
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        279 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 ...
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        5k 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 ...
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        67 views

        Representation of finite differences of order k

        We define recursively finite differences $ g_k (x) $ of order $ k $ of function $ f $ as follows: $g_0(x)=f(x)$, $g_n(x)=g_{n-1}(x+h_n)-g_{n-1}(x) (n\in\mathbb{N})$. It is known that all arguments of ...
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        Differentiability (Hessian) of $\int \log F$ when $\int \log f$ is differentiable?

        For a specific probability density function $f$ with support on ${\mathbb R}$, which is not differentiable everywhere, I have proven that the Hessian matrix of $$g(\theta) = \int \log f(x;\theta)d H(...
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        143 views

        A vector calculus formula

        Let me answer my own question, hoping to be forgiven for that. I asked unsuccessfully that question on Mathematics. Let $A, B$ be vector fields in $\mathbb R^3$. We have $$ \text{curl}\bigl((A\cdot \...
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        151 views

        About $\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$

        An integral has been pushed me over the edge for several weeks. It reads as: $$\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$ I tried ...
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        72 views

        Existence of a certain type of function

        Trying to find functions with the given property: Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...
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        2answers
        153 views

        Dominant root of a family of polynomials

        Let $f(x)=x^5-x^4-x^3-x^2-x-c$, where $c>2$ is a real number. It is easy to prove that there exists a positive real root $\alpha>2$ of $f(x)$ and all the other roots are non real. Also, by ...
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        Looking for example of integral transformations that preserve number of zeros

        Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros. I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$...
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        3answers
        118 views

        Literature request: Function that depends on a linear optimization problem [closed]

        my question is about functions of the following form: $$ f(t) = \max_{\mathbf{x}}~ \mathbf{c^T x} ~ {\rm s.t. \mathbf{Ax} +t \cdot \mathbf{a} \leq \mathbf{b}}, $$ where $\mathbf{x},\mathbf{b}, $ ...
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        1answer
        266 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 ...
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        1answer
        123 views

        A geometric property about certain polynomials in two variables

        Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$ where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...
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        346 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 ...

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