Questions tagged [multivariable-calculus]
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65 questions
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Hessian generating functions
I am looking for a characterization of functions $\Phi: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ such that $\Phi(\mathbf{x}) = \nabla^2 f(\mathbf{x})$ for a function $f$ which is twice continuously ...
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What relationship exists between samples of a function and samples of its vector gradient field?
A real function $f(x)$ is defined on $N$-dimensional real space where $N \ge 3$. $f(x)$ is differentiable and its gradient with respect to x is $g(x)$. So $g(x)$ is a vector field.
Assume we do not ...
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Creating a Multi-Variable equation to create a "Most Favorable Supplier Index" [closed]
I wanted to create a 'most favorable supplier index', where I want to assign a value to each supplier, ranging from 1-5 or 1-10, whichever, from the most and to the least preferred suppliers. There ...
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Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?
This is a cross-post.
Let $U \subseteq \mathbb R^n$ be an open subset, and let $f:U \to \mathbb R$ be smooth. Suppose that $x \in U$ is a strict local minimum point of $f$.
Let $df^k(x):(\mathbb R^n)^...
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Conditions for a function to vanish almost nowhere on its support?
Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support.
Under which conditions is it true that $f≠0$ (...
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Differential entropy under the change-of-variable with additive Gaussian noise
I have two Gaussian random variables $$X \sim \mathcal N(0, I), \ \ \ W \sim \mathcal N(0, \sigma\cdot I)$$ and I known a parametric change-of-variable $Y(\theta) = T(X; \theta)$.
I would like to ...
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Gradient condition implies Hörmander condition
We have tempered distribution $K$ in $\mathbb{R}^n$ which coincides with a locally integrable function in $\mathbb{R}^n\setminus \{0\}$. We call the condition
$$\int_{|x|>2|y|}|K(x-y)-K(x)|dx\leq ...
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Is the minimum of a constraint optimization problem differentiable in the constraint parameter?
Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$.
Let $s&...
- 128k
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Is the optimum of this problem convex in the constraint parameter?
Let $f:\mathbb R^+ \to \mathbb R$ be a smooth function, satisfying $f(1)=0$, and suppose that
$|f|$ grows with the distance from $1$: $|f(x)|$ is strictly increasing when $x \ge 1$, and strictly ...
- 128k
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The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse
Conjecture.
Let
$$ f(r,\alpha,p, \theta) = \ln\left(\left(r\sin\alpha-\sin\theta\right)^{2}\left(1-p\right)^{2}+\left(r\cos\alpha-\cos\theta\right)^{2}\left(1+p\right)^{2}\right). $$
Then for any ...
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Elasticity tensor in terms of principal stretches
Suppose we are given a frame-indifferent isotropic function
$W:GL_+(3) \to [0,\infty)$, where $GL_+(3)$ denotes the set of all real $(3\times 3)$-matrices with positive determinant.
We can write $W(F)$...
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A Curved/Warped Version of Fubini's Theorem
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
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How to compute the volume of a region transformed by a matrix?
This is a rewrite of the OP's question to emphasize what I think are the research level issues here.
Let $\mathscr{R}$ be a bounded convex body in $\mathbb{R}^n$ and let $H : \mathbb{R}^n \to \mathbb{...
- 30.3k
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Integral $ g(a)= \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $ [closed]
I am having trouble calculating this integral:
$$ g(a) = \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $$
I tried calculating $g'(a)$ but then I get stuck.
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A question about multivariable calculus and optimization
Consider the function $f(x) :\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x)\geqslant 0\; \forall x\in \mathbb{R}$, and has a set of extremum points at $x_{j}$.
Consider the integral :
$$\int_{\bar{...
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