# Questions tagged [multivariable-calculus]

The multivariable-calculus tag has no usage guidance.

The multivariable-calculus tag has no usage guidance.

43
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I found that following equation holds for random vector $z \sim N(0,I)$ :
$\mathbb{E} [\frac{zz^T}{z^Tz}] = \frac{1}{n} I$
Proof is very simple that is only calculating integral for each component ...

1
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1
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103
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If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want ...

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I've encountered the following problem during my research, any help would be highly appreciated.
Let a following multivariate equation be given:
$F(x,P,I) = 0$,
where $x$ is a variable and $P,I$ are ...

2
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1
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121
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I am given a graph of a motor's torque vs RPM values at two different current draws, 730A and 300A (see graph). I need to obtain the 3D function to find current as a function of torque and RPM by ...

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39
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This is in the context of Finite Elements on simplices however it hopefully is of more general interest. Basically I am looking for a result that rigorously describes how one can build a basis of $\...

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119
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In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...

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263
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In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears:
\begin{equation}
\int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...

3
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99
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$\newcommand{\S}{\mathbb{S}^1}$
$\newcommand{\la}{\lambda}$Let$$M=\{(x_1,x_2,x_3,x_4) \in (\S)^4\,\, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$
Define $E:M \to \mathbb{R}$ by
$$E(x_1,x_2,...

6
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357
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I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals)
$$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\
g(x)=\frac{\...

2
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1
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128
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I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...

1
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1
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139
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This is concerning Eq. (3.7) of C R Rao's 1945 paper (see p.81 of this article). Can someone help me in figuring out the second equality in Eq. (3.7)?
His claim is (since $\phi(x,\theta) = \Phi(T,\...

6
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1
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314
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Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral:
$$
\int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.
$$
...

2
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1
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186
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Recently I got stuck on a 2 dimensional integral in polar coordinate,
the expression is the following:
$I(x)=\lim_{\xi\rightarrow0^+}\int_0^\infty dr\int_{-\pi/2}^{\pi/2}dt\frac{2\xi ^{2-2 x}r^{2x+1} \...

3
votes

1
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300
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Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$.
Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$:
\...

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154
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While working on some properties of partial derivatives and multiply differentiable functions of several variables, I came across the following Hypothesis 1:
Let $f: \mathbb{R}^n\to\mathbb{R}$, $\...

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45
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Here is the problem:
Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$,...

3
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164
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Let us first define the variation diminishing property for the Gaussian kernel. Consider a function $f: \mathbb{R} \to \mathbb{R}$ that is sufficiently smooth and define
\begin{align}
F(x)= \int_{-\...

11
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3
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537
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I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$:
$$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$
Numerically, I see that ...

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167
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Recently I was trying to construct a counterexample to the statement "If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are ...

1
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1
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368
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Let $\mathbb R^d$ and let $\mu = p(dx)$ be a probability distribution thereupon, with density $p$ (which maybe assumed bounded, etc.). For a continuously differentiable function $f:\mathbb R^d \to \...

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988
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Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...

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77
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Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...

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2
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150
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Suppose $P: \mathbb{R}^n \rightarrow \mathbb{R}_{\ge 0} $ is a differentiable map, with $P(x) = 0 \ \forall x \in \mathcal{X}$ and $P(x) > 0 \ \forall x \in \mathcal{X}^c$. Further, suppose $P$ has ...

6
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1
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320
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I am looking for a function $f:\mathbb C^2 \rightarrow \mathbb C^2$ that satisfies the two equations
$$\partial_{z_2}f_1(z_1,z_2) + \partial_{z_1} f_2(z_1,z_2)=0 \text{ and }$$
$$\partial_{\bar z_1}...

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437
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I am trying to understand the chain rule under a change of variables. Given a function $f : \mathbb R^n \rightarrow \mathbb R$ and a change of variables $G : \mathbb R^m \rightarrow \mathbb R^n$, what ...

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50
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Dealing with one-variable and smooth function $f$ on a real interval $I$ such that $D^m f\in\mathcal{C}^2$, we have by Taylor theorem centered at $a\in I$
$$ D^mf(y)= D^mf(a) + D^{m+1}f(a)(y-a) + \...

2
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1
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163
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Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial ...

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296
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Suppose $x = (x_1,x_2,\dots,x_K) \in \mathbb{Z}^K_{\geq 0}$. For $x,y \in \mathbb{Z}^K_{\geq 0}$, we write $x \succ y$ or $y \prec x$ if $x \neq y$ and
\begin{align*}
x_{i(x,y)} > y_{i(x,y)...

2
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32
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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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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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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 ...

7
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470
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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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71
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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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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 ...

3
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1
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249
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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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1
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161
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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&...

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1
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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 ...

2
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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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0
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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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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{...

0
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1
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264
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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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3
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616
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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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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{...