All Questions
Tagged with multivariable-calculus integration
14 questions
2
votes
2
answers
383
views
Definition of Multivariable Antiderivatives
In the 1-dimensional case antiderivatives $F(x)$ of a function $f(x)$ have the following properties:
$F(x)=\int\limits_0^xf(t)dt$
$\frac{d}{dx}F(x)=f(x)$
$\int\limits_a^bf(t)dt = F(b)-F(a)$
Of ...
- 13.2k
0
votes
0
answers
97
views
An integral over the sphere in $\mathbb{R}^d$
Let $S^{d-1}$ be the unit sphere in $\mathbb{R}^d$.
Let $|x-y|$ denote the euclidean distance between to points $x$ and $y$ in $\mathbb{R}^d$.
Is there a nice expression for the following (maybe ...
- 4,034
2
votes
1
answer
122
views
Analytical solution for a double integral involving logistic functions and Gaussian distributions
I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows:
$$...
- 31
22
votes
1
answer
2k
views
A difficult integral for the Chern number
Cross post from Maths stack exchange
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...
- 333
1
vote
1
answer
152
views
The monotonicity of the bivariate normal with non-isotropic covariance
Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance.
Define $y = (y_1, y_2)$ and let
\begin{align}
F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
- 113
2
votes
0
answers
154
views
Applying 1D integral to matrix integral
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 ...
- 83
2
votes
1
answer
326
views
Evaluation of Gaussian multivariable integral
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 ...
- 83
2
votes
1
answer
330
views
Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
- 162
5
votes
1
answer
344
views
Monotonic dependence on an angle of an integral over the $n$-sphere
Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral:
$$
\int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.
$$
...
- 115
2
votes
1
answer
215
views
A 2 dimensional integral in polar coordinate [closed]
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} \...
- 71
0
votes
0
answers
62
views
Integration question dealing with several variables and Taylor theorem
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 $a\in I$
$$ D^mf(y)= D^mf(a) + D^{m+1}f(a)(y-a) + \...
- 383
2
votes
0
answers
110
views
The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse
Conjecture.
Let
$$ 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). $$
Then for any ...
0
votes
1
answer
366
views
Integral $ g(a)= \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $ [closed]
I am having trouble calculating this integral:
$$ 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.
- 1
2
votes
3
answers
806
views
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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