Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

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9
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96 views

Iterated Limits

If $$\lim_{(x,y) \to (a,b)} f(x,y) = L$$ and if the one dimensional limits - $$ \lim_{x \to a} f(x,y)$$ and $$ \lim_{y \to b} f(x,y)$$ ...
7
votes
0answers
111 views

Closed-form expression for an iterated integral

Does the following iterated integral have a simple closed-form expression in terms of $z$? $$ I = \int_0^\infty \int_0^\infty \sqrt{\frac{1 + x^2 y^2 + x^2 z^2 + y^2 z^2}{(x^2 + y^2 + z^2 + x^2 y^2 ...
7
votes
0answers
185 views

Evaluate $\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} dx$

I found two different approaches, both is giving the same answer. Fubini: $$ \begin{align} \int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \,dx &= \int_0^{\infty} e^{-x} \int_0^a e^{-xy} \,dy\, dx \\ ...
7
votes
0answers
115 views

Other methods for Laplacian equations

Assume $$A^{2}=(x^{2}+y^{2})\cos^{2}\psi+z^{2}\cot^{2}\psi$$ which $A$ is constant. How we can show $\psi(x,y,z)$ satisfies the Laplacian equation $\psi_{xx}+\psi_{yy}+\psi_{zz}=0$ ...
7
votes
0answers
210 views

Computing the volume of a region on the unit $n$-sphere

I would like to compute the surface volume of a region on the unit $n-1$-sphere: $$x_1^2 + \dots + x_i^2 + \dots + x_n^2 = 1,$$ bounded by an ellipsoid $$a_1x_1^2 + \dots + a_ix_i^2 + \dots + ...
6
votes
0answers
166 views

Munkres' Question on Manifolds

In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads: QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$. Let $M$ be the set of all the points ...
6
votes
0answers
229 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
6
votes
0answers
110 views

Proof of inverse function theorem by approximation property

In proving the inverse function theorem using the approximation characterization of the derivative, we are given $F:\mathbb{R}^n \to \mathbb{R}^n$ such that  $$F(p_0 + h) - F(p_0) = DF_{p_0}(h) + ...
6
votes
0answers
121 views

1-form with positive integral over a path

Given any smooth path $\gamma$ on a manifold $M$, when can we construct a 1-form $\omega \in \Omega^1(M)$ so that $\int_{\gamma} \omega > 0$? It is easy to see that such $\omega$ exists if the path ...
6
votes
0answers
231 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
6
votes
0answers
399 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
6
votes
0answers
231 views

Laplace-Beltrami Operator for Euclidean Space

Consider the space $\mathbb{R}^n$ and let $x_1,\ldots, x_n$ be the coordinates. Fix the orientation $dx_1\wedge dx_2\ldots\wedge dx_n$. Let $E^p$ denote the space of smooth $p$ forms and let $d$ ...
6
votes
0answers
152 views

Volume of a sphere by “adding” half-spheres of lower dimension

I'm wondering about different ways to compute the volume of an $n$-sphere. Please see the wikipedia page for one method to compute the volume via hyperspherical coordinates: ...
6
votes
0answers
567 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f = 2\pi+2\tan^{-1}(y,x)$ $y = ...
5
votes
0answers
125 views

Showing some complicated integral expression is bounded

In my research, I come across some expression I need to bound. I wish to show that the following integral is bounded in $t$ and $x$, i.e. the following supremum is finite: $$\sup_{t,x\in ...
5
votes
0answers
182 views

Multivariable Integral, How to compute it?

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm ...
5
votes
0answers
45 views

Inhomogeneous Wave Equation in 3 dimensions

From section 7.5 in this source, I see that, for $\vec{x} \in \mathbb{R}^3$, if $$ \frac{\partial^2 u\left(\vec{x},t\right)}{\partial t^2} - \nabla^2u\left(\vec{x},t\right) ...
5
votes
0answers
186 views

How to prove following integral equality?

Let's have the equality $$ \int \limits_{-\infty}^{\infty} \left[ [\nabla_{\mathbf r'} \times \mathbf A (\mathbf r' )] \times \frac{\mathbf r' - \mathbf r}{|\mathbf r' - \mathbf r ...
5
votes
0answers
79 views

the spectrum and determinant of the Laplacian on $S^3$

I came across the following statement in a paper: On $S^3$, the eigenvalues of the vector Laplacian on divergenceless vector fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell ...
5
votes
0answers
135 views

Algorithm to calculate multiple integral.

One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
5
votes
0answers
67 views

Smoothness in $\mathbb{R}^n$

Embarrasingly simple question, but I got the feeling that I cannot see the forrest for the trees right now: If I have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ and want to show that it is ...
4
votes
0answers
57 views

Question on using Leibniz formula to derive thin-film equation from Navier-Stokes

I am trying to work through the derivation in this paper by Petr Vita, which derives a thin-film simplification of the Navier-Stokes equation, similar to the Reynolds or Lubrication Equation, but ...
4
votes
0answers
32 views

Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n $ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M $ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
4
votes
0answers
204 views

Uniform Differentiability

Let $f:\mathbb{R}^n \to \mathbb{R}$ be differentiable and such that $\nabla f$ is uniformly continuous. Show that $f$ is uniformly differentiable; that is, for any $\epsilon >0$, there is a $\delta ...
4
votes
0answers
133 views

How to solve this special form of multivariate quadratic equations?

Suppose there are $M$ known real vectors $$ x^{i}=[x_{1}^{i},x_{2}^{i},\cdots,x_{N}^{i}],i=1,2\cdots,M $$ in N-dimensional vector space and $M\geq 2N$. I want to know whether they span two orthogonal ...
4
votes
0answers
105 views

vector field as integral

Define a vector field $ \vec{f}(\vec{R}) = \oint_C{|\vec{r} - \vec{R}|^2 d\vec{r} }$ where C is a simple closed curve. show that there are constant vectors $ \vec{P} $ and $ \vec{Q} $ such that $ ...
4
votes
0answers
357 views

Help me correct these properties of : $f_{n}(x)= nx(1-x)^{n}$? Is there maybe a typo in the sequence?

Examine the sequence of functions $(f_n)_{n\in \mathbb{N}}$ on $x\in[0,1]$: $$f_{n}(x)= nx(1-x)^{n}$$ Does $(f_n)_{n\in \mathbb{N}}$ converge pointwise or uniform? I will show that it does ...
4
votes
0answers
101 views

Mean value inequality and Fixed-point theorem

I need help to understand this question, it's not that clear for me: Using the norm $|x|+|y|$ and the Mean value inequality, give a condition over the partial derivatives of $f(x,y)$ and $g(x,y)$ for ...
4
votes
0answers
186 views

How to solve this differential equation for $y$ in terms of $x$ and $k$

$$yy'+\frac yx+k=0$$ How to solve this differential equation for $y$ in terms of $x$ and $k$ where $k$ is a parameter of $x$? $y(x)=y$ is a function and $x(k)=x$ is a gamma function
4
votes
0answers
249 views

Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis. Correlation of the series changes over time and across ...
4
votes
0answers
210 views

History of line integral.

I'm looking for some information about how the line integral was discovered, since I've been looking for a long time for this. I found that Riemann could integer discontinuity functions, then Poisson ...
4
votes
0answers
368 views

local diffeomorphism

I hope this finds you all well. Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in ...
4
votes
0answers
402 views

How to construct a vector space and compute basis?

My professor demonstrated that in vector calculus that you can construct basis vectors for one, two, and three forms using the vectors $dx$, $dx$ and $dy$, as well as $dx \wedge dy$, $dy \wedge dz$, ...
4
votes
0answers
159 views

How do you show that the Laplacian is the square of the (Euclidean) Dirac operator?

If I understand correctly, the Euclidean Dirac operator is given by $$D=\sum_{i=1}^n e_i \frac{\partial}{\partial x_i},$$ where $e_i$ are bases for $Cl_{0,n}(\mathbb{R})$, i.e., the $n$-dimensional ...
3
votes
0answers
28 views

Multivarable limit proof

I came across with this statement and I can't neither prove it right nor find a counterexample. The statement is: Consider two functions $F(x,y)$ and $G(x,y)$ continuous and differentiable around a ...
3
votes
0answers
33 views

How do I integrate $\langle\nabla u,\nabla v \rangle$ in arbitrary dimensions?

I am trying to show that if $u_n$ are eigenfunctions of the Laplacian operator that make up an orthonormal basis of $L^2$, then $u_n\sqrt{\lambda_n}^{-1}$ form an orthonormal basis of $H^1_0$. I ...
3
votes
0answers
34 views

Equivalence of definitions of multivariable differentiability

The usual definition of differentiability for a multivariable function says that $f:\mathbb{R}^2 \to \mathbb{R}$ is differentiable at $(x,y)$ if there is a linear map $d_{(x,y)} f : \mathbb{R}^2\to ...
3
votes
0answers
21 views

double area integral over a Jinc/Bessel

I am having trouble showing the following, which shows up from coherence theory: $\frac{\pi b^2}{\alpha^2}(1-J_0^2(\alpha b)-J_1^2(\alpha b))=\int_0^{2\pi}\int_0^b\int_0^b r_1r_2\frac{J_1\left ...
3
votes
0answers
41 views

Prove that $f$ is continuous at $(0, y_0)$. where $f$ is defined on $\Bbb R^2$.

Prove that f is continuous at $(0, y_0)$ $f(x, y) = \begin{cases} (1+xy)^{1/x} &\mbox{if } x \neq 0 \\ e^y & \mbox{if } x \equiv 0. \end{cases} $ Thank you!
3
votes
0answers
58 views

Understanding the setup for the probability that $Ax^2+Bx+C$ has real roots if A, B, and C are random variables uniformly distribted over (0,1).

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $Ax^2 + Bx + C$ has real roots? First, I set $P(B^2 - 4AC ...
3
votes
0answers
40 views

Homework: Second derivative of $\langle Ax, x \rangle$

So let $A \in M_{n}$ and define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(x) = \langle Ax, x \rangle $. Find f' and f''. After some work, I found the first derivative to be $f'(x)(v) = \langle Ax, v ...
3
votes
0answers
51 views

Intuition behind the Triple Product identity

The well-known identity: $$ \textbf{c}\times(\textbf{a}\times\textbf{b})=(\textbf{c}\cdot \textbf{b})\textbf{a}-(\textbf{c}\cdot \textbf{a})\textbf{b} $$ and its counterpart for a curl: $$ \nabla ...
3
votes
0answers
46 views

What is the domain and range of this multivariable function?

What is the domain and range of the following multivariable function? $g(x,y,z) = {1\over \sqrt{(4 - x^2 - y^2 - z^2)}}$ g is real valued. So far I have that the domain is the set of a vectors ...
3
votes
0answers
62 views

Why does a figure look the same in every coordinate system?

After reading Maximilian M. Answer here: Gauss' Theorem - Can't understand a parameterization I'm trying to figure out why does a figure look the same in every coordinate system I choose. For ...
3
votes
0answers
65 views

Clarification of the Jacobian

Well, that was cool if not tedious but I understand the Jacobian and its application to changing coordinate systems. $${J_{POLAR}= \rho}$$ $$ {J_{cyl}= \rho}$$ and $${J_{sphere}=\rho^2\sin\phi}$$ ...
3
votes
0answers
63 views

Uniform Convergence/continuity

Let $A$ be a compact set of $\mathbb{R}^n$ and $f$ continuous on $A$. Let $F=f_A$, and let $I_0$ be a cube containing $A$. Divide $I_0$ in $2^n$ equal subcubes $I_{1_1},\dots, I_{1_{2^n}}$.On $I_0$ we ...
3
votes
0answers
61 views

Knowing when to use Green/Stokes/Divergence theorem to evaluate line/surface integrals

$\newcommand{\mbf}{\mathbf}$ Evaluate $$ \iint \limits_{S} \mbf{F} \cdot d \mbf{S} $$ where $\mbf{F} = 3xy^2 \mbf{i} + 3x^2y \mbf{j} + z^3 \mbf{k}$ and $S$ is the surface of the unit sphere. I ...
3
votes
0answers
66 views

Invariance of Laplacian under Orthogonal transformations

Let $F=f\circ \bf{L}$, where $\bf{L}$ is a Linear transformation with matrix $(c^{i}_{j})$ of $dim=n\times n$ with $i$ for rows, and $j$ for columns. $F$ and $f$ are $C^2$ real-valued functions. We ...
3
votes
0answers
43 views

Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
3
votes
0answers
51 views

How to calculate hard integral?

How to calculate the integral $$ \int_D \frac {\prod_{i<j}(a_i-a_j)^2\prod_{i<j}(b_i-b_j)^2} {\prod_{i,j} (a_i+b_j)^2}\,d\lambda_{2n-1},$$ where $$D:=\{ (a_1,\dots ...