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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12
votes
0answers
370 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 + ...
11
votes
0answers
125 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
9
votes
0answers
334 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 \\ ...
8
votes
0answers
235 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...
8
votes
0answers
132 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
93 views

Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
7
votes
0answers
147 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 ...
7
votes
0answers
177 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
110 views

Second Variation of Area Functional

This is a follow up question from this one. I have proved that given a parametrized surface ${\bf x}$, the mean curvature is zero if and only if it is a critical point of the area functional. Then ...
6
votes
0answers
174 views

Euler-Lagrange Equation and “Eigen Value ”

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
6
votes
0answers
52 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
6
votes
0answers
174 views

How to take limit *along a path*

So in multivariable calculus for a limit of a function to exist, the limits of the function along all possible paths must exist and equal the same value. But how does one calculate the limit along a ...
6
votes
0answers
211 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 ...
6
votes
0answers
384 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
136 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
259 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
639 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
55 views

Manifold, exist smooth nonnegative function with regular value at 0?

If $X$ is any manifold with boundary, then does there exist a smooth nonnegative function $f$ on $X$, with a regular value at $0$, such that $\partial X = f^{-1}(0)$?
5
votes
0answers
51 views

Continuity ( Functions of 2 variables ).

Given , $$ f(x,y) = \begin{cases} \dfrac{xy^{3}}{x^{2}+y^{6}} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \\ \end{cases} $$ We need to check whether the function is continuous at ...
5
votes
0answers
92 views

Can a differential k-form be integrated on a manifold that is not k-dimensional?

For example, can you integrate a 2-form on some curve, a 1-dimensional manifold, or some 3-dimensional manifold? I know that Stokes's Theorem states that if you integrate $\omega \in \mathcal ...
5
votes
0answers
58 views

Solving a partial differential equation by transformation of variables?

I found an exercise in a book, where one was supposed to transform the differential equation $$y\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}=xyf(x,y)$$ by using the substitutions ...
5
votes
0answers
70 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
5
votes
0answers
64 views

How do I do this change of variables?

Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x = 2$, $z-y = 0$, $z-y = 1$, $z=0$, $z=3$. I set $$u = y-x$$ $$v = z-y$$ $$w ...
5
votes
0answers
141 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
5
votes
0answers
75 views

Is this correct and sufficient to show limit does not exist?

Find limit or show that it does not exist: $$\lim_{(x,y) \to (0,0)} \frac{ 2x^{2}y^{3/2} }{y^{2}+x^{8}}$$ using the path $x=m y^{1/4}$: $$\lim_{(my^{1/4},y) \to (0,0)} \frac{ ...
5
votes
0answers
142 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
102 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 ...
5
votes
0answers
66 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
211 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
122 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
175 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
126 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 $ ...
5
votes
0answers
346 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
24 views

Why is n-th Fréchet derivative symmetric?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$. Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function. Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is ...
4
votes
0answers
39 views

Solving non-linear equations in a chosen subspace

I'm trying to find the root $\mathbf{f(x)=0}$ to the following sets of equations $$ f_1(x,y,z) = x^\prime - \frac{x}{\sqrt{x^2+y^2+z^2}} = 0 \\ f_2(x,y,z) = y^\prime - \frac{y}{\sqrt{x^2+y^2+z^2}} = ...
4
votes
0answers
107 views

closed form is exact in euclidean space

Question is to show that $d(f)=0$ for a $0$ form on $\mathbb{R}^n$ then $f$ is a constant function. See that $$0=df=\sum_i\frac{\partial f}{\partial x_i}dx_i$$ implies that $\frac{\partial ...
4
votes
0answers
40 views

Axysimmetric Poisson equation solution

I'm struggling to find the solution to the follow Poisson problem in spherical coordinates: $$ \Delta\, f\left(r,\theta\right) = \sum_{l=1}^{4}k_{l} \left(r\right) P_{l}^{1}\left( \theta \right)$$ ...
4
votes
0answers
56 views

Integral of a function with an exponentiated inner product

Let $f:\Bbb R^n\to \Bbb R^n$ be a continuous function such that $\int_{\Bbb R^n}|f(x)|dx\lt\infty$. Let $A$ be a real $n\times n$ invertible matrix and for $x,y\in\Bbb R^n$, let $\langle x,y\rangle$ ...
4
votes
0answers
41 views

Construct Curvilinear Coordinates from “Tangent Basis Field”

Let say that you are on a open, simple connected subset of the Euclidian plane (2 dimensional) and you have a second order tensor field defined over it. At every point we have a second order tensor ...
4
votes
0answers
47 views

Exercise from Pugh's Real Analysis Regarding Zero Derivative on Open Subsets of $\mathbb{R}^m$

Assume that $U$ is a connected open subset of $\mathbb{R}^n$ and $f:U\to\mathbb{R}^m$ is differentiable everywhere on $U$. If $(Df)_p=0$ for all $p\in U$, show that $f$ is constant. I immediately ...
4
votes
0answers
63 views

Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
4
votes
0answers
130 views

Convex set of derivatives implies mean value theorem

Let U$ \subset$ $R^{^{n}}\ $be open, $f:U\rightarrow R^{m}$ differentiable on U, and segment $[a,b]\subset U$. Assume that the set of derivatives $\{ f'(x)\in L(R^{^{n}},R^{^{m}}):x\in [a,b] \}$ ...
4
votes
0answers
62 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
4
votes
0answers
85 views

Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
4
votes
0answers
148 views

Application of Cauchy-Schwarz with Sobolev norms

I'm working through the problems in the initial value formulation chapter in Wald's General Relativity. A short summary of the problem. I have to show that $$\sup_{x\in A}|f(x)|\le C||f||_{A,k}$$ ...
4
votes
0answers
67 views

Uniqueness of solution to $u_{t} - \Delta u + |\nabla u|^{2} = 0$

The problem I am working on is as follows: Let $\Omega$ be a connected bounded domain in $\mathbb{R}^{n}$ with smooth boundary and let $f, g: \mathbb{R}^{n} \rightarrow R$ be smooth. Show that there ...
4
votes
0answers
61 views

Unable to evaluate this surface integral

Question is to find the surface integral $$\iint p(x^4 + y^4 + z^4) \, \mathbb dS.$$ The given surface is $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$ and $p$ is the length of the ...
4
votes
0answers
72 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
4
votes
0answers
56 views

The missing vector derivative operation

Generally, the differential operator $d$ takes differential $k$-forms to $(k+1)$-forms. In $\mathbb{R}^n$ (or more generally on an oriented Riemannian manifold), we can identify vector fields with ...
4
votes
0answers
59 views

Improper multivariable integrals

I'm having trouble with the integral $$\iiint_{1\le x^2+y^2+z^2 }\frac{\mathrm{d}x~\mathrm{d}y~\mathrm{d}z}{xyz}$$ this is what I've done so far: $$\lim_{b\to +\infty}\int_1^b \int_0^{2\pi} ...