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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11
votes
0answers
312 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 + ...
9
votes
0answers
301 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
171 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
129 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
294 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 $d$ ...
7
votes
0answers
171 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
165 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
97 views

If the second derivatives $f_{xx}$ and $f_{yy}$ exist, does $f_{xy}$ exist?

If the second derivative with respect to to $x$ exists ($f_{xx}$) and the second derivative with respect to $y$ ($f_{yy}$), does it follow that $f_{xy}$ exists?
6
votes
0answers
77 views

Symbolic manipulation inside integral

I'm an undergrad who has just completed the standard calculus sequence (1, 2, and multivariable). I've done well in the courses, however, things like the following, which is a derivation of kinetic ...
6
votes
0answers
48 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
129 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
210 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 ...
6
votes
0answers
209 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
342 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
131 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
144 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
252 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
633 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
44 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
67 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
178 views

Find the volume of the region bounded by the planes $ z=8-y^2, y = 8-x^2, x=0, y=0, z=0$

I figured out the bounds for z: $z=0$ to $z=8-y^2$ The bounds for y: $y=0$ to $y=8-x^2$ The bounds for x: $x=0$ to $x=\sqrt{8}$ (Since $8-x^2 = 0$) So, the volume by using triple integral: ...
5
votes
0answers
60 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
139 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
96 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
64 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
205 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
112 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
173 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
124 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
195 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
5
votes
0answers
297 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
38 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
53 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 ...
4
votes
0answers
128 views

Application of Cauchy-Schwartz 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
60 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
64 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
49 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} ...
4
votes
0answers
104 views

Differential forms and determinants

2-forms are defined as $du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}$ But what if I have two concret ...
4
votes
0answers
96 views

Second degree partial differential equation with variable-change

Edit: @Etienne mentioned that I did a typo, writing $u_y' = -xye^{-y}$ instead of $u_y' = -xe^{-y}$. I've corrected that in the calculations and now it's closer to being correct! Though I still miss ...
4
votes
0answers
69 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{ ...
4
votes
0answers
74 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
4
votes
0answers
125 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
4
votes
0answers
43 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
99 views

Far-field Poynting vector for time varying source [check my math please]

This is a question from classical electrodynamics, but it's the maths of it I need some help in. I have fields $\rho\left(\vec{r},t\right)$, $\vec{J}\left(\vec{r},t\right)$, ...
4
votes
0answers
355 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
174 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
193 views

Stokes' Theorem problem

Let $M \subset \mathbf{R}^n$ be oriented compact smooth $k$-manifold and $\alpha$ be a $C^1$ diferential $(k-1)$-form defined in a neighborhood of M. Use Stokes' theorem to prove that \begin{align*} ...
4
votes
0answers
281 views

Second derivative of a vector field

I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points ...