# Tagged Questions

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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### 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. $f$ ...
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### partial derivative of a facet normal wrt to one of its vertex

I am struggling to understand the derivation of an equation in a paper (A Bayesian Method for Probable Surface Reconstruction and Decimation, specifically Eqn. 16). Basically they define three ...
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### On the form of injective function $f:\mathbb R^+ \to \mathbb R^+$ such that $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ is a norm

Let $f:\mathbb R^+ \to \mathbb R^+$ be injective such that on $\mathbb R^n , n>1)$ , $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ , where $u=(u_1,u_,\ldots,u_n)$ , defines a norm ; then is it true ...
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### 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 $\partial X = f^{-1}(0)$?
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### 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 ...
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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 $x^2+... 0answers 85 views ### Conditions for Taylor formula I know that, if$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 ... 0answers 78 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 $$u = y-x$$ $$v = z-y$$ $$w = ... 0answers 188 views ### Differential forms and determinants 2-forms are defined as du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix} But what if I have two concret 1-... 0answers 143 views ### Why is a_{n}(x,y)=a_{n}(y)? This particular question is connected (with a slight variation in the definition of g) to an earlier question. The link is here. The specifics are: Given that u(x,y) is the solution of a PDE (x ... 0answers 147 views ### Showing some complicated integral expression is bounded In my research, I come across some expression I need to bound. I wish to show that the following integral is bounded in t and x, i.e. the following supremum is finite:$$\sup_{t,x\in \mathbb{R}}\,... 0answers 121 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 ... 0answers 75 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 $$\frac{\partial^2 u\left(\vec{x},t\right)}{\partial t^2} - \nabla^2u\left(\vec{x},t\right) =f\left(\vec{x},t\... 0answers 227 views ### How to prove following integral equality? Let's have the equality$$ \int \limits_{-\infty}^{\infty} \left[ [\nabla_{\mathbf r'} \times \mathbf A (\mathbf r' )] \times \frac{\mathbf r' - \mathbf r}{|\mathbf r' - \mathbf r |^{3}}\right]d^{3}\... 0answers 160 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 \...
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 ...