# Questions tagged [multivariable-calculus]

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### Lagrange Multipliers Not Identifying All Critical Points [migrated]

Suppose that $x,y,z \in \mathbb{R}_{\ge 0}$. Consider $f(x,y,z) = xyz$ (the volume of some cube) and $g(x,y,z) = x+y+z-1$. I wish to maximise $f$ subject to $g=0$ with Lagrange multipliers. The zero ...
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### Helmholtz decomposition vs Laplacian vector fields on $\mathbb{T}^3$?

I am quite confused between Helmholtz decomposition and Laplacian vector fields in the periodic case. Let $\mathbb{T}^3$ be the $3$-dimensional torus. Then, I thought any divergence-free smooth vector ...
1 vote
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### If $v$ and $w$ are orthogonal smooth vector fields of unit norm and zero divergence, is $\langle w, (\nabla \times v) \times v \rangle=0$?

I am quite confused about manipulating with the curl when the vector field is of unit magnitude and divergence free. For example, let $v, w : \mathbb{T}^3 \to \mathbb{R}^3$ be periodic smooth vector ...
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### 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, ...
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### Differential form of the multidimensional "orthogonal dilation" operator

For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion. ...
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### Multivariable higher-order chain rule

I am trying to understand the chain rule under a change of variables. Given a function $f : \mathbb R^n \rightarrow \mathbb R$ and a change of variables $G : \mathbb R^m \rightarrow \mathbb R^n$, what ...
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) + \... 2 votes 1 answer 215 views ### Question about the implicit function theorem. an example of a homogeneous form for which its implicit function satisfies certain conditions Let F be a homogeneous form with coefficients in \mathbb{R}. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial ... 4 votes 1 answer 299 views ### How to define a function that has these specific properties? Suppose x = (x_1,x_2,\dots,x_K) \in \mathbb{Z}^K_{\geq 0}. For x,y \in \mathbb{Z}^K_{\geq 0}, we write x \succ y or y \prec x if x \neq y and \begin{align*} x_{i(x,y)} > y_{i(x,y)... 2 votes 0 answers 40 views ### Hessian generating functions I am looking for a characterization of functions \Phi: \mathbb{R}^n \to \mathbb{R}^{n \times n} such that \Phi(\mathbf{x}) = \nabla^2 f(\mathbf{x}) for a function f which is twice continuously ... 1 vote 0 answers 32 views ### What relationship exists between samples of a function and samples of its vector gradient field? A real function f(x) is defined on N-dimensional real space where N \ge 3. f(x) is differentiable and its gradient with respect to x is g(x). So g(x) is a vector field. Assume we do not ... 1 vote 0 answers 22 views ### Creating a Multi-Variable equation to create a "Most Favorable Supplier Index" [closed] I wanted to create a 'most favorable supplier index', where I want to assign a value to each supplier, ranging from 1-5 or 1-10, whichever, from the most and to the least preferred suppliers. There ... 7 votes 2 answers 531 views ### Is a function of several variables convex near a local minimum when the derivatives are non-degenerate? This is a cross-post. Let U \subseteq \mathbb R^n be an open subset, and let f:U \to \mathbb R be smooth. Suppose that x \in U is a strict local minimum point of f. Let df^k(x):(\mathbb R^n)^... 1 vote 0 answers 76 views ### Conditions for a function to vanish almost nowhere on its support? Let f:\mathbb{R}^d\rightarrow\mathbb{R} be a continuous function and \mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\} its support. Under which conditions is it true that f≠0 (... 1 vote 0 answers 109 views ### Differential entropy under the change-of-variable with additive Gaussian noise I have two Gaussian random variablesX \sim \mathcal N(0, I), \ \ \ W \sim \mathcal N(0, \sigma\cdot I)$$and I known a parametric change-of-variable Y(\theta) = T(X; \theta). I would like to ... 3 votes 1 answer 277 views ### Gradient condition implies Hörmander condition We have tempered distribution K in \mathbb{R}^n which coincides with a locally integrable function in \mathbb{R}^n\setminus \{0\}. We call the condition$$\int_{|x|>2|y|}|K(x-y)-K(x)|dx\leq ...
Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. Let \$s&...