# Questions tagged [vector-analysis]

Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).

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### A question about integral transformation from the Unit sphere to Euclidean space.

I want to show the following identity about integral on spherical coordinate system: Let $\mathbb{S}^{d-1}$ denote the unit sphere of the Euclidean space $(\mathbb{R}^d,||\cdot||)$ equipped with the ...
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### On the properties of smooth surfaces

My course material has stated some properties for smooth surfaces and I would like to get some intuition or clarification for these. Let $f = (f_1, . . . , f_{n−m}) : U → \mathbb{R}^ {n−m}$ be a $C^1$...
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### Differentiability using gradient of a scalar function [closed]

How should I approach this problem? U and V are differentiable functions of x,y and z. Show that a necessary and sufficient condition that u and v are functionally related by equation F(u,v) =0 is ...
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### Doubt about notation in Robust Optimization

I'm studying the pricing model under Robust Programming here described from page 41 to page 45. I don't understand what's the meaning of subscripts $i$ and "second" $t$ referred to dual ...
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### Norm of a defined function

I know that for a function $f: R^n \times R^n \longrightarrow R$ \begin{align*} f(x,y ): & = \frac{1}{2} \|x- y\|^2_2 \\ &= 1/2(x^\top x - x^\top y - y^\top x +y^\top y) \end{align*} How ...
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### Gradient of norm squared [closed]

I saw this equation in KKT conditions .Could someone please provide a proof for the following rule: enter image description here How it is gradients of Lagrangian in terms of zn ؟
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### Proving spheres are orthogonal

Given two spheres in $\mathbb{R}^3$: $x^2+y^2+z^2=2ax; \ \ \ x^2+y^2+z^2 = 2by$ and $a,b>0$, and $\gamma$ the intersection of the spheres, show that for any $p_0 \in \gamma$, the spheres are ...
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### How can I simplify calcul of gradient?

Goal: Given $f(\alpha) :=\dfrac{1}{2} \|y-X\alpha\|_2^2$ I want to show that $\nabla f(\alpha)=0\iff X^T X\alpha = X^T y$ where: $X\in\mathbb{R}^{n\times p},\alpha\in\mathbb{R}^p,y\in\mathbb{R}^n$. If ...
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### What is wrong with this way of calculating a volume element?

I know that what follows is wrong and I know one needs to use the jacobian to calculate a new volume element $dV$ after a change of variables. However, I don’t understand why the following goes wrong: ...
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### Relationship between Conservative Vector fields and holomorphic functions exerting a primitive

Main thoughts I am currently studying Complex Analysis and I have been noticing the striking resemblance between the requirements for a function to exert a primitive on some open set U and the ...
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