# Tagged Questions

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### Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
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### Partial derivative and derivative.

I want to show that if $f:\mathbb{R}^n\to \mathbb{R}$ and $df_a$ is the derivative of the function at $a$ then $df_a(v)=\displaystyle\frac{\partial f}{\partial v}(a)$. I saw a few proofs of this ...
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### If $f'(z_0)\neq 0$ then $f$ has an holomorphic inverse.

Problem: Let $U\subset\mathbb{C}$ be an open set, $f:U\to\mathbb{C}$ an holomorphic function of class $C^1$ and $z_0\in U$. Prove that if $f'(z_0)\neq 0$ then there exists a neighborhood $V$ of $z_0$ ...
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### minimum and maximum of $f(x,y)=\sin(x)+\sin(y)-\sin(x+y)$

we are asked to find the minimum and maximum of the function$f:A \to A$ $f(x,y)=\sin(x)+\sin(y)-\sin(x+y)$ Where $A$ is the triangle bound by $x=0$,$y=0$ and $y=-x+2\pi$ I'd like someone to review ...
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### Cauchy inequality proof

I am studying cauchy inequality proof from notes I have from my class$$(\forall\vec{x},\vec{y}\in\mathbb{R}^n):|\sum_{i=1}^{n}x_iy_i|\le||\vec{x}||\cdot||\vec{y}||$$ We choose $\vec{x},\vec{y}$. And ...
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### If first 1 by 1 upper left submatrix (principal minor) = 0, conclude straightaway saddle point ? - Question 8

Find all local extremal points for the function $f(x,y) = x^3 - 3xy+y^3$ and classify their type. For $H(f)(0,0),$ I see that $D_1 = \det [0] = 0$. So according to the criteria that I already posted ...
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### For $f(x,y,z,\ w)=x^{5}+xy^{2}-zw$, how is this stationary point $\;$ a saddle point? - Question 14

14. a$)$ Find all stationary points of $f(x,y,z,\ w)=x^{5}+xy^{2}-zw$. $b)$ Classify the stationary points of $f$ as local maxima, local minima or saddle points. Provided Solution a $)$ We compute ...
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### Harmonic map into sphere

Let $B$ be the unit ball and $S$ the unit sphere in $\mathbb{R}^3$. Consider the map $u: B\rightarrow S$ defined as: $$u^j(x)=\frac{x_j}{|x|}\quad\forall \ j =1, 2, 3.$$I ...
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### Find the mistake in my proof of partial derivatives exisiting implying differentiability.

Let $\Omega\subseteq \mathbb{R} ^2$ be open. Let the function $f: \mathbb{R} ^2 \mapsto \mathbb{R}$ have partial derivatives $f_x'$ and $f'_y$ at every point in $\Omega$. I will now try to prove that ...
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### proving existence of diffeomorphism

In my hand out of manifold, I found the following lemma but there is no proof there: Let $U\subseteq\mathbb{R}^m$ be open and pick some $a\in U$. Suppose that $f:U\mapsto \mathbb{R}^n$ is a smooth ...
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### Show that $f$ is harmonic

Let us consider the function: $$f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right]$$ My ...
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### Applying the implicit function theorem

Let $F:\mathbb{R}^2\to\mathbb{R}$ of class $C^2$ and $\displaystyle\frac{\partial f}{\partial v}(u,v)\neq0\; \forall(u,v)\in\mathbb{R}^2$. If $(x_0,y_0,z_0)\in\mathbb{R}^3$ is such that ...
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### If $|\nabla F| > 1$ and $|F| \le 1$, is there a zero nearby?

I saw this claim, stated without much explanation, in an article I'm reading: Let $F:\mathbb{R}^n\to\mathbb{R}$ be a $C^1$ function which satisfies $|\nabla F|>1$ everywhere. We know that ...
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### Compute a line integral with Green's theorem

Consider the two-dimensional vector field $\mathbf{F}=(x^2-y, x+y^2)$, with $\mathbf{x}=(x,y)\in \mathbb{R}^2$. Compute the divergence and the curl of $\mathbf{F}$. Compute the line ...
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Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ... 1answer 225 views ### Teaser or fun calc equation to surprise husband (physicist/EE) at work I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for ... 0answers 121 views ### Proof of inverse function theorem by approximation property In proving the inverse function theorem using the approximation characterization of the derivative, we are given F:\mathbb{R}^n \to \mathbb{R}^n such that$$F(p_0 + h) - F(p_0) = DF_{p_0}(h) + ...
I was solving the following exercise: "Let $x,y \in \mathbb{R}^n$ be nonzero such that if $z$ is orthogonal to $x$ then $z$ is orthogonal to $y$. Prove that $x$ and $y$ are colinear". My idea was: ...
### Calculate the volume of $T = \{(x,y,z) \in \mathbb R^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\}$
Calculate the volume of $T = \{(x,y,z) \in \mathbb R^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\}$ so I said that the integral we need is $\iint_{D} {x^2 + y^2 dxdy}$. But when I ...