# Tagged Questions

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### Solving the Telegraph Equation using Partial Differential Equations and Sturm-Liouville theory

I've been asked to do the following question, and I've got through the brunt of it (so this is going to be a rather long question...), but I'm just having a bit of trouble applying Sturm-Liouville ...
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### geometric interpretation of the norm: $\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$

Let $p:\mathbb R^2 \to \mathbb R$ be a norm so that $$\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$$= $${{\|\vec x\|_1\over 3}}+{2\|\vec x\|_\infty\over 3}$$ The thing is that I need ...
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### Change of variables theorem in the case $L^1_{\textrm{loc}}(U)$?

I'm trying to write a version of the change of variable theorem for the case of locally integrable functions on open subsets of $\mathbb R^n$. Statement: Let $U, V\subseteq \mathbb R^n$ be open sets ...
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### Why are the partial derivatives of $f(x,y)=xy/(x^4+y^4)$ equal to $0$ at $(0,0)$?

Read "$x=0$ or $y=0$" as $x=0\cup y=0$ I don't understand the section of the solution highlighted in green. For instance if: $f(x)=e^{-1/x^2}$ for $x\neq0$ and $f(0)=0$ I cannot just say ...
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### Exchange the order of the two limits

Suppose both limits exist, when is it true that $$\lim_{x\to a}\lim_{y\to b} f(x,y) = \lim_{y\to b}\lim_{x\to a} f(x,y) ?$$ and further when is it true that these two limits are equal to ...
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### Prove that $f(x,y) = 1/(x^2 + y^2)$ has limit $\infty$at $(0,0)$

The question is: "prove that function $f(x,y) =$ $1 \over (x^2 + y^2)$ have limit $\infty$ in point $[0,0]$. This is pretty standard question, and even my book answers it immediately afterwards, ...
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### Average value of function over sphere

Here is another qual problem. Suppose $f:\mathbb{R}^3 \mapsto \mathbb{R}$ is $C^2$, and define the (scaled) average function $A(r)=\int_{S^2} f(rn) \:d\sigma(n)$, where $\sigma(n)$ is the usual ...
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### Determining Lipschitz constant for a special vector field

Let us be given a vector field $v: C \subset \mathbb R^n \to \mathbb R^n$ that has the special structure given by $$v(x) = \alpha(x) \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}$$ with a ...
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### Showing a two-variable function is continuous

The problem asks to show that f(x,y) = \left\{ \begin{align} \frac{x^3y^2}{x^4+y^4}, & (x,y) \neq (0,0), \\ 0, & (x,y) = (0,0), \end{align} \right. is continuous at the origin, however it ...
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### Definition of cluster point

I'm studying if the book Multidimensional Real Analysis by Duistermaat and the definition of cluster point is: A point $a \in \mathbb{R}^n$ is said to be a cluster point of a subset $A$ if for every ...
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### a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$\iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
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### Evaluate integral $\int\int xe^{xy} dx dy$, strange result after rearranging

I have to compute the following integral $$\int_{-1}^0 \int_0^1 x\cdot e^{xy} dx dy$$ It exists according to WolframAlpha. Now I want to evaluate it, let $\varepsilon > 0$, then \begin{align*} ...
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### Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
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### $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
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### Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

The Assignment: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$. A ...
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### Verification of Stokes Theorem

I want to verify Stokes Theorem for the surface $$\Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \}$$ and the vector field $F(x,y,z) := (y,z,x)$. For this I use the ...
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### Find the directional derivative of the scalar field

Find the directional derivative of the scalar field: $f(x,y,z)=\log(x^2+y^2+z^2)$ at $P_0(1,1,1)$ in the direction of the straight line $\ P_0P$ where $P=(3,2,1)$ What I have done: ...
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### Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$\mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}.$$ Prove that ...
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### Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
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### If $a$ is a limit point of $f^{-1}(b)$, then the linear mapping $f'(a)$ is not injective.

Let $U\subset\mathbb{R}^m$ be an open set and $f:U\to\mathbb{R}^n$ a differentiable function. Suppose that there exists $b\in\mathbb{R}^n$ and $a\in U$ such that $a$ is an accumulation point of ...
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### Lagrange multipliers and angle between vectors

Can someone please help me with solving this question? I'm new to learning this and I'm not at all sure if what i've done is correct... The question is: the plane $4x-3y+8z=5$ intersects the cone ...
### limits using $\epsilon - \delta$ to prove two variable function
I'm trying to use the $\epsilon - \delta$ argument to prove $\lim_{(x,y) \rightarrow (1.1)} \frac{2xy}{x^2+y^2} =1$. I know that I need to show that $\forall \epsilon>0, \exists \delta>0$ ...
I want to show that the function $f$ is discontiunous. $f$ is defined as follows: f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...