0
votes
0answers
17 views

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 ...
1
vote
1answer
21 views

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 ...
4
votes
1answer
36 views

Mixed partial derivatives are different

Let $f: \Bbb R^2 \to \Bbb R$ be defined as $$f(x) = \left\{ \begin{matrix} x_1^2 \operatorname{arctan} \left( \frac{x_2}{x_1} \right) - x_2^2 \operatorname{arctan} \left( \frac{x_1}{x_2} \right), ...
1
vote
1answer
39 views

Calculating multi-variable limit.

I am struggling to find a way to approach this limit $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2y+x^2y^3)}{x^2+y^2}$$ I would greatly appriciate if You could explain to me how to solve it or at least show ...
0
votes
0answers
25 views

Sufficient Conditions for Multivariate Decreasing Function

I found the following helpful theorem concerning decreasing functions but it's only valid for $\varphi:\mathbb{R}\rightarrow \mathbb{R}$, I'd like to know if it can be extended to the ...
2
votes
2answers
75 views

Why is continuous differentiability required?

I have two questions. My book proves that if $f:\mathbb{C}\rightarrow \mathbb{C}$ is a holomorphic function, then it satisfies the Cauchy-Riemann equations, and if we look at the function as $F: ...
2
votes
2answers
103 views

A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at http://math.stackexchange.com/a/892212/168832.) Is the following true (for all n)? "If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously ...
2
votes
1answer
16 views

On the Continuity of the Jacobian of a diffeomorphism

Let $\phi:U\longrightarrow V$ be a diffeomorphism between the open sets $U, V\subseteq \mathbb R^n$. Provided $J\phi(x)\neq 0$ for all $x\in U$ we have a map $$J\phi:U\longrightarrow GL_n(\mathbb R), ...
0
votes
1answer
18 views

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 ...
0
votes
2answers
55 views

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 ...
5
votes
1answer
138 views

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 ...
-1
votes
1answer
37 views

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, ...
2
votes
1answer
40 views

direction limits and double limit

Let $f(x,y)$ be a function of two variables. What is the counterexample that there exists $A$ s.t. for all $\theta$, $$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$ but double limit $$ ...
1
vote
1answer
37 views

Improper integral $\int_{B}\frac {1}{|x|^\alpha}dV$

Let B be the ball $|x|\le 1$, $x\in R^n$. For what $\alpha$ does $$\int_{B}\frac {1}{|x|^\alpha}dV$$ exists? I find it hard when it comes to generalize this statement in $R^n$. I've been able to do ...
4
votes
0answers
52 views

Improper multivariable integrals

I'm having trouble with the integral $$\iiint_{1\le x^2+y^2+z^2 }\frac{\mathrm{d}x~\mathrm{d}y~\mathrm{d}z}{xyz}$$ this is what I've done so far: $$\lim_{b\to +\infty}\int_1^b \int_0^{2\pi} ...
1
vote
0answers
42 views

Are the two properties of a function equivalent?

$f(x)$ is a function defined on $\Bbb R^n$. $A$: $\forall x,y$ $$ |f(y)-f(x)-\nabla f(x)^T(y-x)| \le \frac{\beta}{2}\|y-x\|_2^2 $$ $B$: $\forall x,y$ $$ \| \nabla f(y)-\nabla f(x)\|_2 \le \beta ...
0
votes
1answer
53 views

Proof of the rank theorem in Rudin's PMA book

I am studying Rudin's proof of the rank theorem (theorem 9.32 in Principles of Mathematical Analysis.) We have an invertible function $H(x)$ defined on an open set. He claims we can "shrink" the open ...
0
votes
0answers
21 views

Theorem 4.6 in Spivak's Calculus on Manifolds

Could you elaborate on the proof please? This is how I would prove the theorem: Since $\Lambda^n(V)$ is $1$-dimensional, $\omega=\alpha(\phi_1\wedge\phi_2\wedge...\wedge\phi_n)$ for some $\alpha$ ...
5
votes
0answers
39 views

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. $f$ ...
4
votes
1answer
104 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
1
vote
0answers
56 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
4
votes
2answers
88 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
0
votes
1answer
26 views

Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...
2
votes
1answer
23 views

Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
0
votes
0answers
13 views

Expression for volume without changing variables

My question is the same as this: Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables However, my solution, while makes perfect sense to me, is slightly ...
0
votes
1answer
34 views

Surface integral over d-sphere for $|x-y|^{(2-d)}$

I am looking for $\int_{S_{r}(0)}|x-y|^{2-d}dS_{y}$ for $x\neq 0$. The parametrization is hard to work with and the integrand is not rotationally symmetric. I will post any updates. any ideas thank ...
1
vote
2answers
36 views

Finding extremal values on a set

Let $f(x,y)=(x-1)^2+y^2+xy$. Find the maximal and minimal values of $f$ on the set $M=\{(x,y):|x|+|y|\leq4\}$. Attempt: By taking partial derivatives and solving the homogenous algebraic system we ...
1
vote
1answer
39 views

Swapping limit with inner product.

The exercise is: Given $\xi : U \subset \Bbb R^m \rightarrow \Bbb R$, $U$ open, given by $\xi (x) = \langle f(x), g(x) \rangle$, where $f,g: U \rightarrow \Bbb R^p$ are differentiable functions, ...
-1
votes
1answer
66 views

Does the boundedness of partial derivatives imply continuity?

Suppose that the partial derivatives of a given function $f$ are bounded. Does this imply that the given function is continuous? Remark For functions of one variable this is true, because every ...
2
votes
0answers
37 views

Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
4
votes
1answer
78 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
0
votes
0answers
22 views

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 ...
2
votes
1answer
32 views

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 ...
1
vote
2answers
64 views

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 ...
0
votes
1answer
34 views

Finding volume between plane and paraboloid

Find the volume between bounded by $z=4$ and $z=x^2+y^2$.(Answer: $8\pi$) I thouhg using dievergence theorm ($\iint_KdivFdxdydz=\iint_SF\cdot\hat{n}dS$) for $\vec{F}=\big(\frac x 2,\frac y ...
2
votes
1answer
71 views

How to define integration over the boundary of a curve?

When learning about Stokes' theorem ($\int_{\partial \Omega} \omega=\int_{\Omega} \mathrm d \omega$), we are told that it is just a generalization of the 2nd Fundamental Theorem of Calculus $(\int_a^b ...
2
votes
1answer
20 views

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 ...
1
vote
1answer
36 views

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 ...
0
votes
1answer
46 views

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*} ...
1
vote
1answer
57 views

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 ...
2
votes
4answers
66 views

$f$ 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 ...
2
votes
1answer
37 views

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 ...
1
vote
1answer
28 views

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 ...
2
votes
3answers
82 views

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: ...
1
vote
0answers
19 views

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 ...
2
votes
1answer
67 views

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 ...
0
votes
2answers
50 views

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 ...
0
votes
1answer
16 views

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 ...
4
votes
2answers
72 views

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 $\forall \epsilon>0, \exists \delta>0$ ...
0
votes
1answer
43 views

What am I doing wrong in this continuity check?

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 ...