0
votes
0answers
19 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
35 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
96 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
54 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
79 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 ...
1
vote
0answers
114 views

quasi-convexity of a function

Can someone help me identify whether the following function is quasi-convex? Let $p>1$. For $x=(x_1,\dots,x_n),x_i>0,\sum_ix_i=1$, we define $$f(x) = -\log \sum_i (x_i/\|x\|_p)^{p-1}.$$ Plots ...
0
votes
1answer
24 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
18 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
33 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
37 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, ...
0
votes
1answer
59 views

Does boundedness imply continuity?

the partial derivatives of a given function f is bounded. does this implies that the given function is continous ? how can we prove that a function is continous when we only knw that its partial ...
2
votes
0answers
36 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
75 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
29 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
63 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
32 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
70 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
16 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
34 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
44 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
55 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
36 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
81 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
48 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
52 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
41 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 ...
0
votes
1answer
23 views

Let U be open and $f: U \rightarrow \mathbb{R}$ be partial differentiable.

The Assignment: Let $U \subset \mathbb{R}^n$ be open and $f : U \rightarrow \mathbb{R}$ be partial differentiable and let all partial directional derivatives be continous function on $U$. Show ...
1
vote
0answers
51 views

How to show $f(x)=\exp((|x|^2-1)^{-1})$ if $|x|<1$ and $f(x)=0$ if $|x|\geq 1$ is a test function?

What would be the formal argument for showing the function $f:\mathbb R^n\longrightarrow \mathbb R$, $$f(x):=\left\{\begin{array}{ccc} ...
0
votes
1answer
32 views

How to extend a continuous function from domain open in $H^k$ to a domain open in $R^k$?

Assume $f:U\mapsto V$ is continuous and of class $C^r$, and $U$ open in $H^k$ but not $R^k$. How do I extend $f$ to $g:U'\mapsto V$, $g$ is also continuous and of class $C^r$, $U'$ is open in $R^k$, ...
0
votes
0answers
56 views

Multivariable calculus and real analysis in one semester. What is the best way to study for such course?

I am in my first year of EECS and planning on taking a lot of maths classes. I have already taken single variable calculus and linear algebra and did well in them and decided to take multivariable ...
0
votes
0answers
15 views

Poisson equation on a square

Studying PDEs from the notes of my professor, and there's a part I don't understand about seeking a solution for the Poisson equation on a square. Let's start from the beginning though. We want to ...
0
votes
1answer
40 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
0
votes
2answers
68 views

Integral and area in a plane [closed]

Show that the value of the following integral is proportional to the area included in a curve $C$: $$\oint_C3y\,dx+3z\,dy-x\, dz,$$ where $C$ is a smooth, closed curve upon the $2x+2y+z=2$ plain.
4
votes
1answer
141 views

Double integral containing $e^{(b+ic)/z^2}$

I want to solve the two integrals \begin{aligned} I_3\,& = \int_{0}^{\infty} ze^{a/z^2 - z^2} dz\\ I_4\,& = \int_{0}^{\infty} \frac{1}{z}e^{a/z^2 - z^2} dz. \end{aligned} where ...
1
vote
0answers
20 views

Can every simple closed curve be approximated by a simple closed poligonal?

So, I am trying to prove that , if $C$ is a closed simple curved in $\mathbb{R}^2$ parametrized by a regular $\gamma:[a,b] \rightarrow \mathbb{R}^2$ then there exists a $\delta>0$ so that, if ...
1
vote
0answers
27 views

Does every Lipschitz curve have finite length?

So I thought I had proven that every Lipschitz curve had finite length, but then I read what I think is a counterexample: Let $\gamma :[0,1] \rightarrow \mathbb{R}^2$ the parametrization of a path ...
1
vote
2answers
49 views

Differentiability of multivariate functions.

I would appreciate if someone could share some intuition as to the geometric meaning of differentiability condition of functions defined on $\mathbb{R}^n$. Such a function say ...
1
vote
3answers
31 views

Linearity of $D_{v}f(0)$

Exercise: Let $f:\Bbb{R^n}\rightarrow \Bbb{R}$ be homegeneous of degree 1, in the sense that $f(tx)=tf(x)$ for all $x\in\Bbb{R^n}$ and $t\in\Bbb{R}$. Show that $f$ has directional derivatives at 0 ...
0
votes
0answers
17 views

Nature of Hessian of a function of a matrix

If input to a differentiable function is a matrix, what is the nature of Hessian of the function? Is it a tensor or something? This is a simple question, but I guess I am not sure where refer to, to ...
1
vote
3answers
104 views

Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$

I would like to compute the following, $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$ It is obvious that we can rewrite the integral above to, $$ ...
1
vote
1answer
50 views

Continuity conditions for multivariate functions.

Is the following true ? A proof or counter-example or reference would be nice. A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continous at $(0,0)$ if and only if if for all $a, b$, the limits ...
1
vote
1answer
29 views

surface area of the graph of a convex function

I started out with the following question: Say $\Omega$ is a nice bounded domain in $\mathbb{R}^{n-1}$. (One can imagine it being a unit ball in $\mathbb{R}^{n-1}$.) Let $f:\Omega\rightarrow ...