2
votes
1answer
40 views

Inverting a function

I am stuck with the following problem I am supposed to find the inverse of the function $g$ with $2$ variables, where $$\begin{align*}g&: R^2\to R^2 \\ g&(x,y)=(2ye^{2x}, xe^y)\end{align*}$$ ...
2
votes
0answers
31 views

checking whether functions satisfy Inverse Function Theorem.

I've my exam tomorrow and this question is expected to come but donot know how to solve... Here's the INVERSE FUNCTION THEOREM stated in my notes: It says: Let $E\subseteq \mathbb R^n$ be open ...
0
votes
2answers
23 views

Proof of a property of directional derivative

I am stuck with the proof of the following proposition. I am given that the directional derivative of f exists at a with respect to the vector u, and I should prove that f'(a,cu)=cf'(a,u) I tried to ...
3
votes
1answer
157 views

calculation of normal derivative

Suppose $\Omega$ is a bounded region in the plane $\mathbb{R}^2$ with smooth boundary $\partial\Omega$. Suppose $u$ is a smooth function in $\Omega$. I want to calculate ...
1
vote
1answer
26 views

general mean value theorem

Can anyone give me the intuitive explanation of the general mean value theorem stated in my notes as under: Let $f:U\rightarrow \mathbb R$ and $U\subseteq \mathbb R^n$ and let $f$ is differentiable ...
0
votes
1answer
41 views

A point is a saddle point when $D<0$

Show that if $x'=(x,y) \ \ $ is a critical point of a $\mathcal{C}^3$ function $f$ such that: $$D=f_{xx}(x')f_{yy}(x')-(f_{xy}(x'))^2<0$$ Then there are points $x$ and $z$ near $x'$ such that ...
2
votes
1answer
33 views

Polar Coordinates in $\mathbb R^n$

After proving Fubini-Tonelli theorem a formula on polar coordinates in $\mathbb R^n$ is given in my class as follows. Let $f$ be a real-valued integrable function on $\mathbb R^n$ and $S^{n-1}$ be the ...
2
votes
0answers
56 views

Least sum of power of distances

Let $n$ points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of distances $\|A_1X\|^q+ \|A_2X\|^q + ... +\|A_nX\|^q $ (where $q \in \mathbb{Q}$). Are there any general ...
0
votes
0answers
10 views

characterisation of continuity of a function in two variables in polarcoordinates

As the the titel of my question already indicates the question I have is about continuity. I know the "$\epsilon-\delta$ Definition" of continuity and think I have understood it. On the internet I ...
0
votes
0answers
11 views

Show that $ f$ is strongly differentiable at $x_0$ .

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that f is strongly differentiable ...
0
votes
1answer
45 views

Exam Question from multivariable calculus.

This question from a previous multivariable calculus exam.I don't know how to start with this question: Let $f$ be differentiable at every point of line segment joining $x_0$ and $x_0+h$.Show that ...
1
vote
1answer
70 views

Prove the following function is differentiable

I have to prove if this function is differentiable. $$f(x,y)= \begin{cases} (x^2+y^2) \sin\frac 1{(x^2+y^2)} \iff (x,y) \neq (0,0) \\0 \iff (x,y)=(0,0) \end{cases}$$ I tried proving that all of its ...
1
vote
0answers
41 views

Question about milnor's proof of hairy ball theorem

Here is a link about the proof: http://people.ucsc.edu/~lewis/Math208/hairyball.pdf My question is: after lemma 2, Milnor takes the region A to be the region between two concentric spheres. Why can't ...
10
votes
1answer
264 views

Maxima of the function $\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$

I am looking for extrema of the function $$g(a,b):=\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$$ where $a,b >0$ are real parameters. I already plotted this function and got the ...
2
votes
1answer
40 views

How to prove this assertion about $\mathbb{R}^k$?

Suppose $k \geq 3$, $x$, $y \in \mathbb{R}^k$, $|x-y| = d > 0$, and $r > 0$. Then how to prove the following assertions? (a) If $2r > d$, then there are infinitely many $z \in \mathbb{R}^k$ ...
1
vote
1answer
17 views

If $f$ is $C^1(U)$) , are $D_i f_j$ where $i=1,\ldots,n$ and $j=1,\ldots,m$ are all continuous on $U$?

$f$ is a function from an open set $U$ in $R^n$ to $R^m$ then $f=(f_1,f_2,\ldots,f_m)$, I am confused whether the following are true: If $f$ is continuous on $U$, does that imply that ...
2
votes
1answer
57 views

Prove that g is continuous.

Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a continuous function and define $g : \mathbb{R}^{n} \rightarrow \mathbb{R}$ by $g(x) = |f(x)|, x \in \mathbb{R}^{n}$ Prove that g is ...
1
vote
1answer
43 views

Why such a complicated counterexample to differentiable function, which has discontinuous partial derivatives

Here a counterexample is given, that a differentiable function has not necessarily continuous partial derivatives, but I asked myself why such a complicated example is given? Would simply $$ f(x) = ...
1
vote
3answers
44 views

triple integral and limits

$$\iiint\limits_H (x^2+y^2) \, dx \, dy \, dz \\ H=\{(x,y,z) \in R^3: 1 \le x^2+y^2+z^2 \le9, z \le 0 \}$$ I'm using Spherical coordinate system: $$x=r\cos \theta \cos\phi $$ $$y=r\cos \theta ...
0
votes
1answer
20 views

line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied ...
4
votes
1answer
49 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), ...
2
votes
2answers
140 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 ...
0
votes
0answers
30 views

Calculating area using change of variables

Let $X(x,y) = (1,1)$ and let $(u,v) = F(x,y) = (x-y^2,2y)$. Find $F_*X$ in terms of u and v. Find $F^*du\land dv$. Use the change of variables $(u,v) = F(x,y) = (x-y^2,2y)$ to calculate the area in ...
2
votes
1answer
41 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
0answers
43 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 ...
4
votes
1answer
122 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 ...
3
votes
2answers
252 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
73 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
1
vote
1answer
26 views

Tangent line to a curve statement

I am having problems understanding some parts of the proof of some statement related to tangent line to a curve. I'll copy the exact statement and proof and then my doubts. Statement If $\mathcal C$ ...
0
votes
1answer
34 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
56 views

Checking if the Hessian is the derivative of the gradient

Suppose $f: \Bbb R^n \to \Bbb R$. I have a code that computes the gradient of $f$. I have another code that computes the Hessian of $f$ times a vector. Now I want to check if they are correct. ...
2
votes
1answer
49 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
1answer
31 views

Lipschitz function proof

Statement Let $F(t,X)=A(t)X+b(t)$ with $A(t) \in \mathbb R^{n\times n}$ and $b(t) \in \mathbb R^n$. If the coefficients $a_{ij}(t)$ and $b_i(t)$ are continuous functions of the variable $t$ in a ...
3
votes
1answer
51 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
81 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
31 views

Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors?

We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors??
2
votes
1answer
169 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
0
votes
2answers
28 views

Question on derivation of vector identites and using some symbolic manipulations

Let $f,g : \mathbb R^n \to \mathbb R$, then for the gradient we have the product rule $$ \nabla(fg) = (\nabla f) \cdot g + f \cdot (\nabla g). $$ And by $\Delta(f) = \mbox{div}(\nabla(f)) = \nabla ...
1
vote
2answers
67 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 ...
1
vote
1answer
43 views

How to prove that $F(x,y)=(f(x)h(y),g(y))$ is a diffeomorphism?

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be given by $F(x,y)=(f(x)h(y),g(y))$, where $h:\mathbb{R}\to\mathbb{R}$ is a diferentiable function and $f,g:\mathbb{R}\to\mathbb{R}$ are diffeomorphisms. ...
4
votes
1answer
50 views

How to show that $\varphi(x,y)=(x+f(y),f(x)+y)$ is bijective?

Let $f:\mathbb{R}\to\mathbb{R}$ be a $C^1$ function such that $|f'(t)|\leq k<1$ for all $t\in \mathbb{R}$. Let $\varphi:\mathbb{R}^2\to\mathbb{R}^2$ be the function given by ...
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
59 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
65 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 ...
1
vote
1answer
30 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 ...
1
vote
0answers
22 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
69 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
3answers
181 views

Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...
0
votes
2answers
51 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 ...
3
votes
1answer
43 views

$f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$

Is there exists $f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$ for all ...