2
votes
1answer
42 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 $$\begin{align*}g&: R^2\to R^2 \\ g&(x,y)=(2ye^{2x}, xe^y)\end{align*}$$ ...
0
votes
1answer
18 views

Determining when $f(x,y) = x^{4/3} \sin(y/x)$ ($x \ne 0$) is differentiable.

Let $f$ be defined as follows: $$ f(x,y) = \left\{ \begin{array}{lr} x^{4/3} \sin(y/x) & \mathrm{if\ } x \ne 0 \\ 0 & \mathrm{if\ } x = 0. \end{array} \right. $$ I am asked to determine where ...
0
votes
2answers
22 views

Showing that a multivariable function is one to one

I am stuck with the following problem I am given the function $f$ such that $f(x,y)=(x^2-y^2,2xy)$ I am supposed to show that the function is one to one. For a function to be one to one, $f'>0$. ...
0
votes
0answers
24 views

Question about a proof in Lang's undergraduate analysis

This is from page 580 of Lang's undergraduate analysis (2nd edition). I have difficulty in understanding the proof, hope that someone here can enlighten me. My questions are: i) On line 5 of the ...
0
votes
1answer
19 views

Find $f,g$ for a counterexample of multivariable limit

Are there any $f,g : \mathbb R^2 \rightarrow \mathbb R$ such that $\lim_{x \rightarrow 0} f(x) = 0, \lim_{y \rightarrow 0}g(y) = 0$ but $$ \lim_{(x,y) \rightarrow (0,0)} \dfrac{\log(1+f(x)g(y))}{g(y)} ...
2
votes
1answer
32 views

Why do we need an open set to define differentiability? [duplicate]

The general definition of a differentiable mapping is, Let U be an open set in Rn, and let ‘a’ be in U and f:Rp. Then f is a differentiable mapping at ‘a’ if there exists a Df(a) in Hom(Rn, Rp) such ...
1
vote
1answer
24 views

Definition of a functions with respect to partials

I am stuck with the following problem: I am given that $$F(x,y)=f(x,y,g(x,y)) =0.$$ I am asked to show $D_1g$ and $D_2g$ with respect to the partials of $f$ My idea was to write that $DF=DfDg$ ...
1
vote
1answer
18 views

Tangent space and implicit function theorem

Let's say we have a $C^1$-function $f:X\to\mathbb{R}^m$ ($X\subset\mathbb{R}^{n+m}$ an open set) and the rank of the matrix $Df(x)$ is $m.$ We'll let $Z=\lbrace x\in X:f(x)=0\rbrace$ and take some ...
5
votes
1answer
29 views

Multiple Integration order doesn't agree.

Let $0<x,y,t,z<1$ with the additional condition: $$\begin{align*} x &< t\\ \wedge & \ \\ y &<z \end{align*}$$ Call the set of all $x,y,t,z$ satisfying the above conditions ...
0
votes
2answers
29 views

If a continuous function of two variables has finite zero set, then it does not change sign

If $f$ is a continuous function from $\mathbb R^2 \rightarrow \mathbb R$ such that $f(x)=0$ for only finitely many values of $x\in\mathbb R^2$. Can we conclude that $f(x)\leq 0$ or $f(x)\geq 0$ for ...
0
votes
2answers
23 views

The maximum volume of Tetrahedron

A optimization problem: Get the maximum volume of a tetrahedron its 4 vertices on the surface of cube whose edge length is 1 . From the geometrical intutition ,we can get : Selecting ...
2
votes
1answer
31 views

Existence of partial derivative

I know how to compute partial derivatives of functions with more than one variable. But how can i assert that the partial derivatives of a given function exist at a point without computing it? ...
0
votes
0answers
10 views

Visualisation of gradient and computation?

I am learning differentiability in several variables, and I am stuck. I cannot visualize the definition that $\lim: \lim_{f(a+h)-f(a)-ch/h\to 0} = 0$ The book indicates that $ch$ is called gradient. ...
1
vote
2answers
33 views

Epsilon-delta proof with $x$ and $y$ defined

I am stuck with the following problem. The question is as follows: prove that for all $x$ in $[0,2]$, there exists $y$ in $[0,2]$ such that the function $f(x,y)=0$. The function $f$ is defined as ...
1
vote
1answer
19 views

necessity of continuous partial derivatives?

In my old book in calculus, it says that a sufficient condition for the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ to be differentiable at an internal point z, is that the partial derivatives ...
1
vote
2answers
31 views

Projection function preserving distance

Given $n>1$. I wis to construct a function $f:\mathbb{R}^n\mapsto\mathbb R$ such that $$\|X_1-X_2\|\le\|Y_1-Y_2\|\implies|f(X_1)-f(X_2)|\le |f(Y_1)-f(Y_2)|$$ for all $X_1,X_2,Y_1,Y_2\in\mathbb{R},$ ...
1
vote
0answers
31 views

The intuition of the rank theorem

In Rudin's Principle of Math Analysis, I know the proof of the rank theorem. However, I fail to understand its content. 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ ...
0
votes
1answer
15 views

Relation between directional derivatives and derivative? [duplicate]

Is it possible to say that the directional derivatives of a function f at a exists but f is not differentiable at a? If so, why? I cannot get the intuition about it. Could someone please elaborate on ...
0
votes
1answer
45 views

Prove that the range of $f$ is not closed.

I am having trouble computing the range of a function $f : \mathbb{R} \rightarrow \mathbb{R}^2$. My thinking would be that you just find the range of $\frac{2x}{x^2 +1}$ and the range of $\frac{x^2 ...
5
votes
2answers
65 views

How to prove set $S=\{(x,y)\in \mathbb{R}^2~\vert~y>x^2\}$ is open (I need some hints)

Q: Prove $S=\{(x,y)\in \mathbb{R}^2~\vert~y>x^2\}$ is in open in $\mathbb{R}^2$. One of my intuitions: $S=\{(x,y)\in \mathbb{R}^2~\vert~y>x^2\}=\{(x,y)\in \mathbb{R}^2~\vert~y-x^2>0\}$ ...
0
votes
0answers
32 views

Another versions of squeeze theorem

I have two questions regarding squeeze theorem: 1.Can I use squeeze theorem in higher dimensions? (I'm almost sure yes) If we have the strict inequality: $x_n\lt z_n\lt y_n$ when $\lim x_n=\lim ...
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 ...
1
vote
2answers
30 views

Can the components of a continuous injective function fail to be injective?

Can a continuous function $f = (f_1,...,f_m) : \Bbb R^n \rightarrow \Bbb R^m$ be injective while each of its components $f_j : \Bbb R^n \rightarrow \Bbb R$ is not injective?
2
votes
0answers
29 views

What happens when the determinant of the Jacobian is zero?

Let $f \colon \Bbb R^n \rightarrow \Bbb R^n$ be a continuously differentiable function. Let $f'(x)$ denote the Jacobian matrix of $f$ at the point $x \in \Bbb R^n$. I would like to show that if ...
1
vote
0answers
26 views

Questions on Strongly Differentiability.

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 ...
1
vote
1answer
46 views

Proof $\phi(r\cos\theta,r\sin\theta)=\theta$ is not Lipschitz.

Taking $S=\lbrace(x,y)\in\mathbf{R}^2:1<x^2+y^2<9\rbrace\backslash ( [0,\infty)\times\lbrace 0\rbrace),$ and defining $\phi:S\to\mathbf{R}$ as $\phi(r\cos\theta,r\sin\theta)=\theta$ for ...
3
votes
1answer
35 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 ...
1
vote
1answer
64 views

Are the conditions for multivariable integrability the same?

For a single variable function, the function needs to have a finite number of discontinuities and must be bounded over the interval of integration for it to be Riemann integrable over that interval. ...
0
votes
1answer
63 views

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ prove the following:

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ such that $ f(0)\neq -1$ and $\displaystyle\int_{0}^{b} f(t) \, dt=0$ Show that the equation $\displaystyle\int_{x}^{a} f(t) ...
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 ...
1
vote
1answer
49 views

A version of the Fundamental Theorem of Calculus for two variables

Let $f(x,y)$ be differentiable in the rectangle $R=[a,b]\times[c,d]$, show that the function $\displaystyle F(x,y)=\int_{a}^{x} f(t,y) \, dt$ is also differentiable in $R$ and that ...
0
votes
1answer
32 views

Is a multivariable function continuous iff it is continuous with respect to each variable?

I am very uncertain when it comes to understanding the continuity of multivariable functions. If we have, for example, a function $f: \mathbb{R}^{4} \to \mathbb{R}$, and we denote the four variables ...
2
votes
3answers
63 views

Conplex/real Integration and poles of function

So I am working on the following problem: Let $\Delta $ be the unit disk centered at origin, and $f$ is holomorphic on $\Delta-\{0\}$. If $$\int_\Delta|f|dxdy<\infty$$ show that $f$ has at most a ...
2
votes
0answers
12 views

Show that the innerproduct of two vector is a differentiable mapping

The question is to show that the innerproduct is a differentiable mapping. Define $g: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ such that $ g(x_1,x_2) = <x_1|x_2>$. We use the ...
0
votes
1answer
30 views

Question regarding minimum

I'm wondering whether the following should be true. Suppose $f(t)$ is a real valued function (say, on $\mathbb{R}^n$) which attains its minimum at a unique point, say $x^*$ in the closure of a set ...
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 ...
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 ...
1
vote
0answers
19 views

System of PDE's with unknown functions

So by messing around with some stuff in my own research I came across this problem and I have no idea how to proceed. I suspect it may have something to do with solving systems of PDE's but I could be ...
0
votes
1answer
29 views

Differential equation with no nontrivial periodic solution

We are given $f=(f_1,f_2): \mathbb{R}^2 \rightarrow \mathbb{R}^2$, $C^1$ class with the property: $$(1) \ \ \ \forall_{(x,y)\in\mathbb{R}^2} \frac{\partial f_1}{\partial x}+\frac{\partial ...
0
votes
1answer
31 views

Definition of $C^1$ functions with values in $\mathbb R^m$

My analysis textbook defines a $C^1$ function $f:\mathbb{R^n}\to\mathbb{R^m}$ as one in which for each component function $f_i, 1\leq i\leq m$ the partial derivative $\frac{\partial f_i}{x_j}$ exists ...
0
votes
3answers
28 views

Prove that $d_n$ is a Cauchy sequence in $\mathbb{R}$

Let $(x_n$) and $(y_n)$ be Cauchy sequences in $\mathbb{R}^n$ , i.e. lim$_{n,m}$ |$x_n$ − $x_m$| = $0$ and lim$_{n,m}$ |$y_n$ − $y_m$| = $0$. For each n, let $d_n = |x_n − y_n|$. Prove that $d_n$ is a ...
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 ...
0
votes
0answers
20 views

Question concerning what a certain notation means. Its usually in calculus

$x$ln$(1 + \frac{1}{x}) = 1 +$ln$(1 + \displaystyle\sum_{i=1}^n \displaystyle\frac{a_{i}}{x^i}) + O(x^{-n-1})$ for $x \rightarrow \infty$ and $n \in \mathbb{N}$. My question is as follows:I have seen ...
0
votes
0answers
39 views

function differentiable but not continuously differentiable

Can I found an example of a function $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ such that $f$ differentiable but it is not continuously differentiable and it is not invertible on a point??? Any idea? ...
1
vote
0answers
24 views

Are two definitions of differential function equivalent?

Definition 1: From https://www.math.hmc.edu/calculus/tutorials/tangentplanes/differentiability.pdf or Differentiability for a function of two variables A function $f(x, y)$ is differentiable at the ...
2
votes
1answer
46 views

Doubts on Differentiation in $\mathbb{R}^p$

I am currently reading R.G Bartle's "Elements of Real Analysis" for a one semester course in Advanced Real Analysis. In the chapter on Differentiation in $\mathbb{R}^p$, I am confused regarding the ...
2
votes
1answer
72 views

Analysis of $f(x,y,z)=\frac{\sin(xyz)} {x^2+y^2+z^2}$

$f(x,y,z)=\frac {\sin(xyz)} {x^2+y^2+z^2}$ or $0$ if $(x,y,z)=(0,0,0)$ The problem says the following: a) Where is this function continuous, and b) Where is it differentiable? At a quick glance, the ...
1
vote
3answers
62 views

How to prove that $\int_0^b\Big(\int_0^xf(x,y)\;dy\Big)\;dx=\int_0^b\Big(\int_y^bf(x,y)\;dx\Big)\;dy$?

Problem. Let $f:[0,b]\times[0,b]\to\mathbb{R}$ be continuous. Prove that $$\int_0^b\left(\int_0^xf(x,y)\;dy\right)\;dx=\int_0^b\left(\int_y^bf(x,y)\;dx\right)\;dy.\tag{1}$$ My first thought was ...
2
votes
1answer
69 views

If the partial derivatives are $0$ is a function constant?

I am trying to prove that if we have a differentiable function: $f:\mathbb{R}^2\rightarrow \mathbb{R}$, and the partial derivatives of f is 0, then f is constant on a connected set. I am using the ...
0
votes
1answer
37 views

Bounding the average of a vector valued function

Disclaimer: I edited the question so that it fits Daniel Fischer's comments and it becomes more general. I also provide an answer myself in case anyone might be interested in the solution. Question: ...