1
vote
1answer
22 views

The second derivative of g(t)= f(x(t), y(t))

For the chain rule for differentiating a function $g(t) = (f(x), y(t))$ how do you get from the identity $g'(t) = \frac{df}{dx} \cdot \frac{dx}{dt}+ \frac{df}{dy}\cdot\frac{dy}{dt}$ to the identitiy ...
0
votes
1answer
35 views

A function with positive Hessian at a critical point, without having a minimum there

I have a problem with a little instance: $f(x,y) = \begin{cases} (x^4-3x^2y^2+y^2)/(x^2+y^2) & otherwise \\ 0 & \text{(x,y)=(0,0)} \end{cases}$ This is a example of a function which ...
0
votes
1answer
20 views

Quasi-concavity of a function of two variables such as $z=(x^a + y^b)^2$

If I have a function such as $z=(x^a + y^b)^2$ with $a$ and $b$ both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative? The ...
0
votes
1answer
38 views

Finding the critical points of a function

What are the steps in finding the critical points of a function in general? Say for example, the function $$f(x, y) = 2x^3 + 11x^2 + 0.5y^2 - 2xy$$ I can't quite seem to understand the steps/method ...
0
votes
3answers
40 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
0
votes
1answer
55 views

$f(tx,ty,tz)=t^n \cdot f(x,y,z)$ [closed]

For $t>0$ and $(x,y,z)\in \mathbb{R}^3$ show that if $f(tx,ty,tz)=t^n \cdot f(x,y,z)$ then $$x*\frac{\partial }{\partial x}f+y*\frac{\partial }{\partial y}f+z*\frac{\partial }{\partial z}f=n ...
0
votes
1answer
47 views

Prove that $xy\le\frac{x^p}{p}+\frac{y^q}{q}$

Given $x\ge0$, $y\ge0$, $p>0$, $q>0$ such that $\dfrac{1}{p}+\dfrac{1}{q}=1$: Prove that $xy\le\dfrac{x^p}{p}+\dfrac{y^q}{q}$ starting by $xy=1$
1
vote
1answer
41 views

Constructing a function from level sets

Suppose we know what the projection of the level sets into the xy-plane of some function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ looks like. How can I construct a closed form for $f$ by "lifting" the ...
1
vote
2answers
28 views

A question on a multivariable continuously differentiable function

Assume $f(x_{1},x_{2})$ is a real-valued continuously differentiable function, and assume it holds that $x_2D_{1}f(x_1,x_2) - x_1D_2f(x_1,x_2) = 0$ where $D_1$ is the partial derivative with respect ...
1
vote
1answer
29 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
0
votes
1answer
29 views

Proving injectivity and surjectivity

$T(x,y,z) = (xy, yz, xz) \\ R^3 \rightarrow R^3$ My attempt to try showing surjectivity: $xy = a \\ yz = b \\ xz = c$ I want to get $x, y,$ and $z$ in terms of $a, b, c.$ So first I try $x = ...
2
votes
2answers
37 views

How to show this function is surjective

$T(x,y,z) = (2x + y + 3z, 3y - 4z, 5x)$ for $R^3 \rightarrow R^3$ I read the meaning of surjective over and over, but I don't understand how to show it algebraically. So I just guessed: $2x + y + 3z ...
0
votes
3answers
43 views

Prove the following Injective and Surjective function

Prove that the following function is bijective (both injective and surjective): $$f(x,y) = (x^2+y+x-2, x+3)$$ Usually I know how to do these, but the fact that the first x is squared, throws me off ...
1
vote
1answer
44 views

Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...
0
votes
1answer
28 views

Almost continuity implies measurability?

Trying to prove the continuity of $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ $(n>1)$ I got the following property of $ f $: for all $x\in \mathbb{R}^n $ and $(x_k)$ such that $x_k \rightarrow x$ ...
0
votes
1answer
40 views

Partial derivatives and functions equal to 0

If I have the function (family of curves) $$F(x,y,p)=(px)^2+p=0$$ I am under the impression that $$\frac{\partial F(x,y,p)}{\partial p}=2px^2+1$$ Is not always equal to $0$. Please could you ...
3
votes
1answer
41 views

Separate the variables of the function $\frac{x^2}{\sqrt{x^2+y^2}}$

Is there a way to express the function $\frac{x^2}{\sqrt{x^2+y^2}}$ as the product of two functions: $f(x)\cdot g(y)$, i.e. one in each variable? This is becasue I want to apply a convolution whose ...
2
votes
2answers
52 views

Aren't $ f’(xy) $ and $ f’(x/y)$ ambiguous notations? [Stewart P890 14.3.50b, c]

The answers so far have uncloaked a deeper problem of mine: I brook that setting $t = xy$ transforms $f(xy)$ into $f(t)$, but didn't we start with $x, y$ which are 2 separate independent ...
0
votes
1answer
16 views

Where a particular equation meets a particular axis

The question asks where the tangent plane to $z = e^{x - y}$ at $(1,1,1)$ meets the $z$-axis. Without performing any computations or even looking at the given function, based solely on the question ...
1
vote
1answer
66 views

Very interesting multivariable calculus question.

If $\displaystyle z = \frac{f(x-y)}{y}$, show that $\displaystyle z + y \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0$.
1
vote
1answer
88 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...
1
vote
1answer
45 views

Given a set of data points, how to use gradient descent to find the minimum in the function that passes from those data points?

I have a function with n parameters. I don't know the formula of the function but I can generate as many data points as I want using the function that I have. My question is, how can I find the set of ...
0
votes
1answer
29 views

introductory calculus - Help me find a function with a few properties

I was asked to find a function $f: \mathbb R^2 \to \mathbb R$ such that: 1) $f$ is continuous at $(0,0)$. 2) $f$ has directional derivatives at $(0,0)$ (does this mean $f$ is differentiable at ...
1
vote
1answer
114 views

Stacked with this Problem of Calculus

I have been struggling for quite some time with the following problem and I would really appreciate some help: Consider ...
2
votes
2answers
137 views

Demonstrate that the limit of a function of two variables does not exist

From my multivariable textbook: $$\lim_{|x,y|\to|0,0|}\frac{y^2\sin^2 x}{x^4+y^4}$$ (original screenshot) Wolfram indicates that the limit DNE, but does not list the steps used to solve. Is there ...
0
votes
2answers
33 views

multivariable calculus - find a function such that $\lim_{t \to 0} f(tx,ty)=0$

I was asked the following question: Find a function $f:\mathbb R^2 \to \mathbb R$ such that $f(x,y)$ has no limit as $x$ and $y$ approach zero, but $\lim_{t\to 0} f(tv)=0$ for all $v \in \mathbb R^2$ ...
0
votes
1answer
51 views

Total derivative of a function $f:\mathbb R^3\to \mathbb R^3$

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by $$ f(x,y,z) = (x^z,y\tan(x^2z),\frac{1}{x+y})$$ and let $c: \mathbb{R}\rightarrow \mathbb{R}^3$ be given by $$c(t) = (\cos(t), \sin^2(\pi ...
2
votes
1answer
41 views

Continuosly differentation on composite functions

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ a $C^1$ function and defined $g(x) = f(\|x\|)$. Prove $g$ is $C^1$ on $\mathbb{R}^n\setminus\{0\}$. Give an example of $f$ such that $g$ is $C^1$ at the origin ...
0
votes
2answers
114 views

degree of homogeneity

I have the function $$f(x,y)=\frac{y^b}{x^a}+\frac{x^b}{y^a}\quad a,b\gt0$$ The questions I have to answer are For which a and b is the function homogenous? Determine the degree of homogeneity My ...
0
votes
1answer
40 views

Question regarding Continuity of F(x,y)

Let $f(x,y) = \begin{cases} \frac{2(x^3+y^3)}{x^2+2y}&\text{ } (x,y)\not=(0,0)\\ 0 &\text{ }(x,y) =(0,0). \end{cases}$ show that first order partial derivatives of $f$ wrt x and y exist at ...
3
votes
0answers
63 views

What is the domain and range of this multivariable function?

What is the domain and range of the following multivariable function? $g(x,y,z) = {1\over \sqrt{(4 - x^2 - y^2 - z^2)}}$ g is real valued. So far I have that the domain is the set of a vectors ...
1
vote
0answers
81 views

Variants of the bump function.

The title of this question isn't really clear because of the 150 char limit. What I actually want to ask is this: If I would have a bump function for $-1 < x < 1$ and I would have some ...
1
vote
2answers
438 views

Prove $f(x,y) = xy/(x^2 + y^2)$ is continuous everywhere except $(0,0).$

I'd just like to ask you if my proof here is valid. I'll provide you with the method I used and if it seems ok let me know! If not, explanations would be helpful! My main approach to this question ...
1
vote
2answers
50 views

What is the difference between these two functions?

$$r(x) = \langle x, x^2-1 \rangle$$ $$f(x)=x^2-1$$ Their graph is the same, but one is called vector valued function while the other one is a regular one. I think I'll never get to understand this ...
0
votes
1answer
23 views

Boundedness of an implicit function

I have a function $h(u) : \mathbb{R}^N \rightarrow \mathbb{R}^N$ in an implicit form: $$ A(h, u) = 0. $$ What conditions could we put on $A$ to guarantee that $h$ is bounded? And if we have an ...
1
vote
1answer
42 views

differentiability and linear operators

I could give suggestions for this question. $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is differentiable in $a$, then for each $ h \in \mathbb{R}^{n}$ exists a linear transformation ...
0
votes
0answers
34 views

limit images Ball $B[a,r]$

Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ continuous function. I have difficulty proving formally that meets $$\lim\limits_{r \to 0 } f(B[a,r]) =f(a)$$ where $B[a,r]$ is the closed ball in ...
5
votes
1answer
153 views

looking for a diffeomorphism (not C1)

Let $f\colon\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ diffeomorphism with $f(B[0,1])\subset B[0,1]$ and $| \det f^{\prime}(x) |<1/2$ for all $x\in B[0,1]$ then for every continuous function ...
2
votes
2answers
111 views

J-measurable sets and functions of class $C^1$

If $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is $C^1$ class and $det f^{\prime}(0)=0$ show that, when $r\rightarrow 0$ $$\dfrac{Vol(f(B[0,r]))}{ Vol(B[0,r])} \rightarrow 0$$ where $Vol(X)$ is ...
2
votes
1answer
38 views

Quesiton about functions

I'm embarrassed asking this question. I took a long break. main question is : cartesian coordinate system in $R^n$ space is shown as $(x_1,x_2 ...x_n)$. Show that for $1\le i\le n$ each ...
1
vote
1answer
143 views

Determining if the product of two particular harmonic functions is a harmonic function

Let $u$ be a $C^{2}$ harmonic function in $\mathbb{R}^{n}$ and let $g(x) = \left| x \right|^{2-n}$. I would like to show that: $v(x) = g(x)u\left(\frac{x}{\left| x \right|^{2}}\right)$ is also ...
0
votes
1answer
53 views

Discontinuity in multivariable calculus

Show that the function F(x, y) = x(1 - cos(x - y))/(x y)^2 has removable discontinuity along the line x = y. Which values should we assign to this function on the diagonal x = y in order to turn it ...
3
votes
1answer
168 views

Prove function (0, ∞) to (0, ∞) can't exist if $(f(x))^2>(f(x+y))((f(x)+y))$

So we're trying to prove that no function exists $f: (0,\infty) \rightarrow(0,\infty)$ such that $f(x)^2\ge f(x+y) (f(x)+y)\quad x,y\gt0$ I've tried to break it up into 3 cases to show if $x\gt y$ ...
2
votes
0answers
40 views

problem with submersion

Given $\varphi:\mathbb{R}^{m+n}\longrightarrow \mathbb{R}^m$ is $C^{ k}$ class. If there $a\in \mathbb{R}^{m+n}$ with $\varphi^{\prime}(a)$ is surjective. Then there a mergullo $f:V\longrightarrow ...
0
votes
0answers
55 views

Let $A=\{(x,y):x>0,y>0\}$ and $B=\{(x,y):x>0 ∨ y<0\}$…

Let $A=\{ (x,y): x>0, y>0\}$ and $B=\{ (x,y): x>0 \ \vee\ y<0\}$ Show that no there diffeomorphism $f:U \rightarrow V$ such that $\overline{A}\subset U$ and $\overline{B}\subset V$ ...
1
vote
1answer
34 views

we can say about the convergence of $\Lambda_k=\varphi_1\circ\cdots\circ \varphi_{k-1} \circ \varphi_{k}$??

Given a sequence differentiable $\varphi_k:\mathbb{R}^m\longrightarrow \mathbb{R}^m$ such that $\vert \varphi_k^{\prime}\vert \leq \alpha$ with $\alpha \in (0,1)$. that can say about the convergence ...
1
vote
0answers
75 views

Denote “ ∥f−g∥<δ in X ” to mean that ∥f(x)−g(x)∥≤δ and ∥f′(x)−g′(x)∥≤δ …

Let $f,g: U \rightarrow \mathbb{R}^{n}$ differentiable in the open $U \subset \mathbb{R}^{m}$, $\delta >0$ and $X\subset U$. Denote " $\Vert f-g\Vert_{\star} < \delta$ in $X$ " to mean that ...
1
vote
0answers
51 views

Let $U\subseteq \mathbb{R}^{m}$, $V\subseteq \mathbb{R}^{n}$ open, $f:U \rightarrow \mathbb{R}^{n}$ $i$ times differentiable in $x\in U$…

Let $U\subseteq \mathbb{R}^{m}$, $V\subseteq \mathbb{R}^{n}$ open, $f:U \rightarrow \mathbb{R}^{n}$ $i$ times differentiable in $x\in U$, $f(U)\subset V$, and $g:V\rightarrow \mathbb{R}^{p}$ $i$ times ...
2
votes
3answers
64 views

Will the rules of calculus stay the same when a real-valued function is defined over infinite number of variables?

So the question would be: Can we ever talk about a real-valued function that is defined over infinite number of variables? Will the rules of calculus remain the same for such functions described in ...
1
vote
1answer
157 views

How to find the inverse function of $f: \mathbb R^2\to\mathbb R^2$?

How to find the inverse function of $f: \mathbb R^2\to\mathbb R^2$? for example, $u= x+2y, v=xe^y$, how to find the inverse?