Tagged Questions
11
votes
2answers
89 views
$\int_{\mathbb{R}}|f(t)|^2dt=\int_{\mathbb{R}}|f'(t)|^2dt$ implies $f(t)=\mathbb{x}_{i}|f(t)|$
Let $f \in C^{1}(\mathbb{R},\mathbb{R}^m)$ be such that $f$ and $f'$ are square integrable and
$$\{t:f(t)=0\} \subset \{t:f'(t)=0\}$$
$$ |\{t:f(t)=0\}|=n\in \mathbb{N}$$
Prove that if ...
0
votes
1answer
33 views
Work to provide explanation on the definition of the area of a Jordan-measurable set
The problem is as follows:
Given this theorem:
Let $D$ be bounded & Jordan-measurable set
Let $f$ be a bounded function on $D$
And $f$ is continuous except for a set of zero ...
1
vote
0answers
59 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
2
votes
1answer
28 views
Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
The question is as follows:
Given:
(1) function $f: U \subset \mathbb R^n ==> \mathbb R$
(2) $U$ is open and convex set
(3) $f \in C^1$ in $U$
Goal: Show that $f$ is ...
16
votes
6answers
252 views
Why is boundary information so significant? — Stokes's theorem
Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes ...
2
votes
0answers
34 views
Closed curves question
Can you give me some help on the following problem?
Given two closed curves $\alpha, \beta : \mapsto \mathbb{R}^3$ we define $\phi_{\alpha \beta}: I^2 \mapsto \mathbb{R}^3$ as $\phi_{\alpha \beta} ...
0
votes
0answers
25 views
How to show that the partial derivatives exist
In general , how to show that the partial derivatives of a multivariable function exists without comupting it .
1
vote
1answer
26 views
Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1
The task is as follows:
Given: $$F(x,y,z) = 0$$ Goal: Show $\frac{\partial
z}{\partial y}$ (evaluated at $x$) * $\frac{\partial y}{\partial x}$
(evaluated at $z$) * $\frac{\partial ...
1
vote
1answer
43 views
Find local maxima of this quadratic function
How can I find local maxima of this quadratic function?
$$f(x) = \sum _{i=1}^n -\frac{(z_i - x)_+^2}{2} - \left\{((\frac{(z_i - x)_+^2}{2})-(\frac{(y_i - x)_+^2}{2}) ) * c_i\right\} $$
which ...
2
votes
0answers
54 views
Proof on showing F(x,y) is continuous by $\epsilon - \delta$ definition
The task is as follows:
Given: $$F(x,y) = \frac{xy(x^2 - y^2)}{x^2 + y^2}$$
Goal: Prove that $F(x,y)$ is continuous everywhere on the plane
Here is my attempt so far:
(1) By the ...
1
vote
1answer
39 views
Finding Partial Derivative ($n$-dimensional) using implicit differentiation vs explicitly solving
This is a book example (not a homework question) about implicit differentiation on a composite of functions in $n$-dimensional space.
But my book explains this example in a very unclear manner. So I ...
-1
votes
0answers
28 views
Help me understand an implicit function theorem problem?
I'm taking a theoretical multivariable calculus course from MIT OCW (18.024), and one of the questions on the problem set I'm working on pertains to material on the implicit and inverse function ...
-1
votes
1answer
65 views
Continuity of one partial derivative implies differentiability
Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function such that the partial derivatives with respect to $x$ and $y$ exist and one of them is continuous. Prove that $f$ is differentiable.
2
votes
2answers
67 views
Quadratic surface maximization and Hessians
If we have that the contours of a response surface are elliptical and the response is given by the following function:
$$\large \exp\left(-\left(w^2 + \frac{1}{4}l^2 -\frac{1}{4} \cdot w \cdot ...
1
vote
1answer
29 views
Laplacian $\Delta u$ in spherical coordinates
The Laplacian $\Delta u$ in spherical coordinates is $$\Delta u=\frac{\partial^2u}{\partial\rho^2}+\frac{2}{\rho}\frac{\partial ...
1
vote
2answers
52 views
product rule for matrix functions?
Given a real rectangular matrix $X$, and two scalar-valued matrix functions, $f(X)$ and $g(X)$, does the product rule for differentiation of a product of scalar valued functions, hold when ...
4
votes
1answer
182 views
Can one define the derivative of a function using tangent cones? Does such a notion already exist?
I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
4
votes
2answers
67 views
Showing the function $f(x,y)$ is one by one
Yesterday, while teaching geometry, I was faced to a problem saying that the function below is an distance function: $$d(P,Q)=\Big|\ln\frac{\frac{x_1-c+r}{y_1}}{\frac{x_2-c+r}{y_2}}\Big|$$ where in ...
4
votes
1answer
158 views
Gradient of sum of products of matrix traces
For a matrix $X \in \Re_{n\times d}$ find the gradient of
$\sum_{i,j}[\langle X_{i.},X_{j.} \rangle\operatorname{tr}(X^TA_{ij}X)]$ w.r.t $X$,
where $A_{ij}=(e_i-e_j)(e_i-e_j)^T$ using the basis ...
2
votes
2answers
47 views
Existence of limit in $\mathbb R^2$
I want to prove for a function from $\mathbb{R}^2$ to $\mathbb{R}^2$, its limit at 0 exists. Is it enough to prove that the limit exists and same if we approach $0$ through the all the lines starting ...
0
votes
0answers
65 views
Gradient contradicting dimensions. Find the mistake!
$\nabla diag(X^TX)= diag(\nabla(X^TX))=2diag(X)?$ ?- when $X$ is non symmetric rectangular matrix with real entries. $diag(.)$ denotes a diagonal matrix formed with the diagonal elements being the ...
1
vote
0answers
31 views
investigating the negative definiteness of a continuous but not differentiable function
I'm stuck at showing whether the following function is negative definite. Consider the following function $V(x_{1},x_{2}):\mathbb{R}^{2}\rightarrow\mathbb{R}$
$$
...
1
vote
0answers
33 views
Let $E \subset ℝ^n$ open and $f:E→ℝ^m$. Then is $f$ cont. diffb. on $E$ $⇔$ all the partial derivatives $D_jf_i$ exists on $E$ and are cont. on $E$.
Let $E \subset ℝ^n$ open and $f:E→ℝ^m$. Then is $f$ continiuous differentiable on $E$ if and only if all the partial derivatives $D_jf_i$ exists on $E$ and are continuous on $E$.
I don't understand ...
0
votes
2answers
42 views
Let $f:\Bbb R^2→\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$
Let $$f:\Bbb R^2\to\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$$
i) Is $f$ continuous at $(0,0)$?
ii) Is $f$ differentiable at $(0,0)$?
I can prove that $f$ is ...
3
votes
1answer
401 views
Matrix calculus : Find the gradient/derivative?
I know that the derivative of $Tr(Z^TAZ)$ w.r.t $Z$ is $2AZ$. Now I'd like to compute the derivative of $Tr\left[Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)Z\right]$ instead, w.r.t $Z ...
0
votes
1answer
24 views
Analysis: How real functions are related to equation $\ x^3 - y^3 + x - y =0 $ in R3
The question is as follows:
(1) What's the problem of solving for y in the equation $\ x^3 - y^3 + x
- y =0 $ ?
(2) How many real functions, letting $\mathbb -infinity < x < \mathbb ...
2
votes
1answer
40 views
Showing a function is differentiable using definition of derivative
Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be defined by $f(x,y)=(x^2-xy, x+y^2)$. Use the definition of the derivative of a function to show that $f$ is differentiable at the point $p=(1,-1)$.
My ...
0
votes
1answer
24 views
Removable discontinuity for $C^1$ functions on $\mathbb{R}^2$ with uniformly bounded partials.
I'm stuck on the following question: Say I have a function $f\in C^1(\mathbb{R}^2\setminus \{0\})$ with uniformly bounded partials, why must $f$ admit a continuous extension to $\mathbb{R}^2$?
My ...
1
vote
2answers
26 views
How I can calculate $g^{(1)}$?
Let $f$ be an analytic function defined over all complex plane. Now, consider the function $g:ℝ^{r+1}→ℝ$ defined by
$$g(t₁,t₂,...,t_{r+1})=f^{(r+1)}(1-2∏_{j=1}^{r+1}t_{j})$$
where $f^{(r+1)}$ is the ...
3
votes
2answers
115 views
A sequel for Elementary Analysis by Ross?
I've been learning real analysis from this book:
Elementary Analysis, K.A. Ross
I really liked the style of this book. It is quite old, and sometimes very difficult, but I guess I liked the way it ...
6
votes
0answers
140 views
Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?
I've written the question first, then the motivation behind it and lastly some background. Note that the question makes references to definitions and theorems written in the background bit at the end. ...
2
votes
2answers
118 views
Is this function Lipschitz continuous?
Let $\mu \in \mathbb R^d$ be given. Is the function $f:\mathbb R^d \to \mathbb R^d$ defined as $f(x) := \exp(-\|x- \mu\|) (\mu - x)$ Lipschitz continuous?
More specifically, for any $x, y \in ...
1
vote
3answers
75 views
Change of variables in two dimensions
This is from Munkres' Analysis on Manifolds, Section 17, Question 4.
(a) Show that
$$ \int_\Bbb {R^2} e^{-(x^2+y^2)} = \left[ \int_\Bbb R e^{-x^2}\right]^2,$$
provided the first of these ...
1
vote
1answer
44 views
To show that a partial dertivative (of a piecewise function) is continuous at $0$
$$f(z)=\cases{\frac{x^4-6x^2y^2+y^4}{x^2+y^2}
+i\frac{4xy(x^2-y^2)}{x^2+y^2},& $z\ne0$\cr 0, &$z=0$}$$
Let $u=\Re(f)$.
I have shown from first principles that $\frac{\partial ...
1
vote
3answers
91 views
How to use the inverse function theorem?
I have a function $F (x, y) = (x^2+y^2, xy)$ and I need to show that it has an inverse. How do I find the inverse of this function using the inverse function theorem? I have not learned this before in ...
1
vote
1answer
61 views
Finding a $C^1$ surface inside a convex open set (Rudin chapter 10 problem 29)
The problem is as follows (with $n>1)$:
Let $E \subseteq \mathbb R^n$ be a convex open set, and let $F \subseteq \mathbb R^{n-1}$ be it's projection onto the first $n-1$ coordinates. It is clear ...
4
votes
2answers
56 views
Show $\langle x,\nabla f \rangle = pf(x)$
I'm trying to figure out this problem. Perhaps Someone could give me some hints/solve it for me? It would be much appreciated.
Let $U$ be an open subset of $R^n$ and suppose $f:U\rightarrow R$. Then ...
1
vote
1answer
107 views
Problem in understanding orthogonal curvilinear coordinate like spherical coordinate or cylindrical coordinate
Can someone explain your understanding to the orthogonal curvilinear coordinate like spherical coordinate or cylindrical coordinate, which i think is quite strange. I can actually do computation to ...
1
vote
3answers
73 views
Answer Check: Third degree taylor polynomial at the point (1,0,-1)
I found the third degree taylor polynomial $$f(x,y,z) = xy^2z^3$$ at $$(1,0,-1)$$
The answer i got $$p_3(x,y,z)=(\frac{1}{2!}(-2y^2))+ (\frac{1}{3!}(-2(x-1)y^2))+(\frac{1}{3!}(6(z+1)y^2))$$
I am not ...
1
vote
1answer
50 views
Laplacian(F) = (n-1/r)g'(r) + g''(r)
I got one more problem from my self reading of Methods of Advanced Calculus by Edwards, hints and solutions are equally appreciated:
If f(x) = g(r), r= |x|, and n>=3, show that
Laplace(f) = ...
0
votes
1answer
53 views
Prove directly from the definition of mapping that h is differentiable
Hey guys I'm going through some problems on my own, currently going through both chapter 2 and chapter 3 of Advanced calculus of several variables by C.H. Edwards. Anyway I'm having problems with ...
1
vote
2answers
63 views
Show vector (p-q) is orthogonal to the curve at q
Let $f:\mathbb R \to \mathbb R^n$ be a differentiable mapping with $f'(t) \neq 0$ for all $t$ in $\mathbb R$. Let $p$ be a fixed point not on the image curve of $f$. If $q = f(t_0)$ is the point of ...
1
vote
1answer
56 views
Show that f '(a) exists for all a if f is linear
If $f: \mathbb R \to \mathbb R^m$ is linear, prove that $f'(a)$ exists for all $a$ in $\mathbb R$, with $dfa = f$
2
votes
0answers
86 views
Singular derivative matrix implies not one to one?
I am trying to show that if $f:\mathbb{R}^n\to \mathbb{R}^n $ is continuously differentiable and that for all $x\in \mathbb{R}^n$ $Df(x)$ is singular implies that $f$ is not 1-1.
The singularity of ...
0
votes
2answers
62 views
Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.
Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U.
Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C.
Thoughts: Let ...
0
votes
1answer
26 views
$y'=K(y-a)(y-b) ; a,b,k\in \mathbb{R}$ Differential equation , how to decipher it?
$y'=K(y-a)(y-b) ; a,b,K\in \mathbb{R}$
a) Discuss the solutions of this Differential Equation
b) Discuss special solutions to the initial value conditions: $$u(0) < a , a< u(0) < b , b < ...
1
vote
1answer
43 views
HOW to apply Picard-Lindelof to demonstrate that there is ONE solution ONLY for a Initial Value Problem?
Take for exemple: for $y'= 2\sqrt{y}$ the 0 function and the functions :$ (x-c)^2$.
How can somebody say there infinite solutions for the initial value $y(0)=0$ ?
And how to find out that are ...
3
votes
1answer
36 views
A Banach space induced by a maximum sum norm? $||u||_J = \sum \max |u_i(x)|$
Let $J=[a,b] \subset \mathbb{R}$, how does one then see that if $C(J)=C(J,\mathbb{C}^n)$ with the norm $$||u||_J = \sum_{i=1}^{n} ||u_i||_\infty = \sum _{i=1} ^{n} \max _{x\in J} |u_i (x) |$$
is a ...
2
votes
0answers
43 views
Can you point out at my proof that these two norms are truly equivalent what is wrong and can be done better?
Let $J=[a,b] \subset \mathbb{R}$, the slave (me) is asked to show that :
$$u\in C(J,\mathbb{R}^n) \mapsto ||u||_{1} = \sum_{i=1}^{n} ||u_i||_{\infty} = \sum_{i=1}^{n}max_{x\in J}|u_i(x)|$$
and ...
2
votes
1answer
66 views
Multivariable function derivative problem
Let $f : \mathbb{R}^2 \to \mathbb{R}$ be continuous on $\mathbb{R}^2$ and differentiable on $\mathbb{R}^2\setminus\{0\}$. If $Df(x)x = 0$ for all $x \in \mathbb{R}^2\setminus\{0\}$, show that $f$ ...
