Tagged Questions
27
votes
2answers
1k views
Asymmetric Hessian matrix
Are there any functions, $f:U\subset \mathbb{R}^n \to \mathbb{R}$, with Hessian matrix which is asymmetric on a large set (say with positive measure)?
I'm familiar with examples of functions with ...
25
votes
2answers
664 views
Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject(single-variable calculus):
the ...
12
votes
6answers
1k views
References for the multivariate calculus
Maybe due to my ignorance, I find that most of the references for mathematical analysis(real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After ...
10
votes
4answers
2k views
What is the 'implicit function theorem'?
Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
9
votes
1answer
299 views
Information captured by differential forms
My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter ...
8
votes
3answers
480 views
Connectivity, Path Connectivity and Differentiability
I have two questions which pertain to differentiability, connectivity and path connectivity. Ocasionally, I will encounter an author who defines connectivity in the following way:
An open subset $U$ ...
8
votes
3answers
590 views
Does $\lim \frac{xy}{x+y}$ exist at (0,0)?
Given the function $f(x,y) = \frac{xy}{x+y}$, after my analysis I concluded that the limit at $(0,0)$ does not exists.
In short, if we approach to $(0,0)$ through the parabola $y = -x^2 -x$ and $y = ...
7
votes
2answers
506 views
If a function is continuously differentiable and bounded, is its derivative also bounded?
Two questions:
Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable and bounded. Is its derivative $f'$ bounded as well?
What about in the case of ...
7
votes
1answer
621 views
Convergence tests for improper multiple integrals
For improper single integrals with positive integrands of the first, second or mixed type there are the comparison and the limit tests to determine their convergence or divergence. There is also the ...
7
votes
2answers
138 views
Help to finish my proof: inequality with norm and Schwarz ineq
Let $A=\left [ a_{ij} \right ]$ be the matrix of a linear mapping $A\in L\left ( \mathbb{R}^{n},\mathbb{R}^{m} \right )$.
-Prove that: $\left \| A \right \|\leq \left ( ...
7
votes
2answers
102 views
Demonstrate that $F(x,y) = f(x,y)\sin(x^2 + y^2)$ is differentiable at $(0,0)$
Let $f : \mathbb{R^2} \rightarrow \mathbb{R}$ be a bounded function and
$F(x,y) = f(x,y)\sin(x^2 + y^2)$. How can we demonstrate that $F$ is differentiable at $(0,0)$?
I don't how to do this. I ...
7
votes
1answer
286 views
Derivative of a determinant
How can I get that? Is it some application of the Chain Rule considering the determinant as a function of the vector function $f$?
Notation $d_{i j}$ is the cofactor of $\frac{\partial ...
6
votes
1answer
477 views
If derivative of a function is the zero function in $\mathbb R^n$, then the function is constant when the domain is path-connected
Some definitions first. Let $A \subseteq \mathbb R^n$. Let $x,y \in A$. A path between $x$ and $y$ is a continuous function $f: [0,1] \rightarrow \mathbb{R}^n$ with $f(0) = x$ and $f(1) = y$. The set ...
6
votes
2answers
133 views
$C^{2} ( \mathbb{R}^{2}) $ function-proof of inequality
Let $f\in C^{2}( \mathbb{R}^{2} )$. Suppose that $\triangledown f=0
$ on a compact set $A\subseteq \mathbb{R}^{2}$. I want to prove that there is a strictly positive constant $\lambda > 0$ such ...
6
votes
1answer
269 views
Understanding Proof About an Immersion
I am studying the following proof for which an excerpt is provided below:
Update: I have written out a fully-detailed proof of an argument that seeks verify the claim that $\partial \psi$ is ...
