Tagged Questions
1
vote
1answer
28 views
Differentiation - Limits Equal (Possibly MVT, Rolle's or L'Hopital)
Got a quick question from a past exam paper.
If $f:R \rightarrow R$ is differentiable, and $f$ is such that $\lim_{ x\rightarrow \infty } f(x)=\lim_{ x \rightarrow -\infty} f(x)=0$ and there is a ...
1
vote
1answer
29 views
Small question about derivative
how to derive $\int_0^1 G(t,s) e(s)ds$ with respect to $t$
Where $G(t,s)$ is a Green function and $e:(0,1)\rightarrow \mathbb{R}$ continuous and $e\in L(0,1)$
Please help me
Thank you
5
votes
1answer
104 views
Algebraic transformations to continuously extend functions
Lately I was browsing through my analysis lecture notes (since right know I'm somewhat rusty in analysis) and the proof that $x \mapsto \frac{1}{x}$ is differentiable at every $x'\neq 0$ captured my ...
0
votes
1answer
33 views
Nondegenerate critical point
I don't understand this part from the book of Zeidler , can someone help me to understand it ?
Please
Thank you
6
votes
1answer
82 views
Minimal definition of the derivative
The definition of the Fréchet derivative according to Wikipedia is:
Let $V$ and $W$ be Banach spaces, and $U\subset V$ be an open subset of $V$. A function $f : U \to W$ is called Fréchet ...
1
vote
1answer
58 views
Highest derivatives of implicit function
I am learning to use the implicit function theorem (IFT) and met recently the following problem:
Let $F(x,y)=x+y+x^5-y^5$. The given equation defines a smooth function $\phi:U\rightarrow \mathbb{R}$ ...
-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.
1
vote
1answer
94 views
$f$ not differentiable at $(0,0)$ but all directional derivatives exist
Consider the function :
$$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto
\begin{cases}
0 & \text{for } (x,y)=(0,0) \\
\frac{x^3}{x^2+y^2} & \text{for } (x,y) \neq ...
3
votes
0answers
72 views
Green's function for third order boundary value problems
How to find the Green's function $G(t,x)$ for the BVP consisting of the equation :
$$u'''(t)=0 , \quad t\in (0,1)$$
and BC :
$$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$
where $\frac12 < ...
4
votes
1answer
156 views
Gradient of sum of products of matrix traces
For a matrix $X \in \Re_{n\times d}$ find the gradient of
$\sum_{i,j}[\langle X_{i.},X_{j.} \rangle\operatorname{tr}(X^TA_{ij}X)]$ w.r.t $X$,
where $A_{ij}=(e_i-e_j)(e_i-e_j)^T$ using the basis ...
2
votes
1answer
25 views
Differentiability of first derivative of a function
If a function $f$ is differentiable on domain $D$ and $f'$ is increasing on $D$, is $f'$ necessarily continuous on $D$? Is $f'$ necessarily differentiable on $D$? Counterexamples?
From Darboux ...
2
votes
3answers
47 views
Solving $(f'(x))^2 = f(x)f''(x)$ with boundary conditions.
Let $f$ be a continuous real-valued function such that $$(f'(x))^2 = f(x)f''(x).$$ Suppose $f(0) = 1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.
I have this question in my book ...
3
votes
3answers
128 views
Calculation of a derivative
I have to calculate the following derivative
$$\frac{\partial}{\partial{\Vert x\Vert}}e^{ix\cdot y}$$
Then I write
$$e^{ix\cdot y}=e^{i\Vert x\Vert\Vert y\Vert\cos\alpha}$$
andI derive; is this ...
0
votes
2answers
42 views
Let $f:\Bbb R^2→\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$
Let $$f:\Bbb R^2\to\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$$
i) Is $f$ continuous at $(0,0)$?
ii) Is $f$ differentiable at $(0,0)$?
I can prove that $f$ is ...
3
votes
1answer
400 views
Matrix calculus : Find the gradient/derivative?
I know that the derivative of $Tr(Z^TAZ)$ w.r.t $Z$ is $2AZ$. Now I'd like to compute the derivative of $Tr\left[Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)Z\right]$ instead, w.r.t $Z ...
1
vote
2answers
26 views
How I can calculate $g^{(1)}$?
Let $f$ be an analytic function defined over all complex plane. Now, consider the function $g:ℝ^{r+1}→ℝ$ defined by
$$g(t₁,t₂,...,t_{r+1})=f^{(r+1)}(1-2∏_{j=1}^{r+1}t_{j})$$
where $f^{(r+1)}$ is the ...
1
vote
2answers
77 views
Convergence of $\sum\limits^\infty _{k=0} a_k \sin(kx)+b_k \cos(kx)$
Ok, for the infinite series:
$$\sum^\infty _{k=0} a_k \sin(kx)+b_k \cos(kx)$$
How do I show that this converges on any finite interval if $\sum^\infty _{k=0} k(|a_k|+|b_k|)<\infty$?
Also, do the ...
2
votes
2answers
78 views
Why is this function smooth?
Let $f: \mathbb{R}^n\rightarrow \mathbb{R}$ be the following function,
$$f(x)=\begin{cases}
\operatorname{e}^{-\tfrac{1}{1-\|x\|^2}} & \text{if }\|x\|<1,\\\\
0 & \text{otherwise}.
...
1
vote
1answer
100 views
Does a such condition imply differentiability?
Let function $f:\mathbb{R}\to \mathbb{R}$ be such that
$$
\lim_{\Large{(y,z)\rightarrow (x,x) \atop y\neq z}} \frac{f(y)-f(z)}{y-z}=0.
$$
Is it then $f'(x)=0$ ?
1
vote
1answer
42 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
1answer
75 views
Rudin Real and complex analysis question[Differentiation]
At the beginning of the chapter on differentiation, the following theorem is stated without proof. Apparently it is so trivial that it does not require justification. I however don't find it so ...
0
votes
1answer
140 views
Show that this piecewise function is differentiable at $0$
I have shown (from first principles) that the Cauchy-Riemann equations for the following function are satisfied at $z=0$. But to properly prove differentiability at $z=0$, what should I do next? Do I ...
2
votes
0answers
28 views
Looking for an analysis book which uses linear maps notation for multivariable differentiation
I'm taking an analysis course and I find it quite hard to follow what the professor is saying. So far we've been following elementary real analysis by bruckner^2 and Thompson but for the topic on ...
2
votes
2answers
37 views
Holomorphic function $f$ such that $f'(z_0) \neq 0$
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an holomorphic function such that $f'(z_0) \neq 0$ for some $z_0 \in \mathbb{C}$.Prove that there is $r>0$ such that, if $|z-z_0|<r$ and $z \neq z_0 ...
2
votes
2answers
32 views
why constant derivatives?
Really simple question here.
Say $f(x)$ and $g(y)$ then why if
$\frac{d f(x)}{dx} = \frac{d g(y)}{dy}$
then both derivatives are constant?
Thank you all very much
2
votes
1answer
82 views
Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.
Given the function
$$
f(x) = \left\{\begin{array}{cc}
e^{- \frac{1}{x^2}} & x \neq 0
\\
0 & x = 0
\end{array}\right.
$$
show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.
So I have to show ...
2
votes
1answer
198 views
Integration by Parts and Leibniz Rule for Differentiation under the Integral Sign
Basically a friend of mine and I have had this hot debate for a little too long, I contend that these two tools are not only logically unconnected but they require different assumptions (I believe one ...
1
vote
3answers
287 views
Is the function $f(x)=\sin(1/x)$ differentiable at $x=0$?
The function $f$ is defined by $f(x)= \sin(1/x)$ for any $x\neq 0$. For $x=0$, $f(x)=0$.
Determine if the function is differentiable at $x=0$.
I know that it isn't differentiable at that ...
4
votes
1answer
75 views
$\frac{\mathrm d^n}{\mathrm d x^n} e^{-\frac {1}{x^2}} = 0$ at $x=0$ [duplicate]
This is an exercise from David Brannan's Mathematical Analysis. I've proved parts (a) - (c) but need help with Part (d). Any guidance appreciated.
EDIT
I have solved it, by induction using the ...
7
votes
1answer
89 views
Does $f\colon \Omega \to \mathbb R$ differentiable imply $f$ locally Lipschitz?
Let $f\colon \Omega \subseteq \mathbb R^n \to \mathbb R$ be a differentiable function. Is it true that $f$ is locally Lipschitz, i.e. Lipschitz on every compact $K \subset \Omega$?
If $f$ were ...
0
votes
0answers
42 views
variational problem
I have: $\Omega \subset R$, be open and bounded, assume that $q \in L^{\infty}(\Omega)$ satisfies $q\geq 0$ a.e in $\Omega$, and let $f :\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such ...
0
votes
1answer
46 views
A change of variables in the euler equation
If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation:
$$z^2w'' + \alpha zw' + \beta w = 0$$
where $w$ is a function of $z$ and ...
1
vote
1answer
81 views
Legendre's equation polynomial solution
This is a problem on analytic solutions of ordinarry differential equations. Any help will be greatly appreciated. Please, try to be as specific as possible as I don't handle this material very well ...
3
votes
1answer
73 views
Taylor expansion of an integral
I am interested in the Taylor series expansion around $t=0$ of the following expression:
$$I(t)=\int_{0}^{\infty}e^{-x^2}\log\left(e^{-(x-t)^2}+e^{-(x+t)^2}\right)dx$$
Normally, I would proceed by ...
1
vote
2answers
102 views
Solving a second order inhomogeneous differential equation with constant coeffcients
I a seeking to solve the equation: $$u'' + 2vu' + u = cos(\sigma t), \ u(0) = 1, \ u'(0) =0$$ where $0 < v < 1$. Then I have to show that the solutio is purely oscillatory (which I don't know ...
0
votes
1answer
85 views
What are the differences between differential and gradient?
As far as i know, both differential and gradient are vectors where their dot product with a unit vector give directional derivative with the direction of the unit vector. So what are the differences?
1
vote
1answer
129 views
Statements about a twice differentiable function
Can you help me to prove or disprove?
We have a function $f:(0,+\infty)\rightarrow\mathbb{R}$ twice differentiable, such that as $x\rightarrow+\infty$
(a) $xf(x) \rightarrow+\infty$
(b) $xf''(x) ...
0
votes
2answers
42 views
Differentiability on vector values function
I just had my first lecture of my analysis course, and we were introduced the differentiation on general euclidean space where the derivative is regarded as a linear transformation.
Define ...
3
votes
1answer
112 views
Second order linear ODE with variable coefficients
Consider the second-order linear differential equation $u'' + p(x)u' + q(x)u = 0$ where $p$ and $q$ are continuous on the entire $\mathbb{R}$. Suppose that $q(x) < 0 $ everywhere. Show that if $u$ ...
5
votes
3answers
158 views
Find the value of the function at the given point.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying the conditions
$$\begin{align*}
(1)&f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\\
(2)&f(0)=1\\
(3)&f'(0)=-1
...
1
vote
2answers
47 views
Question regarding Cauchy's theorem?
I want to prove the inequality $3x\cdot \mathrm{arccot}{x}\geq\ln(1+x^{2})$... using Cauchy's theorem..I saw this in a book which I bought in a library and ive never seen this type of exercise..can ...
1
vote
1answer
73 views
A question about the second differential
Hi I have a doubt: What is the matrix associated at the second differential?
Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$, differiantable and let $df: \mathbb{R}^n \rightarrow \mathbb{R}$, ...
2
votes
1answer
52 views
Suppose the function $f:\mathbb R \rightarrow \mathbb R$ has left and right derivatives at $0$.
I have been trying to solve the following problem:
Suppose the function $f:\mathbb R \rightarrow \mathbb R$ has left and right derivatives at $0$.Then at $x=0$, which of the following options is ...
0
votes
2answers
81 views
Question about the Fréchet derivative
In a proof to show that $D(L\circ f)_a = L \circ Df_a$, where $f: U \subset E \rightarrow F$ and $L \in L_c(F,G)$.
They take the following limit to show that the frechet derivative exists,
...
2
votes
0answers
34 views
Can I say something about the $f_{xy}$ of such function?
I have a function $f(x,y)$ defining on $x>0,y>0$ satisfying that
(1) $f>0$
(2) for $a>1, f(ax,ay)>af(x,y)$
(3) $f_x,f_y>0$
(4)$f_{xx},f_{yy}<0$
Can I say something about ...
0
votes
3answers
224 views
How to prove this partial derivative?
Consider $u:\mathbf{R}\times\omega\rightarrow\mathbf{R}$, where $\omega\subset\mathbf{R}^{n-1}$ is a bounded domain. For each $y\in\omega$ and each $\lambda>0$, consider ...
1
vote
1answer
60 views
Is inverse use of mean value theorem right?
If we have $f$ is differentiable on $(a,b)$, and continuous on $[a,b]$, then
for any $x\in (a,b)$, exists $y, z \in [a,b]$, such that
$f '(x)=\dfrac{f(z)-f(y)}{z-y}$
Is this right?
2
votes
1answer
140 views
$\delta_{ij}$ and $\delta_{ji}$: relation and meaning
What's the relation between $\delta_{ij}$ and $\delta_{ji}$?
What about their mathematical and physical meanings?
Thank you!
4
votes
2answers
59 views
Prove that $h^{(k)}(0)=\lim_{t\to0}\frac{\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}h(jt)}{t^k}$
Prove that if $h$ is infinitely differentiable in a neighborhood of $0$, then the kth derivative evaluated at 0 is
$$h^{(k)}(0)=\lim_{t\to0}\frac{\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}h(jt)}{t^k}$$
1
vote
1answer
30 views
Showing that a certain function is $C^1$
For an exercise in my analysis course, I have to show that the function
$$\newcommand{\sgn}{\operatorname{sgn}}f: (x,y) \mapsto \begin{cases} \frac{(x \sin y)^2}{|x|+|y|},&(x,y) \neq (0,0) \\
...
