Are all real life signals infinitely continuous and differentiable?
I'm thinking yes because a finite discontinuity in one of the derivatives would imply infinite to take place in the next higher-order derivative. And infinite means infinite energy.
Hello every one
Can one say , that
A globle coordinate chart is a cartesian coordinate
And a local coordinate chart is any kind of curvilinear coordinate ?Thanks
Hey! :o
How could we prove the following rule for differentiable curves in $\mathbb{R}^3$ ?? (Wondering)
$$\frac{d}{dt}[\overrightarrow{\sigma}(t)\times \overrightarrow{\rho}(t)]=\frac{d\overrightarrow{\sigma}}{dt}\times \overrightarrow{\rho}(t)+\overrightarrow{\sigma}(t)\times...
Hello! :o
I am doing my master in the field Mathematics in Computer Science. I am having a dilemma whether to take the subject Partial differential equations- Theory of weak solutions or the subject differentiable manifolds.
Could you give me some information about these subjects...
I've been searching high and low through the Google for a solutions manual to William Boothby's "An Introduction to Differentiable Manifolds and Riemannian Geometry" to no avail. Does anyone know if ∃ such a thing? Thanks.
Suppose that we have this metric and want to find null paths:
ds^2=-dt^2+dx^2
We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$
dx=\pm dt \to x=\pm t
Now switch to the more abstract and rigorous one-forms in differentiable manifolds.
Here \mathrm{d}t (v)...
Suppose that we have this metric and want to find null paths:
ds^2=-dt^2+dx^2
We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$
dx=\pm dt \to x=\pm t
Now switch to the more abstract and rigorous one-forms in differentiable manifolds.
Here \mathrm{d}t (v)...
Hi,
f(X)=\frac{xy^2}{x^2+y^4} is the function in question, this is the value of the function at ##X=(x,y)## when ##x\neq0##, and ##f(X)=0## when ##X=(0,y)## for any ##y## even ##y=0##.
Now, along any vector or line from the origin the directional derivative ##f'(Y,0)## (where ##Y=(a,b)## is...
The problem is to determine whether the function
##f(z) =
\left\{\begin{array}{l}
\frac{\overline{z}^2}{z}~~~if~~~z \ne 0 \\
0 ~~~if~~~z=0
\end{array}\right.##
is differentiable at the point ##z=0##. My two initial thoughts were to show that the function was not continuous at the point...
The function f: R → R is: f(x) =
(tan x) / (1 + ³√x) ; for x ≥ 0,
sin x ; for (-π/2) ≤ x < 0,
x + (π/2) ; for x < -π/2
_
For -π/2 I would say it is not differentiable since both π and 2 are constants and you can not vary a constant.
For 0, I would derivate it by using the first function...
Is anyone familiar with this book?
Differentiable Manifolds: A Theoretical Physics Approach
https://www.amazon.com/gp/aw/s//ref=mw_dp_a_s?ie=UTF8&k=Gerardo+F.+Torres+del+Castillo&i=books&tag=pfamazon01-20
https://www.amazon.com/gp/product/0817682708/?tag=pfamazon01-20
If you are, what's your...
(a) State precisely the definition of: a function f is differentiable at a ∈ R.
(b) Prove that, if f is differentiable at a, then f is continuous at a. You may
assume that
f'(a) = lim {f(x)-f(a)/(x-a)} as x approaches a
(c) Assume that a function f is differentiable at each x ∈ R and also f(x)...
Homework Statement
f(mx)=f(x) + (m-1)xf'(x)+\dfrac{(m-1)^2}{2!} x^2 f''(x)...
Homework Equations
Taylor's Series
The Attempt at a Solution
If I approximate the LHS of the eqn using Taylor's polynomial,
f(mx)=f(mx)+mxf'(mx)+\dfrac{(mx)^2f''(mx)}{2!}+...
But, I'm lost from here...
Theorem: ctsly differentiable at a if the function is cts and its partial derivatives exist and are cts in a neighborhood of a. [1]
- so to be differentiable we can check whether this conditions holds, and if it does ctsly diff => diff.
- the definition of a scalar function being...
Hi there!
How can I prove that a function which takes an nxn matrix and returns that matrix cubed is a continuous function? Also, how can I analyze if the function is differenciable or not?
About the continuity I took a generic matrix A and considered the matrix A + h, where h is a real...
Homework Statement
Problem: Given C is the graph of the equation
2radical3 * sinpi(x)/3 =y^5+5y-3
Homework Equations
(1) Prove that as a set
C= {(x,y) Exists at all Real Numbers Squared | 2radical3 * sinpi(x)/3 =y^5+5y-3
is the graph of a function differentiable on all real...
Homework Statement
Show that if ##f## is a continuously differentiable real valued function on an open interval in ##E^2## and ##\partial^2f/\partial x\partial y=0,## then there are continuously differentiable real-valued functions ##f_1,f_2## on open intervals in ##\mathbb{R}## such that...
Homework Statement
(a) Let \alpha:I=[a,b]→R^2 be a differentiable curve. Assume the parametrization is arc length. Show that for s_{1},s_{2}\in I, |\alpha(s_{1})-\alpha(s_{2})|≤|s_{1}-s_{2}| holds.
(b) Use the previous part to show that given \epsilon >0 there are finitely many two...
Homework Statement
Show that the set of twice differentiable functions f: R→R satisfying the differential equation
sin(x)f"(x)+x^{2}f(x)=0is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f"...
Homework Statement
Where is the function ##f:E^2\to\mathbb{R}## given by ##f(x,y)=\begin{cases}\frac{xy}{|x|+|y|} & , \ \text{if} \ (x,y)\ne(0,0)\\
0 & , \ \text{if} \ (x,y)=(0,0) \end{cases}## differentiable?
Homework Equations
None
The Attempt at a Solution
The function is...
Problem:
I plugged in fx, fy, and f(1,pi) everywhere I could but I have no idea how to move on from here. I'm stuck trying to show that:
(1+Δx) + (1+Δx)sin(pi+Δy) - 1 = Δx - Δy + ε(Δx,Δy)Δx + ε(Δx,Δy)Δy
Please see attached.
I am not sure whether my example of this function is correct.
f(x) = ##sin(\frac{\pi x}{2})##
obviously, f(x) is continuous on [-1,1] and differentiable on (-1,1)
Inverse of f(x) will be ##\frac{2 sin^{-1}x}{\pi} ##
and d/dx (inverse of f(x)) will be ##\frac{2}{π...
Hi! :o
We know that $g$ is differentiable at $x=0$ with $g(0)=g'(0)=0$ and
$$f(x)=\left\{\begin{matrix}
g(x)sin\frac{1}{x} & ,x\neq 0\\
0& ,x=0
\end{matrix}\right. $$
Is $f$ differentiable at $x=0$.If yes,which is the value of $f'(0)$?
That's what I have tried:
If f is differentiable at...
Homework Statement
Suppose that f is differentiable at x . Prove that ƒ(x)=lim[h→0] \frac{ƒ(x+h)-ƒ(x-h)}{2h}
Homework Equations
The Attempt at a Solution
I think that it may be proved by first principle,but I cannot rewrite the limit into the form of lim[h→0]...
The question is to check where the following complex function is differentiable.
w=z \left| z\right|
w=\sqrt{x^2+y^2} (x+i y)
u = x\sqrt{x^2+y^2}
v = y\sqrt{x^2+y^2}
Using the Cauchy Riemann equations
\frac{\partial }{\partial x}u=\frac{\partial }{\partial y}v...
The question is to check where the following complex function is differentiable.
w=z \left| z\right|
w=\sqrt{x^2+y^2} (x+i y)
u = x\sqrt{x^2+y^2}
v = y\sqrt{x^2+y^2}
Using the Cauchy Riemann equations
\frac{\partial }{\partial x}u=\frac{\partial }{\partial y}v...
I'd like to show that if \alpha>\frac{1}{2} then (x^2+y^2)^\alpha is differentiable at (0,0).
The usual way is to show that the partial derivatives are continuous at (0,0).
Yet I am a little confused how to show that 2x\alpha(x^2+y^2)^{\alpha-1} is continuous at (0,0). I have tried working...
Hi! Could you help me with the following?
Let g: R \rightarrow R a bounded function. There is a point z \epsilon R for which the function h: R \ \{z\} \rightarrow R , where h(x)=\frac{g(x)-g(z)}{x-z} is not bounded. Show that the function g is not differentiable at the point z .
My...
Homework Statement
Given the function:
1, x≤0;
\phi={1-3x^2+2x^3, 0<x<1;
0, x≥1.
Show that \phi is continuously differentiable and provide its equation
Homework Equations
The Attempt at a Solution
I have figured...
Homework Statement
Prove that function has directional derivative in every direction, but is not differentiable in (0,0):
f(x,y)=\begin{cases}\frac{x^3}{x^2+y^2},&(x,y)\neq(0,0)\\ \\0,&(x,y)=(0,0)\end{cases}
The Attempt at a Solution
I have already proved that it has directional...
Hello all, I have the following problem from Complex Analysis that I would like for someone to check my understanding on:
Homework Statement
The problem is to find the derivative if it exists of
f(z) = \frac{e^{i\theta}}{r^2} = r^{-2}\cos \theta + i r^{-2}\sin \theta
where I have already...
Hello,
I notice that most books on differential geometry introduce the definition of differentiable manifold by describing what I would regard as a differentiable manifold of class C∞ (i.e. a smooth manifold).
Why so?
Don't we simply need a class C1 differentiable manifold in order to...
Hello,
I read from several sources the statement that the set of points M\inℝ2 given by (t, \, |t|^2) is an example of differentiable manifold of class C1 but not C2.
Is that true?
To be honest, that statement does not convince me completely, because in order for M to be a manifold, we should...
Homework Statement
could you please check if this exercise is correct?
thank you very much :)
##f(x,y)=\frac{ |x|^θ y}{x^2+y^4}## if ##x \neq 0##
##f(x,y)=0## if ##x=0##
where ##θ > 0## is a constant
study continuity and differentiabilty of this function
The Attempt at a Solution...
Homework Statement
Let f(x,y) be a differentiable function with x = rcosθ and y = rsinθ. find the df(x,y)^2/d^2θ (second derivative with respect the theta)
Homework Equations
The Attempt at a Solution
Don't exactly know what I'm doing here.. The notes from class give me this...
Homework Statement
Define f : \mathbb{C} \rightarrow \mathbb{C} by
f(z) = \left
\{
\begin{array}{11}
|z|^2 \sin (\frac{1}{|z|}), \mbox{when $z \ne 0$}, \\
0, \mbox{when z = 0} .
\end{array}
\right.
Show that f is complex-differentiable at the origin although the...
I understand at cusps, corners, etc, because the negative and positive directions do not agree with each other.
But what about at jump discontinuity on a graph? Why wouldn't a function be differentiable there? I understand that from the definition of differentiable that it just isn't, but I...
Hi
I am attempting to self-study Complex Analysis but i am confused over a couple of points.
1 - my book says "if a complex function is differentiable once throughout its domain of definition then it is infinitely differentiable" . How does this apply to z^2 ? If you differentiate it once...
Homework Statement
Let V be the linear space of all real functions Differentiable on (0,1). If f is in V define g=T(f) to mean that g(t)=tf'(t) for all t in (0,1). Prove that every real λ is an eigenvalue for T, and determine the eigenfunctions corresponding to λ.Homework Equations
The Attempt...
Homework Statement
Prove that if a function f is once-differentiable on the interval [a, b], then
Vf = \int ^{b}_{a} | f'(x) | dx,
where Vf = sup_{P} \sum^{i=n-1}_{0} | f(x_{i+1}) - f(x_{i}) | where the supremum is taken over all partitions P = \left\{ a = x_{0} < x_{1} < ... < x_{n}...
Using the definition of the derivative find at which points the function f(z) = Im(z)/z conjugate is complex differentiable.
I know that it is not complex differentiable anywhere but I need to show it using the definition and not the Cauchy Riemann equations.
Homework Statement
Fx= ax^2 + bx + c -infiniti<x<0
= D x=0
=x^2 sin(1/x) - 2 0<x<infiniti
a) FInd all vlaues of a, b, c and d that make the function f differntiable on the domain -∞<x<∞
b) Using the values founbd in part a, determine lim x->...
If f is a differntiable function, find the expression for derivatives of the following functions.
a) g(x)= x/ f(x)
b) h(x) [f(x^3)]^2
c) k(x)= sqrt (1 + [f(x)]^2)
First off, I am not even sure what they are asking. I am assuming that they want the derivative for each component of the...
A problem in mathematical analysis that I have problems getting to grips with (need it to characterize the direction field of a differential equation)
Suppose f(x) is a continuously differentiable function on ℝ with f(0)=f'(0)=0. Suppose that for any ε>0 there is some x in (0,ε) such that...
Let $u:\mathbb{R}^n\rightarrow \mathbb{R}$ be a twice differentiable function. The 2-dimensional wave equation is
$\frac{\partial ^2u}{\partial x^2}=\frac{\partial ^2u}{\partial t^2}$, where $(x,t)$ are coordinates on $\mathbb{R}^2$. Prove that if $f,g:\mathbb{R}\rightarrow \mathbb{R}$ are...
The question is:
Using the chain rule to prove that a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ which is polynomial in the coordinates is differentiable everywhere.
(The chain rule is for the use under function composition circumstances, how to apply it here to prove that the function $f$...