Differentiable Definition and 291 Threads

Hello mates, is the function ## f(x) = \frac{(2^x - 1)}{x} ## differentiable at x = 0? For it to be differentiable it has to be continuous? From the graph f(0) is undefined although limit exists. I have read that at points like a corner, gap and vertical tangents it is not differentiable. So...

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

jbunniii submitted a new PF Insights post A Continuous, Nowhere Differentiable Function: Part 2 Continue reading the Original PF Insights Post.

jbunniii submitted a new PF Insights post A Continuous, Nowhere Differentiable Function: Part 1 Continue reading the Original PF Insights Post.

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.

I could really use some help on this problem as well! 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...

What is a good book on 2-dimensional surfaces (3-spheres, etc.)? I need to know about geodesics, etc.

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...