1. ### I Integrating with the Dirac delta distribution

Given \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y) \end{split} \end{equation} where ##\epsilon > 0## Is the following also true as ##\epsilon \rightarrow 0## \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon}...

Hi everyone. I am studying Chebyshev Polynomials to solve some differential equations. I found in the literature that if you have a function being expanded in Chebyshev polynomials such as $$u(x)=\sum_n a_n T_n(x),$$ then you can also expand its derivatives as $$\frac{d^q u}{dx^q}=\sum_n... 3. ### I Rotation of functions I was solving a problem for my quantum mechanics homework, and was therefore browsing in the internet for further information. Then I stumbled upon this here: R is the rotation operator, δφ an infinitesimal angle and Ψ is the wave function. I know that it is able to rotate a curve, vector... 4. ### Derivative of a vector Hello everyone, i'm stuck doing this problem, i've tackled the partial derivative but i can't figure out the derive for x component part, i solved the partial derivative part, i came to this result: What do can i do from here on, thank you! 5. ### Tension T in a parabolic wire at any point I am unsure how to go about this. I tried following the suggestion blindly and end up with with some cumbersome terms that are not the answer. From what I understand the derivative at each point would equal to T? Answer: I just can seem to get to this. I think I'm there but can't get it in... 6. ### Covariant derivative of a (co)vector field My attempt so far:$$\begin{align*} (\nabla_X Y)^i &= (\nabla_{X^l \partial_l}(Y^k\partial_k))^i=(X^l \nabla_{\partial_l}(Y^k\partial_k))^i\\ &\overset{2)}{=} (X^l (Y^k\nabla_{\partial_l}(\partial_k) + (\partial_l Y^k)\partial_k))^i = (X^lY^k\Gamma^n_{lk}\partial_n + X^lY^k{}_{,l}\partial_k)^i\\...
7. ### Matt & Hugh play with a Brick and derive Centripetal Acceleration

Matt and Hugh play with a tennis ball and a brick. Then they do some working out to derive the formula for the centripetal force (a = v^2/r) by differentiati...
8. ### A Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))

Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w)) Hello to my Math Fellows, Problem: I am looking for a way to calculate w-derivative of Fourier transform,d/dw (F{x(t)}), in terms of regular Fourier transform,X(w)=F{x(t)}. Definition Based Solution (not good enough): from...
9. ### I Understanding the definition of derivative

As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...
10. ### I Idea about single-point differentiability and continuity

Many have probably seen an example of a function that is continuous at only one point, for example ##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x, &...
11. ### I Velocity, acceleration, jerk, snap, crackle, pop, stop, drop, roll...

Edit: I see this was discussed in the related thread sorry for a repost. If acceleration causes a change in velocity, and jerk causes a change in acceleration, snap causes a change in jerk, crackle causes a change in snap, pop causes a change in crackle, stop causes a change in pop, drop causes...
12. ### Infinite series to calculate integrals

For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
13. ### B Justification for cancelling dx in an integral

In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))... ## \int_0^\phi \frac {d} {dx} f(x) dx =...
14. ### Evaluating This limit

<Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x\rightarrow 0} (x-tanx)/x^3$$ I solve it like this, $$\lim_{x\rightarrow 0}1/x^2 - tanx/x^3=\lim_{x\rightarrow 0}1/x^2 - tanx/x*1/x^2$$ Now using the property $$\lim_{x\rightarrow 0}tanx/x=1$$,we have ...

23. ### Derivative of expanded function wrt expanded variable?

Homework Statement If I have the following expansion f(r,t) \approx g(r) + \varepsilon \delta g(r,t) + O(\varepsilon^2) This means for other function U(f(r,t)) U(f(r,t)) = U( g(r) + \varepsilon \delta g(r,t)) \approx U(g) + \varepsilon \delta g \dfrac{dU}{dg} + O(\varepsilon^2) Then up to...
24. ### I Two questions about derivatives

In Rudin, the derivative of a function ##f: [a,b] \to \mathbb{R}## is defined as: Let ##f## be defined (and real-valued) on ##[a,b]##. For any ##x \in [a,b]##, form the quotient ##\phi(t) = \frac{f(t) - f(x)}{t-x}\quad (a < t <b, t \neq x)## and define ##f'(x) = \lim_{t \to x} \phi(t)##, if the...
25. ### I Is the exponential function, the only function where y'=y?

Hello, I was wondering. Is the exponential function, the only function where ##y'=y##. I know we can write an infinite amount of functions just by multiplying ##e^{x}## by a constant. This is not my point. Lets say in general, is there another function other than ##y(x)=ae^{x}## (##a## is a...
26. ### B Understanding this graph

Could someone explain to me how from this graph you can deduce that ##\tan(\theta) = \frac {df} {dx}##. Thanks
27. ### I Relationship between force and potential energy

I am aware that the negative derivative of potential energy is equal to force. Why is the max force found when the negative derivative of potential energy is equal to zero?
28. ### Calculus derivatives word problem

Homework Statement Is it possible to accurately approximate the speed of a passing car while standing in the protected front hall of the school? Task: Determine how fast cars are passing the front of the school. You may only go outside to measure the distance from where you are standing to the...
29. ### B When do we use which notation for Delta and Differentiation?

I was taking notes recently for delta y/ delta x and noticed there's more than one way to skin a cat... or is there? I saw the leibniz dy/dx, the triangle of change i was taught to use for "difference" Δy/Δx, and the mirror six ∂f/∂x which is some sort of partial differential or something...
30. ### Derivative of x(t)?

Homework Statement Homework Equations The Attempt at a Solution I am trying to repair my rusty calculus. I dont see how du = dx*dt/dt, I know its chain rule, but I got (du/dx)*(dx/dt) instead of dxdt/dt, if I recall correctly, you cannot treat dt or dx as a variable, so they dont...