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...
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...
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!
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...
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...
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...
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...
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, &...
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...
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.
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 =...
<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$$,we have ...
Hi forum.
I'm trying to prove a claim from Mathematical Analysis I - Zorich since some days, but I succeeded only in part.
The complete claim is:
$$\left\{\begin{matrix} f\in\mathcal{C}^{(n)}(-1,1) \\ \sup_{x\in (-1,1)}|f(x)|\leq 1 \\ |f'(0)|>\alpha _n \end{matrix}\right. \Rightarrow \exists...
Hi guys, I am reading my lecture notes for Mechanics and Variations and I am trying to understand the maths here. From what I can see there we differentiated with respect to a derivative. Could you tell me how do we do that? Thanks
Homework Statement
##f(x) = (5x+6)^{10} , f'(x)=?##
Homework Equations
##\frac{d}{dx}x^n = nx^{n-1}##?
3. The Attempt at a Solution [/B]
I do know the solution ##f'(x) = 50(5x+6)^9##,but I don't know how this solution came to be.I downloaded this problem from the web and it only comes with...
A total derivative dU = (dU/dx)dx + (dU/dy)dy + (dU/dz)dz. I am unsure of how to use latex in the text boxes; so the terms in parenthesis should describe partial differentiations.
My question is, where does this equation comes from?
Homework Statement
Two balls of mass m are attached to ends of two, weigthless metal rods (lengths l1 and l2). They are connected by another metal bar.
Determine period of small oscillations of the system
Homework Equations
Ek=mv2/2
v=dx/dt
Conversation of energy
2πsqrt(M/k)
The Attempt at a...
Recently I came up with a proof of “ for a nth degree polynomial, there will be n roots”
Since the derivative of a point will only be 0 on the vertex of that function,and a nth degree function, suppose ##f(x)##has n-1 vertexes, ##f’(x)## must have n-1 roots.
Is the proof valid?
We define the differential of a function f in
$$p \in M$$,
where M is a submanifold as follows
In this case we have a smooth curve ans and interval I $$\alpha: I \rightarrow M;\\ \alpha(0)= p \wedge \alpha'(0)=v$$.
How can I get that derivative at the end by using the definitions of the...
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...
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...
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...
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?
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...
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...
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...