In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. It is customarily denoted by letters such as f, g and h.If the function is called f, this relation is denoted by y = f (x) (which reads "f of x"), where the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (e.g., f is a function of the variable x).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
I am solving a problem of the boundary condition of Dirichlet type, in order to solve the problem, the functions within the differential equations suppose to be harmonic.
I have a function f(x,y,z) (the function attached) which is not harmonic. I must find an equivalent function g(x,y,z) which...
Because the Taylor series centered at 0, it is same as Maclaurin series. My attempts:
1st attempt
\begin{align}
\frac{1}{1-x} = \sum_{n=0}^\infty x^n\\
\\
\frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^\infty (1-x)^n\\
\\
\frac{1}{x^2} = \sum_{n=0}^\infty (1-x^2)^n\\
\\
\frac{1}{(2-x)^2} =...
Hi, I'm reading "Wave Physics" by S. Nettel and in chapter 3 he introduces the Green's function for the 1-dimensional wave equation. Using the separation of variables method he restricts his attention to the spatial component only. Let ##u(x)## be the spatial solution to the wave equation and...
I want to compute the gradient of some smooth function at many points by taking the value of the function at point x(i) subtracted from the value of the function at point x(i+1) and then divide the result by ( x(i+1)-x(i) ). My function has a struct as an argument and within that struct I have...
Introducing the spacetime spherical symmetric lattice, I use the following notifications in my program.
i - index enumerating the nodes along t-coordinate,
j - along the r-coordinate,
k - along the theta-coordinate,
l - along the phi-coordinate.
N_t - the number of nodes along t-coordinate.
N_r...
##r,\theta,\phi## are the usual spherical polar coordinate system.
##\int_v\nabla•(\frac{\hat r}{r})dv## over a spherical volume of radius ##R## reduces to ##\int_s(\frac{\hat r}{r})•\vec ds=4\pi R##
Now ##r## runs from 0 to ##R,\theta## from 0 to ##\pi## and ##\phi## from 0 to ##2\pi##.
In...
The singularities occur at ##z = \pm i\lambda##. As ##\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0##, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to...
I set the location of the particle (x,y,z); therefore,
→
the force F_1 is (z^2/root(x^2+y^2) * x/root(x^2+y^2) , z^2/root(x^2+y^2) * y/root(x^2+y^2), 0), since cosΘ is x/root(x^2+y^2).
→...
This is the form of the function above:
I started by equating (1) to 1/2:
$$T(\varphi)=\frac{r^{2}+\tau^{2}-2\tau\cos\varphi}{1+\tau^{2}r^{2}-2\tau r\cos\varphi} = \frac{1}{2},$$
which can be rearranged to:
$$2r^{2}+2\tau^{2}-1-\tau^{2}r^{2}=2\tau\left[2-r\right]\cos\varphi$$
using...
Hi colleagues
This is a very very simple question
I can show when $f$ is integrable and is even i.e. $f(-x)=f(x)$ then
$\int_{-a}^{a} \,f(x)\,dx=2\int_{0}^{a} \,f(x)\,dx$
what about improper integrals of even functions, like the function ${x}^{2}\ln\left| x...
WHAT HAPPENS IS That I need to model the example of A Protein G example, using a function f in Matlab, but when I execute the script, the graphics I get do not correspond to those of the example.
The problem is that I can not understand what the model seeks to represent, besides that I do not...
Hi PF!
Do you know what a strictly convex function is? I understand this notion in the concept of norms, where in the plane I've sketched the ##L_1,L_2,L_\infty## norms, where clearly ##L_1,L_\infty## are not strictly convex and ##L_2## is. Intuitively it would make sense that any...
The question before asked to find the net force as a function of time which I got:
F = 4.83×10¹⁰ (1.65×10⁻⁸ t − 7.41×10⁻⁶) N
I just have no idea how to do it with the y directed force since I only have a horizontal acceleration equation.
Thanks so much to anyone that helps, I appreciate it!
Hello, I have the force defined as a function of time, where F=A-Bt and A=100N, B=100Ns-1. I have to determine, how long it will take for object to stop, if t0=0s and v0=0,2ms-1 and mass of the object is m=10kg. Can somebody please help me with this, because I'm having hard time with this task.
Section ##3.8## talks about the gradient and smooth surfaces, defining when the directional derivative ##(\partial f/\partial\mathbf{u})(\mathbf{p})## takes maximum value and that when it equals ##0##, then ##\mathbf{u}## is a unit vector orthogonal to ##(grad\ f)(\mathbf{p})##.It also says that...
Well doesn't ##u(x) = 0.4 x## work? Seems too easy, but the phrasing at the end "for all ##x\in I##" makes me think since ##0.4x = x## only at ##x=0##, and not all of ##I##, that this is okay. But am I wrong?
All I've done so far is think about F_net. Since F=ma, and a is a vector, I was thinking that I should find the x and y components of a and then try to calculate F_net that way, but I'm confused as to where I should use x(t) and y(t). Or instead, thinking about it as the change in momentum over...
Hello All,
I have a question regarding the simple function in MATLAB.
My textbook talks about it and it looks very useful, it will show you a bunch of steps of how to simplify an expression or equation.
I am using MATLAB R2018b and it looks like the function is gone. I am wondering if something...
I recently had to find what f(7) equals if f(x) = \frac{x^2-11x+28}{x-7}. I first tried \frac{x^2-11x+28}{x-7} \cdot \frac{x+7}{x+7}, and it seemed like a perfect fit since I eventually got to \frac{x^2(x-4)-49(x+4)}{x^2-49}=(x-4)(x+4), but that gave me f(7)=33, instead of the right answer...
Hi,everyone. Recently, I am studying green's function in many body physics and suffer from trouble.Following are my problems.
(1) What is the origin of the definition of green's function in many body physics?
(2) What is the physical meaning of self energy ? It seems like it is the correction...
Very often, the term "Green's function" is used more than "correlations" in QFT. For example, the notation:
$$<\Omega|T\{...\}|\Omega> =: <...>$$
appears in Schwartz's QFT book. And it seems very natural, basically because the path integral definition of those terms "looks like" the...
If I have a Lagrangian of the form
\mathcal{L}=-\frac{1}{2} (\partial \phi)^2 - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{3!} \phi^6,
in 3 dimensions, what is the one-loop correction to the 4-point function? Am I correct in thinking that the following Feynman diagram is the representation of the...
I just went over analysis of a data set that was analzed using Linear Regression (OLS, I believe) and I saw Newton's method was used. Just curious, how is it used? I assume to minimize the cost function, but this function was not made explicit. Anyone know?
Thanks.
I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement:
$$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the...
If the derivative of a function f is given by
$$f'(x)=\frac{1}{5}(x^2-4)^5-x^2$$
how many points of inflection will the graph of the function have?solution find $f"(x)$
$$f''(x)=2x((x^2-4)^4-1)$$
at $f''(x)=0$ we have factored
$$2 x (x^2 - 5) (x^2 - 3) (x^4 - 8 x^2 + 17) = 0$$
then...
The Feynman propagator:
$$D_{F}(x,y) = <0|T\{\phi_{0}(x) \phi_{0}(y)\}|0> $$
is the Green's function of the operator (except maybe for a constant):
$$ (\Box + m^2)$$
In other words:
$$ (\Box + m^2) D_{F}(x,y) = - i \hbar \delta^{4}(x-y)$$
My question is:
Which is the operator that...
I was able to find the maximum value for this function by differentiating and equating it to zero and find the time t and substitute it back to the original expression to get the max amplitude.
tm = -0.001012 s
v(tm) = 56.6
Another method that was presented in my book was can you explain how...
It is obvious that the function f is not injective. From the given equation, we get f(f(2))=6.And since,there is an inequality given in the problem, I think we can use that to find f(6).But I have got stuck here and can't move.Do I have to find what is f(x) first?Then how?
205.8.9 Find the derivative of the function
$y=\cos(\tan(5t-4))\\$
chain rule $u=\tan(5t-4)$
$\frac{d}{du}\cos{(u)} \frac{d}{dt}\tan{\left(5t-4\right)}\\$
then
$-\sin{\left (u \right )}\cdot 5 \sec^{2}{\left (5 t - 4 \right )}\\$
replacing u with $\tan(5t-4)$
$-\sin{(\tan(5t-4))}\cdot 5...
My attempt :
Given ##f(x)## and ##g(x)## for ## -1.6 < x < 1.6## we get ##0\leq f(x)<1.6##
Thus, for ##f(g(x))## we get ## -3 \leq g(f(x)) < -1.4##
Thus the required set should be the interval ##[-3, -1.4)##?
My Questions :
1. What have I missed since my answer does not match the given...
Quantum fields have wave functions that determine a particle position in space. It solves non-locality, double-slit paradox, tunnel effect, etc. What if the wave function is also in time? Won't it solve the breaking of causality at quantum level? (Delayed Choice/Quantum Eraser/Time)
Not much...
Show that the value of ##\int_0^1\sqrt(1-cosx)dx## is less than or equal to ##\sqrt2##
##1\ge cos x\ge-1##
The problem is a worked one but I am just confused by a simple thing. We integrate the function f ##\int_0^1\sqrt(1-cosx)dx in the interval [0,1] but I don't understand that what stands...
To check if it is injective :
##h'(x) = 3(x^2-1)##
##\implies h'(x) \geq 0## for ##x \in (-\infty, -1]##
Thus, ##f(x)## is increasing over the given domain and thus is one-one.
To check if it is surjective :
Range of ##f(x) = (0, e^4]## but co-domain is ##(0, e^5]## thus the function is into...
Say I have a function `func` that does a certain task, returning some expression `exp`. Can I use this expression in another function without having to call `func` again, which I suppose will redo all the steps needed to derive `exp` in the first place?
E.g double func() {...
1.To shift the graph of a function :
Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative.
Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...
If I'm given a function ##f(x)##, say it has continuos first derivative, then I expand it as ##f(x + \Delta x) = f(x) + (df / dx) \Delta x##. If instead, I'm given ##f^{-1}(x)## how do I go about expanding it? Will this be just ##f^{-1}(x + \Delta x) = f^{-1}(x) + (df^{-1} / dx) \Delta x##?
Hi,
I want to program an GARCH model for exchange rates. To do this, I calculated the residuals. Next, I did the following (in python)
def main():
vP0 = (0.1, 0.05, 0.92)
a = minimize(garch_loglike, vP0, eps, bounds = ((0.0001, None), (0.0001, None), (0.0001, None))...
Is there anyone here that know and understand the OVGF method who can help me? I have some doubts about it, and there is almost nothing about it in literature.
From ##\vec r''(t)## we integrate to get
$$\vec r'(t) = \left(-\sin(t)+C_1\right)\hat i + \left(6\cos(2t)+C_2\right)\hat j - \left(9.8t+C_3\right)\hat k$$
Solving for the C constants using ##\vec r'(0) = 1\hat i + 6\hat j + 0\hat k##,
##\vec r'(0) = <C_1, C_2, C_3>##
##=<1, 6, 0>##
So we now...
I know that due to causality g(t-t')=0 for t<t' and I also know that for t>t', we should get
g(t-t')=\frac{sin(\omega_0(t-t'))}{\omega_0}
But I can't seem to get that to work out.
Using the Cauchy integral formula above, I take one pole at -w_0 and get
\frac{ie^{i\omega_0(t-t')}}{2\omega_0}
and...
For 1) I found two ways but I get difference results.
The first way is I use P(A|B) = P(A and B)/P(B).
I get P(X<1|Y<1)=(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗)/(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗+∫_1^2▒∫_0^(2-x)▒〖3/4 (2-x-y)dydx〗)=6/7
The 2nd method is I use is
f(x│y)=f(x,y)/(f_X (x)...
Let $m_n$ be the smallest value of the function:
$$f_n(x)=\sum_{k=0}^{2n}x^k.$$
Show, that $m_n\to\frac{1}{2}$ as $n \to \infty$.
Source: Nordic Math. Contest
Hi, I am curious about:
$$x^m \delta^{(n)}(x) = (-1)^m \frac {n!} {(n-m)!} \delta^{(n-m)}(x) , m \leq n $$
I understand the case where m=n and m>n but not this. Just testing the left hand side with m=3 and n=4 and integrating by parts multiple times, I get -6. With the same values, the...
I am working with a polynomial and wish to integrate over one of it's branch surfaces with high precision. The function is:
## -z^2 + z^3 + w (-4 z + 3 z^2) + w^3 (-2 + 8 z + 4 z^2 - 4 z^3) + w^2 (-z^3 - 9 z^4) + w^4 (6 - 8 z^2 + 7 z^3 + 8 z^4)=0##
So I first solve the associated...
As you can see, I've tried using KCL at node A to find the 2nd order ODE that describes this circuit in terms of the capacitor voltage. The problem I run into, however, is that I can't find anything to put the node voltage at A in terms of. I've tried (not shown here) doing mesh current as well...
Form solid state physics, we know that the volume of k-space per allowed k-value is ##\triangle{\mathbf{k}}=\dfrac{8\pi^3}{V}##
##\sum_{\mathbf{k}}F(\mathbf{k})=\dfrac{V}{(2\pi)^3}\sum_{\mathbf{k}}F(\mathbf{k})\triangle{\mathbf{k}}##...
OK, so I'm trying to work out a few ideas regarding locality. I've studied at the undergrad level in the past (including quantum), but with professors that slaved away at proving math constructs and never bothered to indulge in clarifying the context of any concepts, so I'm pretty weak here...
The graph of the Planck blackbody function has an interesting feature:## \\ ## ## \rho_o=\frac{\int\limits_{0}^{\lambda_{max}} L_{BB}(\lambda,T) \, d \lambda}{\int\limits_{0}^{+\infty} L_{BB}(\lambda, T) \, d \lambda} \approx .2500 ##,
where ## \lambda_{max} ##, in an exact derivation of...