Functions Definition and 1000 Threads

Hi all, I understand that the mixed partial derivative at some point may not be equal if the such mixed partial derivative is not continuous at the point, but are the actual functions of mixed partial derivatives always equal? In other words, if I simply compute the mixed partial derivatives...

Hello all, First of all, I am aware that dissonance and consonance between pitches also depend to an extent by culture and musical origin but there also seems to be some degree of objective perception among people that can be explained scientifically. Also, I'm very new to this so I could be...

Homework Statement The water depth in a harbor is 21m at high tide and 11m at low tide. Once cycle is completed every 12 hrs. (a) Find equation for the depth as a function of time. (b) Draw a graph for 48 hrs after low tide, which occurred at 14:00. (c) State the times where the water...

Homework Statement Find the following limit: Homework EquationsThe Attempt at a Solution My lecturer has said that rational functions which are a ratio of two polynomials are continuous on R^2. He also said that the limits of continuous functions can be computed by direct substitution. The...

first of all assume that I don't have proper math knowledge. I came across this idea while I was studying last night so I need to verify if it's valid, true, have sense etc. orthogonality of function is defined like this: https://en.wikipedia.org/wiki/Orthogonal_functions I wanted to...

Hi, As is commonly known, u = u(T,v) h = u(T,p) I've worked with some maths proofs of this a while ago, but do you guys have an intuitive way of understanding this without the maths, that is, why the state function for internal energy is defined by intensive volume and enthalpy with pressure...

What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...

I've been reading about canonical transformations in Hamiltonian mechanics and I'm a bit confused about the following: The author considers a canonical transformation $$q\quad\rightarrow\quad Q\quad ,\quad p\quad\rightarrow\quad P$$ generated by some function ##G##. He then considers the case...

The questions is asking me to find \frac{f}{g} basically , the question is asking me to find the answer , even though i know it, i can't get my head around it. the composite function is f(x)=x^2+1 g(x)=1/x we need to find foG (f of g) [composite functions].

Homework Statement Let ##f(t)=\int_{t}^{t^2} \frac{1}{s+\sin{s}}ds,t>1.##Express ##f## as a composition of two differentiable functions ##g:ℝ→ℝ^2## and ##h:ℝ^2→ℝ##. In addition, find the derivative of ##f## (using the composition). Homework EquationsThe Attempt at a Solution Honestly, I have...

Hi I am writing my final Mathematics exams for Grade 12 in South Africa in 5 days. I am well prepared with an aim of getting 100%, but one concept in functions might prevent that - the concept of how the nature of roots are affected by vertical/horizontal shifts in a function, and how to...

Temperatures can be converted from Fahrenheit to Celsius using the function f(x) = 5 /9 (x − 32). (a) Calculate f(59). (b) Find f −1 (x), and verify that f −1 (f(59)) = 59. (c) Let K be the set {x : f(x) = x}. Find all elements of K and list K

Homework Statement I have a series of 12 values that I need to calculate the Theoretical Intensity, I, using the formula below. I have found values for all variables and their uncertainties, and have calculated the I value for each set using the formula. Now I need to calculate the...

Hi PF! I am wondering why we define velocity for polar coordinates as $$\vec{V} = \nabla \times \frac{\psi(r,z)}{r} \vec{e_\theta}$$ and why we define velocity in spherical coordinates as $$\vec{V} = \nabla \times \frac{\psi(r,\phi)}{r \sin \phi} \vec{e_\theta}$$ The only thing I don't...

Homework Statement Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. Homework Equations Hyperbolic sine: ##\sinh (u)=\frac{1}{2}(e^u-e^{-u})## Hyperbolic...

Prove: ##(\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)## Newton's binomial: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## and: ##(a-b)^n~\rightarrow~(-1)^kC^k_n## I ignore the coefficients. $$(\cosh(x)+\sinh(x))^n=\cosh^n(x)+\cosh^{n-1}\sinh(x)+...+\sinh^n(x)$$...

Here is the question: This is the step I came to after taking the derivatives and doing some simplification: ^ I did the work myself on paper, I just couldn't type out the whole thing clearly so that anyone else can see what I'm referring too... so I used some online tool to show that...

Homework Statement Design an combinational circuit using a decoder and external gates defined by the boolean functions F1, F2, F3(see picture) Homework EquationsThe Attempt at a Solution I'm quite confused as to the exact method in doing this. I understand that a decoder takes n inputs and...

Hi! For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. I'm reading Gasiorowicz's Quantum Physics and, as you can see in the attached image of the page, he works his way to find the momentum operator. My...

I have this exercise on my book and I believe it is quite simple to solve, but I'm not sure if I did good, so here it is Homework Statement given a vector B ∈ ℝn, B ≠ 0 and a function F : ℝ → ℝn such that F(t) ⋅ B = t ∀t and the angle φ between F'(t) and B is constant with respect to t, show...

On wikipedia it says the following, "...the Green's function of the Hamiltonian is a key concept with important links to the concept of density of states." https://en.wikipedia.org/wiki/Green%27s_function Can anyone explain why?

Homework Statement I have the two functions below and have to find the convolution \beta * L Homework Equations Assume a<1 \beta(x)=\begin{cases} \frac{\pi}{4a}\cos\left(\frac{\pi x}{2a}\right) & \left|x\right|<a\\ 0 & \left|x\right|\geq a \end{cases} L(x)=\begin{cases} 1 &...

What is the relation between the correlators ##\langle 0 | T\phi(x_{1})\phi(x_{2}) | 0 \rangle## and ##\langle 0 | T\phi(p)\phi(x=0) | 0 \rangle##? I can derive the momentum space Feynman rules for ##\langle 0 | T\phi(x_{1})\phi(x_{2}) | 0 \rangle##. Are the momentum space Feynman rules for...

When working in the complex domain (##z = x + iy##), how does one write the equation of a line? I have attached a problem I was working on (and have the solution), but am curious as to why the definition of a line is given by ##ax + by = c##. Are not ##x## and ##y## also variables that take on...

$\textbf{10)} \\ f(x)\text{ is continuous at all } \textit{x} \\ \displaystyle f(0)=2, \, f'(0)=-3,\, f''(0)=0 $ $\text{let} \textbf{ g } \text{be a function whose derivative is given by}\\ \displaystyle g'(x)=e^{-2 x} (3f(x))+2f'(x) \text{ for all x}\\$ $\text{a) write an equation of the...

Is v(r) ≡ v(x,y,z)

I have some conceptual issues with functions in vectors spaces. I don't really get what are really the components of the vector / function. When we look at the inner product, it's very similar to dot product, as if each value of a function was a component : So I tend to think to f(t) as the...

Through experimental observations, I have found that the two functions ##f\left(x\right)=x^{a\left(a-x^3\right)}-a## and ##g\left(x\right)=a^{x\left(x-a^3\right)}-x## will always intersect at ##a## when ##x>0##. Is there a way to mathematically prove this? For instance, simultaneously solving...

Homework Statement This problem concerns the surface determined by the graph of the equation ##x^2 + xy -xz = 2## a) Find a function ##F(x,y,z)## of three variables so that this surface may be considered to be a level set of F. b) Find a function ##f(x,y)## of two variables so that this...

Hi all, I am working on the following integral ## \int_0^{2\pi}\frac{1+\cos[\alpha x]}{1+\sin x}dx ## where ##\alpha## is odd integer. Unless I set the ##\alpha## to a number then I can find the integral with mathematica easily. For general case with symbolic ##\alpha##, I cannot find the...

Everyone "knows" that \lim_{x\rightarrow ∞}\frac{2^x}{x^2} = ∞. We "know" this because 2x grows faster than x2. I use quotes because this is just what we're told in basic calculus classes. But what about a theorem for this? I've searched through google, looked through various university homework...

My undergrad probability theory course just got to random variables and distribution functions. Up until this point, the material was very straightforward and I could understand what was being done, but I feel that I am just not seeing the jump between probability with sets and probability with...

Note from mentor: this thread was originally posted in a non-homework forum, therefore it lacks the homework template. I was wondering if the electric charge density ##\rho({\bf{r}})## of a point charge ##q## at position ##{\bf{r}}_{0}## is given by...

Homework Statement Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$ a) Show ##T## is a linear transformation from ##C## to ##C##. b)What...

Homework Statement Given r(t)=\left< \frac { sint }{ t } ,\frac { { e }^{ 2t }-1 }{ t } ,{ t }^{ 2 }ln(t) \right> Re-define r(t) to make it right continuous at t=0 Homework EquationsThe Attempt at a Solution This is probably the simplest problem ever, but I don't even know what it's asking...

Homework Statement Show that sin 600° . cos 330° + cos 120° . sin 150° = - 1 Homework Equations I know that sinΘ = opposite/hypotenuse and cosΘ = adjacent/hypotenuse. The Attempt at a Solution I am equipped with knowledge about what sinΘ and cosΘ is from right angled triangle. I stand in...

Homework Statement Is there a physical difference between the following wave functions? If yes, why? If no, why not? \Psi(x,0) =5e^{-ax^2} \Psi(x,0) =\frac{1+i}{\sqrt{3}}e^{-ax^2} \Psi(x,0) =e^{i\pi/7}e^{-ax^2} Homework Equations - The Attempt at a Solution They only differ in the...

This is not a homework question but a general doubt. Suppose we have a function y = pcosx, where 'p' is an arbitrary constant. So my question is how will the graph of this function change with different values of 'p'? This doubt can also be extended for other functions like y = pex, y = p...

Homework Statement Determine all primitive functions for the function: 2x(x^2+3)^4 2. The attempt at a solution When i expanded i got the primitive to be: 2(x^9/9)+3x^8+18x^6+54x^4+81x^2But this was wrong. I am not sure I have understood the question. Help?

What's the difference between f(x)=3 and f(x)=3x^0 ? and why Limit of the second function when x\rightarrow0 exists ? and is the second function continuous at x=0 ?

Hello! (Wave) I want to show that the domain of any partially defined recursive function is equal to the range of some ( totally defined ) recursive function. I haven't understood which is the difference between a partially defined recursive function and a totally defined recursive function...

So I have to either prove that these functions are 1 - 1 or show a counter example to prove they are not. I believe that I have proven that these functions are 1 - 1, but I am not 100% sure: For each of the following functions, either prove that the function is 1 – 1 or find a counterexample...

A tad embarrassed to ask, but I've been going in circles for a while! Maybe i'll rubber duck myself out of it. If f(t) = f(t+T) then we can find the Fourier transform of f(t) through a sequence of delta functions located at the harmonics of the fundamental frequency modulated by the Fourier...

Homework Statement Let f(x) = 1−3x−2x^2 , x ∈ [−2, −1]. Use the Horizontal Line Test to show that f is 1–1 (on its given domain), and find the range R of f. Then find an expression for the inverse function f −1 : R → [−2, −1]. The Attempt at a Solution I have already done the horizontal line...

In QFT, the commutation relation for the field operator \hat{\phi} and conjugate momentum is [\phi(x,t),\pi(y,t)] = i\delta(x-y) Maybe this is obvious, but what would the commutator of \phi or \pi and, say, e^{i k\cdot x} be?

Hi, Is the Dyson's equation basis independent (for instance, I construct the basis set where the elements are atomic orbitals and those orbitals are non-orthogonal) ? What is the unperturbed retarded Green's function for one-particle case in matrix notation if the basis functions are not...

Hi everybody! I'm preparing an exam of "Analysis II" (that's how the subject's called in German), and I have trouble understanding how to find the limit of a multivariable function, especially when it comes to proving the uniform convergence. Here is an example given in the script of my teacher...

Homework Statement Homework Equations none The Attempt at a Solution -amplitude is 3 -period is 180° -right 60° -down 1 Rough sketch of graph: I would like to know if the graph looks right, is there any improvements to be made? Thanks :)

Homework Statement Homework Equations 3. The Attempt at a Solution a) The height of the high tide is 4.5 m b) The height of the low tide is 0.25 m c) Period = 12.5 hours k= 360/12.5 = 28.8 amplitude = 2.125 m vertical shift = 2.375 m phase shift = it doesn't look like there is any...

Homework Statement I want to prove this relation ##J_{n-1}(x) + J_{n+1}(x)=\frac{2n}{x}J_{n}(x))## from the generating function. The same question was asked in this page with solution. http://www.edaboard.com/thread47250.html My problem is the part with comparing the coefficient. I don't...