Partial Definition and 1000 Threads

I have another dilemma with terminology that is puzzling and would appreciate some advice. Consider the following truncated Taylor Series: $$\begin{equation*} f(\vec{z}_{k+1}) \approx f(\vec{z}_k) + \frac{\partial f(\vec{z}_k)}{\partial x} \Delta x + \frac{\partial f(\vec{z}_k)}{\partial...

$$f(x,y)=\left\{\begin{array}{ccc} (x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & , & (x,y)\neq (0,0) \\ 0 & , & (x,y)=(0,0) \end{array}\right.$$ This function is differentiable at (0,0) point but ##f_x## and ##f_y## partial derivatives not continuous at (0,0) point. I need another...

I've got a PDE that I derived from a physical problem, so I suppose it has a solution and that it is unique. I am solving for streamlines in the region having a quarter of an annulus shape, so ##\theta## ranges between ##0## and ##\pi/2## and ##r## ranges between ##r_i## and ##r_o##. The...

Apologies this is probably a very bad question but it's been a while since I have seen this. I have ##z=x+iy##. I need to convert ##\frac{\partial \psi(z)}{\partial z}## , with ##\psi## some function of ##z##, in terms of ##x## and ##y## I have ##dz=dx+idy##. so ##\frac{\partial \psi }{\partial...

The ####x partial derivative is equal to $$L \frac{4x}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$ and the partial for ##y## is $$L \frac{4y}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$ Using the limit definition of partial derivatives I got the partial wrt ##x## is $$L \frac{h^{\frac{4}{5}}}{h}$$ which doesn’t exist...

Solve the given PDE for ##u(x,t)##; ##\dfrac{∂u}{∂t} +10 \dfrac{∂u}{∂x} + 9u = 0## ##u(x,0)= e^{-x}## ##-∞ <x<∞ , t>0## In my lines i have, ##x_t = 10## ##x(t) = 10t+a## ##a = x(t) - 10t## also, ##u(x(t),t)= u(x(0),0)e^{-9t}## note this is from, integrating ##u_t[u(x(t),t] =...

Solve the given PDE for ##u(x,t)##; ##\dfrac{∂u}{∂t} +8 \dfrac{∂u}{∂x} = 0## ##u(x,0)= \sin x## ##-∞ <x<∞ , t>0## In my working (using the method of characteristics) i have, ##x_t =8## ##x(t) = 8t + a## ##a = x(t) - 8t## being the first characteristic. For the second...

For this problem (b), The solution is, However, I don't understand how they got their partial fractions here (Going from step 1 to 2). My attempt to convert into partial fractions is: ##\frac{2s + 1}{(s - 1)(s - 1)} = \frac{A(s - 1) + B(s - 1)}{(s - 1)(s - 1)}## Thus, ##2s + 1 = A(s - 1) +...

The PDE is $$ \frac{1}{a^2 x^2} (u_y)^2 - (u_x)^2 =1$$ I know the solution, its ## u=x senh(ay) ##, but I dont know how I can get it. I've tried variable separation and method of characteristics but they dont seem to work.

Using the concepts of Summability Calculus but generalized such that the lower bound for sums and products is also variable, we can prove that the solution to the following PDE: $$P^2\frac{\partial^2P}{\partial x\partial y}=(P^2+1)\frac{\partial P}{\partial x}\frac{\partial P}{\partial...

According to my notes, the absorption law states that p ∨ (p ∧ q) = p, p ∧ (p ∨ q) = p I have found a video where they were discussing a partial absorption such as ¬q ∧ (¬p∨q) = ¬q ∧ ¬p This is not in my notes, but is this correct? specifically, is the terminology used to decribe this property...

I’m having a little confusion about part b of this question as to why I am allowed to use the limit definition of a partial derivative. Here’s what I think: I know that y^3/(x^2+y^2) is undefined at the origin but it does approach 0 when it GETS CLOSE to the origin. So technically defining...

How can the following term: ## T_{ij} = \partial_i \partial_j \phi ## to be written in terms of Kronecker delta and the Laplacian operator ## \bigtriangleup = \nabla^2 ##? I mean is there a relation like: ## T_{ij} = \partial_i \partial_j \phi = ?? \delta_{ij} \bigtriangleup \phi.## But...

Hello, Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$ Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be...

Some may say that ##\frac{ \partial g }{ \partial t }## is correct because it is a term in a partial differential equation, but since ##g## is a one variable function with ##t## only, I think ##\frac{ dg }{ dt }## is correct according to the original usage of the derivative and partial...

I am studying a 2D material using tight binding. I calculated density of states using this method. Can I also calculate partial density of states using tight binding?

Hi everyone! It's about the following task. Partial molar quantities a) How are partial molar quantities defined in general? b) If X is an extensive state variable and X̅ is the associated partial variable, what types of variables must X̅ have? c) Is the chemical potential of component i in a...

I study genotype-environment associations in alpine species. I frequently see altitude as the sole predictor of partial pressure of oxygen in the literature concerning hypoxia adaptations. However, I understand that partial pressure of oxygen is also influenced by temperature, humidity, and...

Hello everyone, I seem to be majorly lacking in regards to intuition with partial derivatives. I was studying the Euler-Lagrange equations and realized that my normal intuition with derivatives seems to lead me to contradictory or non sensical interpretations when reading partial derivatives...

Can someone please help me to find out what happened here ?

In the coursebook the question says: The reaction below was carried out at a pressure of 10×10⁴ Pa and at constant temperature. N2 + O2 ⇌ 2NO the partial pressures of Nitrogen and Oxygen are both 4.85×10⁴ pa  Ccalculate the partial pressure of the nitrogen(ll) oxide, NO(g) at equilibrium. In...

Looking at pde today- your insight is welcome... ##η=-6x-2y## therefore, ##u(x,y)=f(-6x-2y)## applying the initial condition ##u(0,y)=\sin y##; we shall have ##\sin y = u(0,y)=f(-2y)## ##f(z)=\sin \left[\dfrac{-z}{2}\right]## ##u(x,y)=\sin \left[\dfrac{6x+2y}{2}\right]##

From Rand Lectures on Light, we have, in the interaction picture, the equation of motion of the reduced density matrix: $$i \hbar \rho \dot_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle = \Sigma_b \phi_b | \langle V \rho_{AB} | \phi_b...

This is part of the notes; My own way of thought; Given; ##U_{xy}=0## then considering ##U_x## as on ode in the ##y## variable; we integrate both sides with respect to ##y## i.e ##\dfrac{du}{dx} \int \dfrac{1}{dy} dy=\int 0 dy## this is the part i need insight...the original problem...

Since we are adding numbers produced according to a fixed pattern, there must also be a pattern (or formula) for finding the sum. Hi, We use this method to find the ##S_n##. I don't understand how the sum will also be in a pattern. Can someone please explain this line in bold?

I'm stuck on this problem, I've tried to follow techniques for similar questions, namely I seem to be struggling with these questions where I have to use an equation inside an equation. I've attached photos of my process so far, but obviously, I'm not getting the right answer because what I'm...

He draws an n-manifold M, a coordinate chart φ : M → Rn, a curve γ : R → M, and a function f : M → R, and wants to specify ##\frac d {d\lambda}## in terms of ##\partial_\mu##. ##\lambda## is the parameter along ##\gamma##, and ##x^\mu## the co-ordinates in ##\text{R}^n##. His first equality is...

ok I posted this a few years ago but replies said there was multiplication in it so I think its a mater of format ##\dfrac{\partial u^2}{\partial x\partial y}## is equivalent to ##u_{xy}## textbook

In my book it is written "Ends of dipole possesses partial charges. Partial charges are always less than the unit electronic charge (1.6×10−19 C)". Suppose in a double bond(two electron is shared by each atom) or triple bond(three electrons are shared by each atom), can the electronegative atom...

I just started to study thermodynamics and very often I see formulas like this: $$ \left( \frac {\partial V} {\partial T} \right)_P $$ explanation of this formula is something similar to: partial derivative of ##V## with respect to ##T## while ##P## is constant. But as far as I remember...

In the 128 pages of 《A First Course in General Relativity - 2nd Edition》:"The covariant derivative differs from the partial derivative with respect to the coordinates only because the basis vectors change."Could someone give me some examples？I don't quite understand it.Tanks！

Rate of effusion

Hello, I am trying to solve the following problem: If ##z=f(x,y)##, where ##x=rcos\theta## and ##y=rsin\theta##, find ##\frac {\partial z} {\partial r}## and ##\frac {\partial z} {\partial \theta}## and show that ##\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z}...

If the right-hand side is zero, then it will be a wave equation, which can be easily solved. The right-hand side term looks like a forced-oscillation term. However, I only know how to solve a forced oscillation system in one dimension. I do not know how to tackle it in two dimensions. I have...

I am going through this page again...just out of curiosity, how did they arrive at the given transforms?, ...i think i get it...very confusing... in general, ##U_{xx} = ξ_{xx} =ξ_{x}ξ_{x}= ξ^2_{x}## . Also we may have ##U_{xy} =ξ_{xy} =ξ_{x}ξ_{y}.## the other transforms follow in a similar manner.

Say you want to find the following Integrals $$\int \frac{1}{(x-1)(x+2)} (dx)$$ $$\int \frac{1}{(x-1)(x^2 + 2)} (dx)$$ The easiest way to solve them will be by using partial fraction decomposition on both the given functions. Decomposing the first function, $$\frac{1}{(x-1)(x+2)} =...

Attempt at question No. 1: ΔD = ∂D/∂h * Δh + ∂D/∂v * Δv ∂D/∂h = 3Eh^2/(12(1-v^2)) ∂D/∂v = 2Eh^3/(12(1-v^2)^2) Δh = +- 0,002 Δv = 0,02 h = 0,1 v = 0,3 ΔD = 3Eh^2/(12(1-v^2)) * Δh + 2Eh^3/(12(1-v^2)^2) * Δv Because the problem asked for maximum percentage error then I decided to use the...

Hi Pfs Partial tracing maps what occurs in a big Hilbert space toward a smaller one. We have to use it when degrees of freedom are physically unobservable or when we have only a coarse grained view of the environment. it is like in Flatland , where the two dimensional inhabitants has no access...

Hello! Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...

Hello! Consider this partial differential equation $$ zu_{xx}+x^2u_{yy}+zu_{zz}+2(y-z)u_{xz}+y^3u_x-sin(xyz)u=0 $$ Now I've got the solution and I have a few questions regarding how we get there. Now we've always done it like this.We built the matrix and then find the eigenvalues. And here is...

Greetings! I want to caluculate the summation of this following serie I started by removing the 4 by and then and I thought of the taylor expansion of Log(1-x)=-∑xn/n but as the 2 is not inside (-1,1) I couldn´t use it any hint? thank you! Best !

Let $$y=\frac {1+3x^2}{(1+x)^2(1-x)}= \frac {A}{1-x}+\frac {B}{1+x}+\frac {C}{(1+x)^2}$$ $$⇒1+3x^2=A(1+x)^2+B(1-x^2)+C(1-x)$$ $$⇒A-B=3$$ $$2A-C=0$$ $$A+B+C=1$$ On solving the simultaneous equations, we get ##A=1##, ##B=-2## and ##C=2## therefore we shall have, $$y=\frac {1}{1-x}+\frac...

How is the order of a partial differential equation defined? This is said to be first order: ##\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0## And this second order :##\frac{d}{d t}\left(\frac{\partial L}{\partial...

I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second...

What the hell is this and is it solvable?

The first plot shows a large number of terms of Zeta(0.5 + i t) plotted end to end for t = 778948.517. The other plots are two zoomed-in regions, including one ending in a Cornu spiral. Despite all sorts of vicissitudes, the plot generally spirals outwards in a "purposeful" sort of way. It is...

Hi For a function f ( x , t ) = 6x + g( t ) where g( t ) is an arbitrary function of t ; then is it correct to say that f ( x , t ) is not an explicit function of t ? For the above function is it also correct that ∂f/∂t = 0 because f is not an explicit function of t ? Thanks

$$\sum_i (\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j}\dot{q}_i)+\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j})\ddot{q}_i)+\frac{\partial}{\partial t}(\frac{\partial T}{\partial \dot{q}_j})$$ They wrote that above equation is equal to...

Intersecting the graph of the surface z=f(x,y) with the yz -plane. This is the option I have chosen, but it's wrong. I don't understand why. x is fixed so I thought the coordinates: y and z are left. I thought this source may be helpful...