Derivative Definition and 1000 Threads

I read that ##d/dx## is the direction that x increases (paraphrased from Griffiths e-mag). Why is this? I can't find any information online when I search this up. Original text:

When a current flows into or out of the capacitor plates, a magnetic field is created between them. Even though there are no charges flowing in the space between the plates, there is still a magnetic field and the source of that field is an uncharged current called a “displacement current”...

I would like to evaluate expressions with Simpy, but unfortunately I am unable to get a simple answer, the one I would get by hand if I had the time to perform all the computations. As far as I understand, Mathematica does it and yields 4 times the Simpy result, which is a big worry since I wish...

y'(t)/x'(t) = cos t/-sin t = -cot t. But as t is the angle, and the derivative is the slope, then isn't the slope supposed to be tan t?

Drawing : I draw a diagram explaining the situation to the right. The vectors ##\mathbf{A}## and ##\mathbf{B}## are drawn at some time ##t## making an angle ##\theta## between them. At some later time ##t+dt##, the same vectors have changed to ##\mathbf{A}(t+dt)## and ##\mathbf{B}(t+dt)## while...

1. How to get f’(1), f’(2), f’(3), and f’(5) 2. How to calculate average speed change of y to changes in x in the interval [0,6] 3. Estimate value of f’+(0) and f’-(6) pls help me about this graph, i dont know how to read this 🙏

Can someone please give as simple an example as possible to show what U substitution is about? I know basic integration rules but don't understand the point of u-substitution. I've read that it's used to "undo the chain rule", but I don't see how, and don't see how we can spot when we'd need to...

In special relativity, is the derivative with respect to coordinate time of relative position equal to relative velocity? Does it matter if constant velocity is used?

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...

I believe that ##f'(\Phi(z))=\frac{df(\Phi(z))}{dz}##, I get confused with the prime in ##z'## and is it really just this derivative? I wonder how many people read this 1983 book.

I have a 5x5x5 set of grid points in space. I can describe each point with p(x,y,z), and I can convert them to spherical or other coordinates. At each point, I have a quaternion assigned to it. So, numerically, I can describe a q(x,y,z) quaternion field. The goal is to obtain a functional form...

For this problem, The solution is, However, does someone please know why we allowed to assume that the derivative exists for f i.e ##f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}##? Thanks!

Is this wrong?

If ##f(g(x)) =f(x+h)##, then ## \lim_{h\to 0} \frac{f(g(x))-f(x)}{h}= \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}=f'(x) ##. What if we let ##g(x) = x+\sin(h)##? Then ## \lim_{h\to 0} \frac{f(g(x))-f(x)}{h}= \lim_{h\to 0} \frac{f(x+\sin(h))-f(x)}{h}## This is not equivalent to that the standard...

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...

Suppose I have a Cartesian Coordinate system (x,y) and a polar coordinate system (##r, \theta##). The position vector (3,4) and (5, ##\arctan \frac{4}{3}##) are the same except the representation. The position vector is a tensor, how does the position vector follow the tensor transformation...

The Riemann curvature tensor contains second derivatives of metric and squares of the first derivatives. The second derivatives have to be there because they are the ones that cannot be eliminated locally by a choice of coordinates. But other than being a mathematical artifact, is there a...

My attempt: Since ##\frac{f(a+h)-f(a-h)}{2h}-f'(a)=O(h^2)## as ##h \to 0##, then: $$\lim_{h \to 0} \frac{\frac{f(a+h)-f(a-h)}{2h}-f'(a)}{h^2} < \infty$$ So $$\lim_{h \to 0} \frac{\frac{f(a+h)-f(a-h)}{2h}-f'(a)}{h^2} = \lim_{h \to 0} \frac{f'(a)-f'(a)}{h^2}=0 < \infty$$ Because the value of...

Good evening, I'm running into some trouble with this problem, and I have a hint as to why, but I'm not completely sure. Please see the steps below for context. I've been able to set up the proper equation representing the density as a function of distance from the center which looks like this...

My doubt arises over the definition of L'(v^2). If we are using ##x= v'^2##, shouldn't the derivative be made with respect to that very term? In essence, shouldn't it be: L'(v^2) = \frac{\partial L(v^2)}{\partial (v'^2)}? In the article I read, L'(v^2) = \frac{\partial L(v^2)}{\partial (v^2)} is...

(ΔSA/ΔD) = 2πHΔD Something is wrong I guess as I get wrong value.

Can't figure it out, here's a screenshot with better typography.

Problem summary I have the characteristic function of a probability distribution but I'm having difficulty obtaining its derivative. Background I am reading the following paper: Schwartz, Lowell M. (1980). On round-off error. Analytical Chemistry, 52(7), 1141-1147. DOI:10.1021/ac50057a033. The...

Let's see how messy it gets... ##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##...

Hi,PF The book is "Calculus" 7th ed, by Robert A. Adams and Christopher Essex. It is about an explained example of the first conclusion of the Fundamental Theorem of Calculus, at Chapter 5.5. I will only quote the step I have doubt about: Example 7 Find the derivatives of the following...

In https://www.math.drexel.edu/~tolya/derivative, the author selects a domain P_2 = the set of all coefficients (a,b,c) (I'm writing horizontally instead off vertically) of second degree polynomials ax^2+bx+c, then defines the operator as matrix to correspond to the d/dx linear transformation...

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...

Here is the problem that I'm trying to solve. I've done the first part that is prove but I'm. facing difficulty in finding nth derivative I'm attaching pics of my attempt of solving this problem as I've no idea that how to type all these mathematical expressions. Can anyone please guide me...

I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...

What is the expression for the covariant derivative of a Weyl spinor?

Setting: a plane with the standard Cartesian coordinate system. A particle is constrained to the x axis, with position x and moving at speed x dot. Another particle is constrained to the y axis, with position y and moving at speed y dot. The distance between them at any moment is s. It is...

This is probably a stupid question but, ## \frac{d\partial_p}{d\partial_c}=\delta^p_c ## For the notation of a 4D integral it is ##d^4x=dx^{\nu}##, so if I consider a total derivative: ##\int\limits^{x_f}_{x_i} \partial_{\mu} (\phi) d^4 x = \phi \mid^{x_f}_{x_i} ## why is there no...

idk how to start after finding the second derivative

TL;DR Summary: I attempt to find the derivative of uv with respect to x using non standard analysis, hyperreals, and the standard part function st; I take u to be a function of x, and I also take v to be a function of x. Hello everyone! I've been learning about non standard analysis concepts...

The answer is E. I was initially very confused as to why the answer was not A but realized that the graph was velocity vs position (rather than velocity vs time) which means I can't simply take the derivative of the given graph. One thing I tried was writing out the equation first(c being a...

The name of this molecule is 1‐(3‐nitrophenyl)ethanol. I'm confused why ethanol is treated as the parent chain in this case, not the phenyl group. If the ring is composed of more atoms, should it be the parent chain? Thank you.

Hi everyone, I'm in a graduate level mechanical engineering thermodynamics class. We're working on derivative reductions using the gibbs and maxwell relations. I was wondering if anyone has any good sources of practice problems that I could use. I've looked through my textbook and there are...

What should I do when the f(x, y) function's second derivatives or Δ=AC-B² is zero? When the function is f(x) then we can differentiate it until it won't be a zero, but if z = some x and y then can I just continue this process to find what max and min (extremes) it has? What I've done is...

I'm having a hard time understanding some concepts and would really appreciate some help(not super smart so I need some things basically dumbed down). In my physics lab we're going over Newton's Second Law. There's a statement in the lab papers I don't understand. It states "As you should know...

Hi, I'm trying to calculate my own physics problem but didn't get it something. When I'm trying to calculate the impulse of the object when it's hit by F=10N force in the smallest possible time, then should I write: dP/dt = Fnet => dP = Fnet*dt ? Another question: In general, if I calculate...

My goal is to develop the equation 21. You should asume that \delta(r_2-r_1)^2 =0. These is named renormalization. Then my question is , do my computes are correct with previous condition ?

So the chain rule for second derivatives is $$ \frac {d^2 y} {d t^2} = \frac{d}{dx}(\frac {dy} {dx}) \cdot \frac {dx} {dt} \cdot \frac {dx} {dt} + \frac {dy} {dx} \cdot \frac {d^2 x} {d t^2} = \frac{d^2 y}{d x^2} \cdot (\frac {dx} {dt})^2 + \frac {dy} {dx} \cdot \frac {d^2 x} {d t^2}$$ Today I...

This isn't a homework problem exactly but my attempt to derive a result given in a textbook for myself. Below is my attempt at a solution, typed up elsewhere with nice formatting so didn't want to redo it all. Direct image link here. Would greatly appreciate if anyone has any pointers.

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 book Quantum Field Theory for the Gifted Amateur, they define the functional derivative as: $$ \frac{\delta F}{\delta f(x))} = \lim_{\epsilon\to 0} \frac{F[f(x') + \delta(x'-x)) ] - F[ f(x') ]}{\epsilon} $$ Why do they use the delta function and not some other arbitrary function?

My try: ##\vec s = \vec \theta \times \vec r## ##\frac {d}{dt} (\vec s) = \frac {d}{dt} (\vec \theta \times \vec r)## ##\vec v = \frac {d}{dt}(\vec \theta) \times \vec r + \vec \theta \times \frac {d}{dt}(\vec r)## ##\vec v = \vec \omega \times \vec r + \vec \theta \times \vec v## And I...

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！

Hi. I was reading Lighthill, Introduction to Fourier Analysis and Generalised Functions and in page 17 there is an example/proof where I can't make sense of the following step: $$ \left| \int_{-\infty}^{+\infty} f_n(x)(g(x)-g(0)) \, \mathrm{d}x \right| \le \max{ \left| g'(x) \right| }...