Derivative Definition and 1000 Threads

Is ##\text{d}^{2}=\text{d}\wedge\text{d}## a definition of the exterior algebra, or can it be derived from more fundamental mathematical statements?

I am trying to numerically solve a PDE, and just had a question as to the validity of a certain approach. For example, given the PDE: $$ \frac {\partial ^2 E}{\partial t^2} = - k\frac {\partial E}{\partial t} + c^2 \frac {\partial ^2 E}{\partial z^2} - c\frac {\partial E}{\partial z}$$ If I...

I know that taking the derivative of certain functions that explain physical phenomena can lead to another statement describing the physical system, the most famous being the derivatives of position. That is, position-->velocity-->acceleration-->jerk-->jounce...and taking any other further...

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...

Please refer to the below image (Example 5). Do anyone know how 5x^4 > 1 > 0?

Homework Statement Find y'' Homework Equations 9x^2 +y^2 = 9 The Attempt at a Solution y' 18x+2y(y')=0 y'=-18x/2y y'=9x/y For the second derivative, I get the correct answer (same as the book) up until the very last step. Here's where I'm left at: -9( (-9x^2 - y^2) / y^3 ) The book then...

What does a function's smallest non-zero derivative say about the function? For example, say we have a function that looks like 6x^3, if you keep taking the derivative of this function until you get the smallest non-zero derivative, in this case 6x^3 -> 18x^2 -> 36x -> 36, what is the...

Homework Statement Let $$f(x,y)=\begin{cases} \frac{x^2y}{x^2+y^2} \space & \text{if} \space(x,y)\neq(0,0)\\0 \space & \text{if} \space(x,y)=(0,0)\end{cases}$$ a) Use the definition of the partial derivative to find ##f_x(0,0)## and ##f_y(0,0)##. b) Let a be a nonzero constant and let...

Homework Statement Hi, this is a question that has been bothering me for a while. (Im in calculus II at the moment) Why do i need to derivate some functions by definition and other times i dont? for example if somebody asks me to calculate the partial derivatives of a branch function in a a...

Homework Statement My homework problem is a proof in orbital mechanics, but I'm not looking for specific help on that just yet, I'd like to work through it on my own. In doing so however, I'm having a hard time conceptualizing the idea of derivatives of vectors at a specified time. If r is a...

Mod note: Changed title from "Differential Euler's Number" 1. Homework Statement Find the derivative. f(t)=etsin2t The Attempt at a Solution f'(t)=etsin2t(sin2t)(cos2t)(2) However the book seems to say that there should be an extra "t" in the solution. Some help?

How does Hubble parameter and scale factor's derivative differ geometrically? I am reading S. Caroll's GR book. But I cannot get the full representation of these two parameters. On the book, it says How can \dot{H} and \ddot{a} be opposite of each other on the same instance if both are...

Homework Statement Let ##f(x,y) = \|x \| - \|y\| - |x| - |y|## and consider the surface defined by the graph of ##z=f(x,y)##. The partial derivative of ##f## at the origin is: ##f_{x}(0,0) = lim_{h \rightarrow 0} \frac{ f(0 + h, 0) - f(0,0)}{h} = lim_{h \rightarrow 0} \frac {\|h\| -|h|}{h} =...

I just realized there's a little difference between the differential and integral forms of Faraday's law I didn't notice earlier. In the differential form, it is the partial time derivative that is written, while in integral form, it is simply the time derivative. Why is that ?

Hello, and thank you for your time. I just started my A-levels derivatives/differentiation , and I would be more than happy if you could help me clarify it. For example I know that y is a function in terms of x right? y=f(x) The derivative of it is f'(x)=dy/dx . This means it is the rate of...

Homework Statement Find the directional derivative using ##f\left(x,y,z\right)=xy+z^2## at the point (4, 2, 1) in the direction of a vector making an angle of ##\frac{3π}{4}## with ##\nabla f(4, 2, 1)##. Homework Equations ##f\left(x,y,z\right)=xy+z^2##The Attempt at a Solution I found the...

Homework Statement (a) Find the directional derivative of z = x2y at (3,4) in the direction of 3π/4 with the x-axis. Give an exact answer. (b) Find the directional derivative of z = x2y at (3,4) in the direction that makes an angle of 3π/4 with the gradient vector at (3,4). Give an exact...

F = f(x*f(x^2)), such that  f (4) = 6,  f '(4) = 1, and  f '(12) = 3. Find F '(2) I know the format looks weird, but that's exactly how the function was written, which is why I'm not sure how to proceed with this one.

Consider the principal branch of the function f(z)= z7/3 Find f'(-i) and write it in the form a+bi My attemp is : I know zc = exp(c logz) and the derivative of that is : (c/z) * exp(c Logz) That is in this case (7/3)*(i) *exp((7/3)*Log-i) = f'(-i) I know that Log(-i) = Log(1) + i(-pi/2)= -i...

Hello, I have a maybe unusual question. In a paper, I recently found the equation $$\mathcal{L}_v(v_i dx^i) = (v^j \partial_j v_i + v_j \partial_i v^j) dx^i$$ Where v denotes velocity, x spatial coordinates and \mathcal{L}_v the Lie derivative with respect to v. Now I'm an undergraduate who...

There has been SO much talk of HP and HP vs torque, it can make your head spin. I've been trying to help with some clarification for those that seem to be very confused of the physics and concepts that i came up with a new "definition" to augment what is commonly read as the definition of HP...

Hello, I am working with the mass flow rate equation which is:$$\frac{d \dot{m}}{dt}=\dot{m}_{in}-\dot{m}_{out}$$ To determine the change of the height of water in a reservoir. Assuming m_in = 10 and m_out = sqrt(20h), then : $$\frac{d (\rho \cdot Q) }{dt}=\rho \cdot Q_{in} - \rho\cdot...

Homework Statement E(t) = 100te1+cos((2π*t)/365 What is the derivative of this function? Homework EquationsThe Attempt at a Solution 100*e1+cos((2π*t)/365) + 100t*e1+cos((2π*t)/365) * -(2π/365)sin(2πt/365) I have tried to use the rules for derivative of products, and also used the chain rule.

Homework Statement Homework Equations d/dx The Attempt at a Solution d/dx (T) = d/dx(1/2mx'2) = mx'' d/dx(U) = d/dx(1/2kx2) = kx' ≠ kx It's probably me who made an error because I know that that equation (2.3) is the one I should be getting, but I don't understand how they did it because...

I'm working through the discussion of calculus of variations in Taylor's Classical Mechanics today. There's a step where partial differentiation is involved that I don't understand. Given: $$S(\alpha)=\int_{x_1}^{x_2} f(y+\alpha\eta, y'+\alpha\eta', x)\,dx$$ The goal is to determine ##y(x)##...

Find $f'(x)$ $\displaystyle f(x)={3}^{2x+5}+\log_3(x^2+4)$ Didn't know how to do the $\log_3(x^2+4)$

Homework Statement The normal approach using the fundamental theorem of calculus seems inapplicable. I define a function B(R) based on a definite integral with one of the limits being R. One factor in the definite integral has R in it and that function vanishes to 0 at x = R. Using the...

Consider this equation: f(x(t),y(t))=2(x(t))^2+x(t)y(t)+y(t) One way to calculate df/dt is directly using the chain rule: \frac{df}{dt}=4x(t)\frac{dx}{dt}+\frac{dx}{dt}y(t)+\frac{dy}{dt}x(t)+\frac{dy}{dt} \frac{df}{dt}=(4x(t)+y(t))\frac{dx}{dt}+(x(t)+1)\frac{dy}{dt} Another way is by using...

I am a 7th grader who is interested in Quantum mechanics and I'm learning schroninger's equation and there is a partial derivative in it and I looked it up but the best I could find was that it was a function of variables of the variables derivatives, but that didn't make much sense. Can someone...

I had posted a question earlier which this is related to, but a different equation. $$\frac{d}{dt} \int_0^t H(t,s)ds = H(t,t) + \int_0^t \frac{\partial H}{\partial t}(t,s)ds$$ This was another formula needed in a proof however I don't see how this one holds either. I tried following a proof of...

Hello. We have the derivative of a function: d F(x)/dx. If we substitute x = au, how can I show that d F(x)/dx = (1/a) (dF/du) ?

Find the derivative of the function $f(y)$ $$f(y)=\tan^{-1}\left({8{y}^{3}+1}\right)$$

Hello so if we have geodesic equation lagrange approximation solution: d/ds(mgμνdxν/ds)=m∂gμν∂xλdxμ/ds dxν/ds. So if we have schwarzschild metric (wich could be used to describe example sun) which is:ds2=(1-rs/r)dt2-(1-rs/r)-1dr2-r2[/SUP]-sin22. But that means that ∂gμν/∂xλ=0. So that means that...

$\large{242.7.3.83}$ Differentiate $$\displaystyle f(x)=\ln\left[{\frac{(2x+3)(x+6)^5}{(1-2x)^3}}\right]$$ Assume first step is expansion.. $$f(x)=\ln\left({2x+3}\right) +5\ln\left({x+6}\right) -3\ln\left({1-2x}\right)$$

If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##. Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial...

Hey all, I've had this point of confusion for a bit and I have thought that with time I may be able to clear it out myself. Nope, hasn't happened. I think I need help. Let us say we have the following \phi_{k+1}=\phi_{k}+v_k where, v_k\overset{iid}{\sim}\mathcal{N}(0,\sigma^2) and...

find ∫ln x dx. i can't work this out but i know its integration. why is there a dx here. there is usually no dx when i am differentiating something

I am trying to create a function of A and x which has the following properties. A is a scaling parameter that determines the shape of the function. I write the function below in f(A,x) form 1) f(A,1)=1 always 2) For all x>1, 0<f ' (x)<1 3) As A approaches some upper bound (which could be...

I am really struggling with one concept in my study of differential geometry where there seems to be a conflict among different textbooks. To set up the question, let M be a manifold and let (U, φ) be a chart. Now suppose we have a curve γ:(-ε,+ε) → M such that γ(t)=0 at a ∈ M. Suppose further...

Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...

Supposing we have $f(x) = {2}^{lnx}$, how would we find $f'(x)$?

I have a set of data as follows, How can I calculate the second derivative of the curve obtained from these data. x=[0.1;0.07;0.05;0.03;0]; r=[-98.9407;-105.7183;-111.2423;-116.0320;-120.0462];

Hi, friends! I read that, if ##f\in L^1[c,d]## is a Lebesgue summable function on ##[a,b]## and ##g:[a,b]\to[c,d]## is a differomorphism (would it be enough for ##g## to be invertible and such that ##g\in C^1[a,b]## and ##g^{-1}\in C^1[a,b]##, then...

Homework Statement I would just like to know if this statement is true. Homework Equations \frac {\partial^2 f}{\partial x^2} \frac{\partial g}{\partial x}=\frac{\partial g}{\partial x} \frac {\partial^2 f}{\partial x^2} The Attempt at a Solution I've thought about this a bit and I haven't...

Can someone help me with this? (dA/dt)=1cm/s (cm^2 whatever...leave out trivial corrections). A=pir^2 (dA/dt)=2pir(dR/dt) Multiply through by (1/2pir) (dA/dt)/(2pir)=dR/Dt What is the rate of change of the radius for a circumfrance of 2 I just used the related rates formula that I derived for...

Ok so when differentiating 1/(1-x)= Σ xn from n=0 to infinity the book says it is 1/(1-x)^2 = Σ n*(x)n-1 from n=1 to infinity i don't understand why the original sum starts at 0 and then the derived sum starts at 1

Hi guys! i have a problem, and I'm unable to solvie it :/ I have this two variable function: it is 0 in {0,0} while it is (x^3 y^2)/(x^2+Abs(y)^(2a)) elsewhere. do...given the vector {l1,l2} they are asking me: for which "a" the directional derivative along that vector exist in {0,0}? and when...

Homework Statement ## \frac{d}{dt}\gamma(t)\vec{u(t)} ## Homework Equations See above The Attempt at a Solution This comes from trying to verify a claim in Chapter 12 of Griffiths Electrodynamics, 4th. edition (specifically Eq. 12.62 -> Eq. 12.63, if anyone has it on hand). I would have...

Homework Statement I have been working on the problem below and I am stuck. I am stuck primarily because of the part where is says x=0. If x-0, it should cancel everything out. The derivative of 0 is 0 so will cancel everything out I think, so I am not sure if that is the reasoning and the...

I'm sorry if it is a silly question, but I've looked elsewhere and I haven't found the answer. Please tell me if any of the following is true: ∂μ† = -∂μ ∂i† = -∂i ∂μ† = -∂μ ∂i† = -∂i I'm studying QFT and I don't know how to proceed when I have to take the adjoint of a derivative. Sometimes it...