Derivative Definition and 1000 Threads

Let's say I have two vector fields a(x,y,z) and b(x,y,z). Let's say I have a scalar field f equal to a•b. I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b. Ideally, I would like to be able to express...

Hi everyone, Given a vector-valued function ##\vec{A}##, how do I show that: $$\vec{\nabla} \times \left(\frac{\partial \vec{A}}{\partial x}\right) = \frac{\partial}{\partial x}(\vec{\nabla} \times \vec{A})$$ In other words, are the cross product and derivative commutative w/ each other? I...

Homework Statement Can someone please check my working, as I am new to Einstein notation: Calculate $$\partial^\mu x^2.$$ Homework Equations 3. The Attempt at a Solution [/B] \begin{align*} \partial^\mu x^2 &= \partial^\mu(x_\nu x^\nu) \\ &= x^a\partial^\mu x_a + x_b\partial^\mu x^b \ \...

Homework Statement The problem statement is in the attachment Homework Equations E[/B] = -∇φ ∇ = (∂φ/∂r)er The Attempt at a Solution I am confused about how to do the derivative apparently because the way I do it gives E = - (∂[p*r/4πε0r3]/∂r)er = 3*(p*r)/4πε0r4er

Homework Statement How do you find the derivative of the radial vector r Homework Equations r [/B]= ru'_r + ru_r r = \frac{dr}{dt}u_r + r\frac{du_r}{dt} can't get latex to work either The Attempt at a Solution [/B] If r is the magnitude of r, how would you find the derivative of it...

Homework Statement "Derive the equations for position (in terms of acceleration, initial position, initial velocity, and time) and velocity (in terms of constant acceleration, a, initial velocity, v0, and time, t) from the definitions of position, velocity, and acceleration (derivative...

Hello all, I have a complicated function: \[f(x)=\left ( e^{x}+x \right )^{^{\frac{1}{x}}}\] I need to find it's derivative and it's limit when x goes to infinity. As for the derivative, I thought maybe to use LN, so that I can get rid of the exponent, am I correct? How should I approach...

Homework Statement Hi, I am trying to follow the working attached which is showing that the average energy is equal to the most probable energy, denoted by ##E*##, where ##E*## is given by the ##E=E*## such that: ##\frac{\partial}{\partial E} (\Omega (E) e^{-\beta E}) = 0 ## MY QUESTION...

Homework Statement $$f:\mathbb{R^2}\to\mathbb{R}$$ a differentiable function in the origin so: $$f(t,t) =t^3+t$$ and $$f(t,-2t)=2t$$ Calculate $$D_vf(0,0)$$ $$v=(1,3)$$ Homework Equations 3. The Attempt at a Solution [/B] I have no idea on how to approach this problem. I know that...

I've been studying a bit of differential geometry in order to try and gain a deeper understanding of the mathematics of general relativity (GR). As you may guess from this, I am approaching this subject from a physicist's perspective so I apologise in advance for any lack of rigour. As I...

Homework Statement I'm trying to find a formula for the nth derivative for the function f(x)=x1/3 The Attempt at a Solution I know that it has alternating signs so it start with (-1)n+1 and I know the exponent for it is x(1/3-n) but I'm having a hard time figuring out the coefficient of x...

Homework Statement Can someone explain why for the first derivative 3 point forward difference formula is 4f(x-h) - f(x-2h) ?? Homework EquationsThe Attempt at a Solution Why it's not f(x-h) - f(x-2h) ? Is there anything wrong with the notes ?

Hey! :o Let $a\in \mathbb{R}$. Find the derivative of the function $f:\mathbb{R}\rightarrow \mathbb{R}$ $$f(x)=\left\{\begin{matrix} x^ae^{-\frac{1}{x^2}} & \text{ if } x>0\\ 0 & \text{ if } x\leq 0 \end{matrix}\right.$$ in all the points $x\in \mathbb{R}$, where it exists. So, first we have...

I'm going through a basic introduction to tensors, specifically https://web2.ph.utexas.edu/~jcfeng/notes/Tensors_Poor_Man.pdf and I'm confused by the author when he defines vectors as directional derivatives at the bottom of page 3. He defines a simple example in which ƒ(x^j) = x^1 and then...

Homework Statement The question is given just like this: ##\frac{d}{dx}(exp\int_1^x P(s)\ ds)## = ? I assume they want me to find the derivative of the whole thing. Homework EquationsThe Attempt at a Solution I'm thinking the first step is: ##\frac{d}{dx}(exp\int_1^x P(s)\ ds) = (exp\int_1^x...

Hello, What is the numerical formula for a derivative, considering the x's are not in the same distance?

Homework Statement Show ##DF/Dt=0##. ##F = x-a-e^b\sin(a+t)## and ##a## is given implicitly as ##y=b-e^b\cos(a+t)## where ##a=f(y,t)## and ##b## is a constant. Also, velocity is $$u=e^b\cos(a+t)\\v=e^b\sin(a+t)$$ Homework Equations ##DF/Dt=F_t+v\cdot\nabla F## The Attempt at a Solution...

$\tiny{242.2q.3}$ $\textsf{find the derivative}\\$ \begin{align} \displaystyle y&=\frac{x \, \sqrt[]{x^2+1}}{(x+1)^{2/3}} \\ \ln{y}&=\ln x + \frac{1}{2}\ln(x^2+1) - \frac{2}{3}\ln(x+1)\\ \end{align} $\textit{thot this would help but what next??}$

$\tiny{242.2q.3}$ $\textsf{Find the derivative}\\$ \begin{align} \displaystyle y&=\frac{1+\ln{(t)}}{1-\ln{(t)}} =-\frac{1+\ln(t)}{\ln(t)-1}=\frac{f}{g}\\ f&=1+\ln(t) \therefore f'=\frac{1}{t}\\ g&=\ln(t)-1 \therefore g'=\frac{1}{t}\\ y'&= \frac{f\cdot g' - f'\cdot g}{g^2}\\...

Hi, I am struggling to derive the relations on the right hand column of eq.(4) in https://arxiv.org/pdf/1008.4884.pdfEven the easy abelian one (third row) which is $$D_\rho B_{\mu\nu}=\partial_\rho B_{\mu\nu}$$ doesn't match my calculation Since $$D_\rho B_{\mu\nu}=(\partial_\rho+i g...

Homework Statement [/B] I've tried to search this up but to no avail. How am I suppose to solve this: d2y/dx3Homework Equations N/A The Attempt at a Solution Here's what I think I need to do: 1: Square and cube y and x respectively. 2: Find the second and third derivative of y and x...

I am trying to learn GR, primarily from Wald. I understand that, given a metric, a unique covariant derivative is picked out which preserves inner products of vectors which are parallel transported. What I don't understand is the interpretation of the fact that, using this definition of the...

Hello, I am having trouble with solving the problem below The problem Find all primitive functions to ## f(x) = \frac{1}{\sqrt{a+x^2}} ##. (Translated to English) The attempt I am starting with substituting ## t= \sqrt{a+x^2} \Rightarrow x = \sqrt{t^2 - a} ## in $$ \int \frac{1}{\sqrt{a+x^2}}...

Homework Statement The two graphs are possible legitimate representations of ##y=\sec^{-1}(x)##. The derivative is positive on all the domain and so is graph A, but graph B has negative tangent when x<-1 Homework Equations Derivative of inverse secant...

Homework Statement -here is the problem statement -here is a bit of their answer Homework Equations Chain rule, partial derivative in spherical coord. The Attempt at a Solution I tried dragging out the constant and partial derivate with respect to t but still I can't reach their df/dt and...

Homework Statement $$\int\frac{dx}{x\sqrt{x^2-1}}=\left\{ \begin{array} {lc} \sec^{-1}(x)+C_1 & {\rm if}~x>1 \\ \sec^{-1}(-x)+C_2 & {\rm if}~x<-1 \end{array} \right.$$ Why the second condition ##\sec^{-1}(-x)+C_2~~{\rm if}~x<-1## ? Homework Equations Derivative of inverse secant...

I am not that great at vector calculus , etc. Can someone show me how to take the time rate of change of a potential gradient? (Not homework) Thx.

An inductor and resistor are arranged in parallel to a constant voltage source. There is a switch connected to a terminal on the inductor that can create a closed loop that includes either the voltage source, or the resistor. The switch is left connecting the source and inductor for a long...

Homework Statement Hi all! I'm having trouble understanding the implementation of some derivatives in the expression (1) of this article: https://www.ncbi.nlm.nih.gov/pubmed/26248210 How do I implement ∑(ij) ∂ijw ? Thank you all in advance. Homework Equations w is a square matrix(120x120)...

1. Homework Statement Find derivative of y=e^(cos(t)+lnt) Homework EquationsThe Attempt at a Solution So just using the chain rule: y'=e^(cos(t)+lnt)*(-sin(t)+1/t) The answer in the back of the book is y'=e^(cos(t))*(1-tsin(t))

Good Morning Could someone please distinguish between the Frechet and Gateaux Derivatives and why one is better to use in the Calculus of Variations? In your response -- if you are so inclined -- please try to avoid the theoretical foundations of this distinction (as I can investigate that by...

Hi all, According to wikipedia: Can someone explain to me with a mathematical proof the following: $$ \frac {\partial f(x)} {\partial v} = \hat v \cdot \nabla f(x) $$ I don't get this identity except the special example where the partial derivative of f(x) wrt x is a special kind of a...

Desperate times call for desperate measures. I hope someone can show me how to do this. I don't want to offend anyone, but the truth is i have no work to show. I have exam on monday and i know a task like this will be given, exactly the same just different numbers. I have no vision on studying...

Assume we known that f(0) = 1 and f'(0)=2 Find $$xf^7(x)''(0)$$ Will chain rule work here? is the $$ u=xf^7$$ and $$ y = u^7$$ I don't know if I am going in the right direction.

How does this equation: $$\dfrac{12x\sqrt{2x^3+3x+2}-\frac{\left(6x^2+3\right)^2}{2\sqrt{2x^3+3x+2}}}{2\left(2x^3+3x+2\right)}$$ becomes this equation $${12x^4+36x^2+48x-9}frac{4\left(2x^3+3x+2\right)^\frac{3}{2}}$$

if (d/dx) cos(x) = -sin(x) then (d/d{1/x}) cos(x) = ? i.e. the derivative of cos(x) with respect to 1/x

I know it seems pretty self explanatory, but I've tried to do this question and I've apparently gotten the wrong answer twice. If anyone can give me a clear solution to the problem, that would be greatly aooreciated. I initially tried to follow a video I saw online, but I think there is...

Homework Statement So I know I have to take the derivative with respect to x, then respect to y, then respect to z, but I am not getting the right answer. I know that the answer is 0 and my professor did it with very few steps that I do not understand. Can someone please guide me through it?

Hi, So, in order to calculate a Jacobian, I need to evaluate a partial derivative of a total derivative, i.e. Let's say I have a function f(x), how do I calculate something like: ∂(df/dx)/∂f?

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...

I'm going through the book "Elementry Differnetial Equations With Boundary Value Problems" 4th Eddition by William R. Derrick and Stanley I. Grossman. On Page 138 (below) ) The authors take the derivative of a definite integral and end up with a definite integral plus another term. How did...

According to Faraday's Law, Time-Changing magnetic field creates an induced current in a closed conducting loop. This is the equation: ##\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}## 1-) Does this current (##\nabla \times \mathbf{E} ##) have to be an alternate...

Consider the covariant derivative ##D_{\mu}=\partial_{\mu}+ieA_{\mu}## of scalar QED. I understand that ##D_{\mu}\phi## is invariant under the simultaneous phase rotation ##\phi \rightarrow e^{i\Lambda}\phi## of the field ##\phi## and the gauge transformation ##A_{\mu}\rightarrow...

i could only find g'(x) which is 1 but can't find 4f(x) may anyone help me again.

Does it want me to plug in G'(3) into the equation?

Hi PF I try to understand how we get get a Taylor expansion of a non linear functional. I found this good paper here F maps functions to scalars. F[f] is defined. It has not scalars as arguments. I agree with A13 and A18. In another paper (in french) skip to page 9 the fisrt term is ##\int dx...

Consider the Lie derivative of the vector field ##\bf{Y}## with respect to the vector field ##\bf{X}## on manifold ##M^{n}(x)## defined as ##\displaystyle{[\mathcal{L}_{\bf{X}}Y]_{x}:=\lim_{t\rightarrow 0} \frac{[{\bf{Y}}_{\phi_{t}x}-\phi_{t*}{\bf{Y}}_{x}]}{t}}## Now, I understand that...

Homework Statement http://prntscr.com/czcn8h Homework Equations n/a The Attempt at a Solution I know that if you derive x^2sin(1/x) you get -cos(1/x) + sin(1/x)(2x). But what do I do from here? If I use the limit definition, i'll end up getting something like h(sin(1/h)) after evaluating. I...

Homework Statement Develop aprogram that will determine the second derivative of pi(16 x^2 - y^4) at y=2 with step sizes of 0.1, 0.01, 0.001…. until the absolute error (numerical-analytical) converges to 0.00001. Use the 2nd order Central Difference Formula. User Input: y, tolerance Output: h...

I am trying to solve a system of equations and have a question regarding the validity of my approach when implementing a fifth-order Cash-Karp Runge-Kutta (CKRK) embedded method with the method of lines. To give the questions some context, let me state the problem I am attempting to solve: $$...