Derivative Definition and 1000 Threads

I now study general relativity and have a few questions regarding the mathematical formulation: 1) What ist the relation between an connection and a covariant derivative? Can you explain the exact difference? 2) One a lorentzian manifold, what ist the relation between the...

Homework Statement The displacement of a particle can be modeled by the function x(t)=\frac{2x-5}{4x^2+2x}, where t is in seconds, x is in meters, and t ∈ [1,10] a) Determine the derivative of the function without using the quotient rule. b) Hence, find exactly when the particle is...

Homework Statement derive this function f(x) = (2x + x^3) / sqrt(x)Homework EquationsThe Attempt at a Solution how can i derive this without using quotient rule? my prof is asking to do so without using it. I

Hello, I am having trouble finding the derivative of this function. Any help would be appreciated! $$\frac{x\cos(x)+\sqrt{x}}{x+\sin(x)}$$ What I tried was expanding it first to like $$(x+\sin(x)(x\cos(x)+\sqrt{x})'-(x\cos(x)+\sqrt{x}$$ But I ended up with a long weird answer and doesn't seem...

Homework Statement Evaluate the derivative of the function y = (1/x) + sqrt(cos x) at the given point, (pi/2, 2/pi) Homework EquationsThe Attempt at a Solution I used power rule on (1/x) and chain rule on sqrt(cos x) but when I was simplifying, there is a sqrt(cos x) in one of the...

Hi guys...i'm having a bad time understanding the concept of vector derivative with respect to different frames. Suppose i have the vector displayed in the picture, the frame A and B are in motion with respect of each other, i can understand the concept of the derivative of the vector R in the...

Homework Statement Homework Equations Chain rule, partial derivation The Attempt at a Solution dv/dt=dv/dx*dx/dt+dv/dy*dy/dt dx/dt=-4t -> evaluate at (1,1) =-4 dv/dt=-4dv/dx+4(-2) dv/dt=-4dv/dx-8 How can I find the missing dv/dx in order to get a value for dv/dt? Thanks!

Homework Statement Homework Equations dz/dt = dz/dx⋅dx/dt + dz/dy⋅dy/dt The Attempt at a Solution [/B] I am getting : =[-sin(x+7y) ⋅ 10t] + [-sin(x+7y) ⋅ 7 ⋅ (-1/ t2)] then changing x and y terms: =[-sin((5t2)+7(1/t)) ⋅ 10t] + [-sin((5t2)+7(1/t)) ⋅ 7 ⋅ (-1/ t2)]

Homework Statement Hi all, I currently have a modified Einstein-Hilbert action, with extra terms coming from some vector field A_\mu = (A_0(t),0,0,0), given by \mathcal{L}_A = -\frac{1}{2} \nabla _\mu A_\nu \nabla ^\mu A ^\nu +\frac{1}{2} R_{\mu \nu} A^\mu A^\nu . The resulting field...

Homework Statement The lagrangian is given by: L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi + \frac{\alpha}{2} \phi \partial_{\mu} \phi \partial^{\mu} \phi And the question is to find the feynman rules. Homework EquationsThe Attempt at a Solution I started by using the...

Homework Statement Homework EquationsThe Attempt at a Solution part a) finding partial derivatives: and plugging in (2,0,1) into each, I get the gradient which is <0,-2,0> to find the directional derivative, it is the dot product of the gradient and unit vector of (3,1,1): part b)...

After doing a couple courses in physics as well as calculus and differential equations, I was starting to wonder about splitting a derivate, such as ## \frac{dy}{dx} ##, into separate pieces ##dy## and ##dx##. I know we've never done it in calculus or differential equations because it isn't...

Hello I'm currently trying to solve these two problems: 1) Find the partial derivatives ∂m/∂q and ∂m/∂h of the function: m=ln(qh-2h^2)+2e^(q-h^2+3)^4-7 Here, I know I should differentiate m with respect to q while treating h as a constant and vice versa. But I'm still stuck, and I'm not sure...

Homework Statement At 9 P.M. an oil tanker traveling west in the ocean at 18 kilometers per hour passes the same spot as a luxury liner that arrived at the same spot at 8 P.M. while traveling north at 23 kilometers per hour. If the "spot" is represented by the origin, find the location of the...

Homework Statement After a wild ride on a windsurfer, you head to a lifeguard tower to fly a kite. You stand on the tower, which is 3 meters above the ground and release your kite. You let out 10 meters of string and the kite begins moving according to the following equation: y(t) = (0.9...

I want to know how to do this so after I find the equation by hand, I can check to see if I am correct.

Homework Statement f(x,y) = (xy) / (x2 + y4), when (x, y) ≠ (0,0) 0, when (x,y) = (0,0) Homework Equations Explicitly show that f(x,y) does not satisfy lim h -> 0 [ E(v,h) / ||h|| ] = 0 when v = 0 (h, v, and 0 are all vectors; I'm not sure how to put a hat on them) The Attempt at a Solution I...

I was looking over a derivation to find the laplacian from cartesian to cylindrical and spherical coordinates here: http://skisickness.com/2009/11/20/ Everything seems fine, but there is an instance (I have attached a screenshot) where implicit differentiation is done to find $$ \frac {\partial...

Homework Statement If f(x) =(x^46 + x^45 + 2)/(x + 1) , calculate f(46)(3), the forty-sixth derivative of f(x) at x = 3. Express your answer using factorial notation: n! = n (n 1) (n 2) 3 2 1. Homework Equations Quotient rule The Attempt at a Solution I have tried trying to find a pattern...

Hi this is my first post ever in MHB, and I'm in Calculus 1 wondering which formula to use to find derivatives. There are 2 as far as I know: (1) and the one one at the beginning of this: (2) Example Problem: How would i know which formula to use? Is there a particular reason the...

Hi, I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is...

Homework Statement ##F'(x)##=##[2f(x)+g(x)]'## ##F'(0)=## Given g'(0)=2 and f'(0)=5 Homework Equations The Attempt at a Solution I can solve this if the questions is as follows ##F'(x)##=##[f(x)+g(x)]'## by applying sum rule but I don't know what to do about...

A basic question, not a homework problem. Say I have the expression: 5x = 10 Can I apply the derivative operator, d/dx, to both sides? d/dx(5x)=d/dx(10) would imply 5=0. I thought you can apply operators to both sides of an equation. Why can't you not do it in this case?

Just something that has been bugging me. Can someone bestow why the first derivative is velocity and the second derivative is acceleration. I want to conceptually understand this. Thank you

If R is Ricci scalar ∇i∇j F(R) = ? , where ∇i is covariant derivative.

https://www.particleincell.com/2012/curvilinear-coordinates/ http://www.jfoadi.me.uk/documents/lecture_mathphys2_05.pdf Hi, I have a question about the curvilineare coordinate system. I wonder why is normal to the isosurfaces?isnt ei a tangent vector to the surface ui since "With these...

Homework Statement differentiate ∫ e^(-x*t^4)dt from -x to x with respect to x.[/B]Homework Equations erf(x) = (2/sqrt(π)) ∫e^(-t^2)dt from 0 to x. Leibniz rule. I know that ∫t^2e^(-t^2)dt from 0 to x = (√π/4)*erf(x) - (1/2)*x*e^(-x^2)[/B]The Attempt at a Solution By using Leibniz rule...

F(x)=5/(x^2)-9 Find F'(x)&F''(x) Find the Vertical Asymptote, Horizontal, and Slant

y = a / (x - b)3 a = 77.1 ± 15.2 b = -1.78 ± 1.18 x = 21 ± 1 --- y' = -3a / (x - b)4 How do I find the uncertainty of y'? Thanks.

Homework Statement Find f'(x) if f(x) = 8^(sin^2(3x)) Hint: you will need to use the double angle formula for trig functions and your answer should only have one trig function in it. Homework Equations if y=a^u then y' = ln a * a^u * du sin(2x) = 2sinxcosx The Attempt at a Solution We're...

Suppose [FONT=Verdana]u is a vector-valued function. Is it true that [FONT=Verdana] (∇×u)⋅( (u⋅∇)u ) = (u⋅∇)(∇×u)⋅u ? [FONT=Verdana]Please note the lack of a dot product on the first two terms of the RHS and the parenthesis around the second term of the LHS. I'm trying to understand whether...

Homework Statement \frac{d}{dx}7.5sin(\frac{pi}{10}x) The Attempt at a Solution 7.5(\frac{pi}{10})cos(\frac{pi}{10}x) Maximum: f'(x) = 0 7.5(\frac{pi}{10})cos(\frac{pi}{10}x) = 0 7.5(\frac{pi}{10})cos^{-1}(0)= \frac{pi}{10}x ** (\frac{pi}{10}\frac{10}{pi})7.5(90) = x (1)(7.5)(90) = x...

Homework Statement I need to find the partial derivative of the action S with respect to the generalized coordinate q(tf) and according to my textbook, it should equal the generalized momentum p(tf). How can I derive this? Homework Equations S = integral of L dt, with boundary ti to tf. (ti...

I know this is probably the least of my worries at the moment but my quantum textbook solves ##\frac{\mathrm{d}\phi (t) }{\mathrm{d} t}=\frac{iC}{h}\phi (t) ## as ##\phi (t) = e^{-i(\frac{C}{h})t}##. Is this not off by a sign? Its really bugging me.

Why is $$ \left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n-1} $$? This is in a textbook and says that its proof is left as an exercise. It seems to be a difficult equality. I believe this should just be $$ \left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n =...

Hi, Suppose I have a function on a graph with a vertical axis is Force and the horizontal axis is distance. Then the area under the curve is given by F*d = Work = Energy, correct? If so, then what would the slope of the curve represent? F/d = ? Thank you.

Bourbaki defines a function as follows: We give the name of function to the operation which associates with every element x the element y which is in the given relation with x; y is said to be the value of the function at the element x, and the function is said to be determined by the given...

How does the quaternion derivative work in the presence of a quaternion product. More specifically, does the standard product rule apply for quaternion derivatives? Say, I have a function f(q) = q* x a x q [where q -> quaternion, a -> const vector x-> quat prod] what is the result of the...

[I asked this question over a year ago, but I thought I'd try again.] Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R## be a ##C^\infty## function. I have the following characterizations: 1) ##f'\geq 0## everywhere iff ##f## is increasing. 2) ##f''\geq 0## everywhere iff...

Hello, I am wondering about the derivative of the Dirac delta distribution ##\delta(t)##. I know: $$\frac{d}{dt} u(t) = \delta(t)$$ So what is ##\frac{d}{dt} \delta(t)##? How do we take the derivative of a distribution? I've heard about distributional derivatives, but I don't think any of...

Homework Statement Write the derivative of y = (x2+4x+3)/(x1/2) I got the correct answer, but my question is, why can't I rewrite this as: y = (x^2+4x+3)*(1/x1/2) Then see my attempted solution for the result... Homework Equations y = (x2+4x+3)/(x1/2) The Attempt at a Solution y =...

Homework Statement Q: The position of a dragonfly that is flying parallel to the ground is given as a function of time by r⃗ =[2.90m+(0.0900m/s2)t^2]i^− (0.0150m/s3)t^3j^ . At what value of t does the velocity vector of the insect make an angle of 30.0 ∘ clockwise from the x-axis? . Homework...

Mod note: Thread moved from Precalc section Homework Statement F(x)=sqrt(-2x^2 +2x+4) 1.discuss variation of f and draw (c) 2.find the equation of tangent line to (c) that passes through point A(-2,0) The Attempt at a Solution I solved first part I found the domain of definition and f'(x) and...

Homework Statement Find the distance between the origin and the line tangent to ##x^\frac{2}{3}+y^{\frac{2}{3}}=a^{\frac{2}{3}}## at the point P(x,y) Homework Equations [/B] Distance= ##\frac{\left |a_{0}+b_{0}+c \right |}{\sqrt{a^{2}+b^{2}}}## The Attempt at a Solution To begin I find...

Hi, I feel sometimes when I'm doing calculus I lose the logic and intuition behind what I'm doing, especially when integrating. I have yet to find a way to think about it in a way it makes sense to me why the definite integral would tell us the area under a curve. Same with why the second...

Hey, I don't understand why emf generated is not the first derivative. For example I have a graph of magnetic flux through a wire against time. I thought that emf generated was the rate of change of flux, however this doesn't work. Emf=-Dflux/Dtime . In this example I have a y=sinx curve for...

If my understanding is correct the definition of a derivative is lim h->0 (f(x+h)-f(x))/h However, I've also seen this used: lim x->c (f(x)-f(x))/(x-c) are these both considered valid definition for the derivative or does the derivative have to tend towards zero? I am a bit confused because I...

I was trying to compute the time derivative of the following expression: \mathbf{p_k} = \sum_i e_{ki}\sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)!} \mathbf{r_{ki}}(\mathbf{r_{ki}\cdot \nabla})^n \delta(\mathbf{R_k}-\mathbf{R}) I am following deGroot in his Foundations of Electrodynamics. He says...

Homework Statement The displacement of a machine is given by the simple harmonic motion as x(t) = 5cos(30t)+4sin(30t). Find the amplitude of motion, and the amplitude of the velocity. Homework Equations x''(t) = -4500cos(30t)-3600sin(30t) The Attempt at a Solution [/B] I should note that...

Can anyone explain how to take the derivative of (Aδij),j? I know that since there is a repeating subscript I have to do the summation then take the derivative, but I am not sure how to go about that process because there are two subscripts (i and j) and that it is the Kronecker's Delta (not...