Derivative Definition and 1000 Threads

I need some guidance regarding the directional derivative ... Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows: Theodore Shifrin: Multivariable Mathematics and Susan Jane Colley: Vector Calculus (Second Edition)Colley...

I am trying to reconcile apparent differences between the definitions of the derivative of a vector-valued function $$f: U \rightarrow \mathbb{R}^n$$ (where $$U \subset \mathbb{R}^m$$ ) of a vector variable from two textbooks ... The textbooks are as follows: Andrew Browder: "Mathematical...

Homework Statement "The functional ## J[f] = \int [f(y)]^pφ(y)\, dy ## has a functional derivative with respect to ## f(x) ## given by: $$ \frac {δJ[f]} {δf(x)} = \lim_{ε \rightarrow 0} \frac 1 ε \left[ \int[f(y) + εδ(y-x)]^pφ(y)\, dy - \int [f(y)]^pφ(y)\, dy\right] $$ $$ =...

We have ##B=(1, X, X^2, X^3)## as a base of ##R3 [X]## and we have the endomorphisms ##d/dX## and ##d^2/dX^2## so that: ##d/dX (P) = P'## and ##d^2/dX^2 (P) = P''##. Calculating the matrix in class, the teacher found the following matrix, call it ##A##: \begin{bmatrix} 0 & 1 & 0 & 0...

Homework Statement Find the unit vectors along which the given functions below increase and decrease most rapidly at P0 . Then find the derivatives of the functions in these directions. Homework Equations solution: The Attempt at a Solution why are the derivatives’ values along these...

Homework Statement find derivative of (x-2)(x-3)^2 Homework Equations using product rule. The Attempt at a Solution 1(x-3)^2+2(x-3) x^2-6x-9 +2x-6 x^2-4x-15 doesn't factor.

I have been given that r = 1/u and that r(dot) = (-1/u^2) *(du/dt) How is r(dot) calculated? I don't understand the steps of how to get from r to r(dot) From my understanding r(dot) should be the derivative of (1/u) with respect to time, but I don't understand how to get to the final answer...

Dear Members, I was going through some video lecture (Quantum Mechanics) when I encountered a definition of momentum as shown in the attached picture. I do not understand how iota and ħ is ignored ? There are some negligible terms after plus sign. What are those ? In short how they have...

Is this the correct partial derivative of B? ##\vec{B} = \frac{g \vec{r}}{4 \pi r^3}## ##\frac{\partial \vec B}{\partial r}## = ##-3\frac{g \vec{r}}{4 \pi r^4} + \frac{g}{4 \pi r^3 }(\frac{\partial r_r \hat r}{\partial r})##

What the heck. MathJax is back up and I'm feeling lucky... It's not an easy one. I'm just looking for some input. How do you take this derivative? \frac{d}{d(\phi ^* \phi )} ( \phi ^* + \phi ) where * is the complex conjugate and \phi is complex. -Dan

Hi initially I am aware that christoffel symbols are not tensor so their covariant derivatives are meaningless, but my question is why do we have to use covariant derivative only with tensors? ?? Is there a logic of this situation? ?

Homework Statement Let f(x) be the function whose graph is shown below (I'll upload the image) Determine f'(a) for a = 1,2,4,7. f'(1) = f'(2) = f'(4) = f'(7) = Use one decimal. Homework Equations f(x+h)-f(x)/h The Attempt at a Solution Hi everybody I was trying to do this function...

Homework Statement Need to prove that the derivative of a rotation matrix is a skew symmetric matrix muktiplied by that rotation matrix. Specifically applying it on the Rodrigues’ formula.Homework EquationsThe Attempt at a Solution I have shown that the cubed of the skew symmetric matrix is...

Homework Statement So my problem is mainly intuitive one, in that this *feels* wrong, and am mostly looking for insight. If we have uniform 1D motion of a particle along ##x## with constant velocity ##v##, what is the rate of change (first derivative with respect to time) of the variable...

Homework Statement Homework Equations Derivative as a limit: $$y'=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ The Attempt at a Solution $$f'(x)=\lim{\Delta x\to 0}\frac{f(x)f(\Delta x)-(1+xg(x))}{\Delta x}=\bigstar$$ $$\left\{ \begin{array}{l} f(\Delta x)=1+\Delta x...

Homework Statement Find the divergence of the function ##\vec{v} = (rcos\theta)\hat{r}+(rsin\theta)\hat{\theta}+(rsin\theta cos\phi)\hat{\phi}## Homework Equations ##\nabla\cdot\vec{v}=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2v_r)+\frac{1}{r sin\theta}\frac{\partial}{\partial...

Hello all, I was just wondering if there is any rules for integrating a function with respect to it's own derivative. That is to say ##\int _{ }^{ }f\left(x\right)d\left(f'\left(x\right)\right)## or ##\int _{ }^{ }yd\left(\frac{dy}{dx}\right)## Thank you in advance for your time :)

Homework Statement Let ##\omega \in \Omega^r(N)## and let ##f:M \to N##. Show that ##d(f^*\omega)=f^*(d\omega)## Homework Equations ##\Omega^r(N)## is the vector field of r-form at a given point in the manifold N, ##f^*## is the pullback function and ##d## is the exterior derivative...

Homework Statement The radius of a circle increases from 3 to 3.01 cm. Find the approximate change in its perimeter. Here's a link to the actual question, in case you need the answer for 6(a) to solve 6(b) http://imgur.com/a/nQt6M Homework Equations Perimeter of circle = 2πr Area of circle =...

Homework Statement Only the second part Homework Equations Second derivative: $$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$ The Attempt at a Solution $$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$ Do i differentiate the differential dt? $$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$...

Homework Statement Homework Equations Differential of a product: $$d(uv)=u\cdot dv+v\cdot du$$ The Attempt at a Solution $$dV=\pi \left[ -\frac{1}{x}x^2+2x\left(-\frac{x}{3} \right) \right]dx=-\pi x^2dx$$ If dx>0 dV<0, it's wrong, the volume increases

Homework Statement Use the definition of the derivative to find dy/dx for ##~y=\sqrt{2x+3}## Homework Equations Derivative as a limit: $$y'=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$The Attempt at a Solution $$\lim_{\Delta x\rightarrow 0}\frac{\sqrt{2(x+\Delta...

Homework Statement $$y=1+t^2,~~x=\frac{t}{1+t^2}$$ What is dy/dx Homework Equations Parametric equation's derivative: $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$ The Attempt at a Solution $$\frac{dx}{dt}=\frac{1-t^2}{(1+t^2)^2}$$ $$\frac{dy}{dx}=\frac{2t(1+t^2)^2}{1-t^2}$$ I can't translate it back...

Homework Statement Find dy/dx for ##~y=\sec^2(5x)## Homework Equations Secant and it's derivative: $$\sec\,x=\frac{1}{\cos\,x}$$ $$\sec'\,x=\tan\,x\cdot\sec\,x$$The Attempt at a Solution $$y=\sec^2(5x)~\rightarrow~y'=2\cdot 5 \cdot \sec(5x)\tan(5x)\sec(5x)=10\tan(5x)\sec^2(5x)$$ The answer...

Homework Statement Isn't the derivative of y with respect to x Defined as ##~\frac{dy}{dx}##? What and how do i have to prove? Homework Equations The chain rule: $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$ The Attempt at a Solution $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$

Homework Statement In textbook i was given formula to calculate error. I know that: E(t) = f(t)- L(x) = f(t) - f(a)- f'(a)(t- a) [L(x) is linear approximation]; [Lets call this Formula 1] I understand that, but that I have formula: E(x) = f''(s)/2 * (x-a)^2 [lets call this Formula 2] Here...

Homework Statement The question asks to calculate ∂f/∂x for f(x,y,t) = 3x2 + 2xy + y1/2t -5xt where x(t) = t3 and y(t) = 2t5 Homework Equations The answer is given as ∂f/∂x = 6x + 2y - 5t The Attempt at a Solution I'm confused because the answer given seems to treat x,y ,t as...

Hi, May someone helps me regarding this!? i have a controller which will control AC motor as attached. in this controller, a stage comes where I need to use a Derivative Block before point 'B' as shown in the attached picture " controller block diagram" [...

Hi PF! I'm trying to put the first derivative of the modified Bessel function of the first kind evaluated at some point say ##\alpha## in a sum where the ##ith## function is part of the index. What I have so far is n=3; alpha = 2; DBesselI[L_, x_] := D[BesselI[L, x], {x, 1}] Sum[BesselI[L...

Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...

This link shows us how to derive Hamilton's generalised principle starting from D'Alembert's principle. While I had no trouble understanding the derivation I am stuck on this particular step. I can't justify why ## \frac{d}{dt} \delta r_i = \delta [\frac{d}{dt}r_i] ##. This is because if I...

What is time derivative of angular velocity ( measured w.r.t. an inertial frame ) of a rotating co. system w.r.t. the same rotating co. system? I think a person sitting in a closed rotating box feels the an object at rest w.r.t. him as rest. He doesn't observe the angular velocity of the...

Homework Statement Homework Equations ##V=V^u \partial_u ## I am a bit confused with the notation used for the Lie Derivative of a vector field written as the commutator expression: Not using the commutator expression I have: ## (L_vU)^u = V^u \partial_u U^v - U^u\partial_u V^v## (1)...

Homework Statement the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0 Homework EquationsThe Attempt at a Solution I have problems regarding finding the equation of tangent line to the part of parabola because the question not specifically...

Firstly I know how to do this with first derivatives in differential equations - for example say we had ##\frac{dy}{dx}=4y^2-y##, and we're also told that ##y=1## when ##x=0##. ##\frac{dy}{dx}=4y^2-y## ##\frac{dx}{dy}=\frac{1}{4y^2-y}=\frac{1}{y\left(4y-1\right)}=\frac{4}{4y-1}-\frac{1}{y}##...

$\tiny{7.6.16}$ $\textsf{Find the derivative of y with respect to x}$ \begin{align*}\displaystyle y&=\ln{(\tan^{-1}(4x^3))} \\ y'&= \end{align*} didn't get the arctan ? ☕

Is there a spatial derivative that uses the del operator and the double cross product? If so, is it used in physics?

Homework Statement How to show that lie deriviaitve of metric vanish ##(L_v g)_{uv}=0## <=> metric is independent of this coordinate, for example if ##v=\partial_z## then ##g_{uv} ## is independent of ##z## (and vice versa) 2. Relevant equation I am wanting to show this for the levi-civita...

Homework Statement Question attached. Homework Equations 3. The Attempt at a Solution [/B] I'm not really sure how to work with what is given in the question without introducing my knowledge on lie derivatives. We have: ##(L_ug)_{uv} =...

$\tiny{242.7x.25}$ $\textsf{Find the derivative of y with respect to x}$ \begin{align*}\displaystyle y&=8\ln{x}+\sqrt{1-x^2}\arccos{x} \\ y'&=\frac{8}{x}+? \end{align*} the first term was easy but the second😰

Let $f(x)$ be a function satisfying \[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\] Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.

Homework Statement I am working on a project dealing with the velocity and acceleration of satellites based on their distance from Earth. I was recommended to include some calculus in this project. Originally I thought I could just take the derivative of the orbital speed equation to find...

Homework Statement I understand the derivation it showed that included the sin (15.7 in the image) I just don't understand the following (15.8 in the image). Does "t" get pulled out of the equation? If so what do we derive for then? Does it become 0? If so, it would remain 0 and sin(0) is just...

Dear all, I'm asking why there is no higher derivative than two in physics ? I never encountered a third (time or space) derivative in physics. Have you some litterature about this? Thank you. Regards.

Homework Statement Evaluate the derivative of the following function: f(w)= cos(sin^(-1)2w) Homework Equations Chain Rule The Attempt at a Solution I did just as the chain rule says where F'(w)= -[2sin(sin^(-1)2w)]/[sqrt(1-4w^(2)) but the book gave the answer as F'(w)=(-4w)/sqrt(1-4w^(2))...

Wikipedia defines hessian of Difference of Gaussians as and earlier in the page uses D for difference of gaussians, So do i just need D(x,y) or do i need d/dx D(x,y) for the elements? If so how does one go about differentiating DoG? Any help appreciated

What is the result of this kind of partial differentiation? \begin{equation*} \frac{\partial}{\partial x} \left(\frac{\partial x}{\partial t}\right) \end{equation*} Is it zero? Thank you in advance.

I'm working through a proof in my differential equations book, but I think I'm hung up on a basic calculus derivative. If we have a function ##f(x,y)## and we substitute ##v=\frac{y}{x}## , rearrange to get ##y=vx##, and then take the derivative, supposedly by the product rule we get...

In Hawking-Ellis Book(1973) "The large scale structure of space-time" p69-p70, they derive the energy-momentum tensor for perfect fluid by lagrangian formulation. They imply if ##D## is a sufficiently small compact region, one can represent a congruence by a diffeomorphism ##\gamma: [a,b]\times...