Derivative Definition and 1000 Threads

If given a function ##u(x,y) v(x,y)## then is it correct to write ##\frac{\partial }{\partial x}u(x,y)v(x,y)=\frac{u(x+dx,y)v(x+dx,y)-u(x,y)v(x,y)}{dx}##??

I first solved the first two terms and then i solved the resulting term with the third term and so on.At last i was left with x^n/((x-a1)(x-a2)...(x-an)) .Thrn i took log on both sides and then differentiated both sides with respect to x.I got 1/y dy/dx=n/x -1/(x-a1)-1/(x-a2)...-1/(x-an).But now...

Problem Statement: Use the definition of the total time derivative to a) show that ##(∂ /∂q)(d/dt)f(q,q˙,t) = (d /dt)(∂/∂q)f(q,q˙,t)## i.e. these derivatives commute for any function ##f = f(q, q˙,t)##. Relevant Equations: My approach is given below. Please tell if it is correct and if not ...

The gradient is < (2x-y), (-x+2y-1) > at P(1,-1) the gradient is <3, -4> Since ∇f⋅u= Direction vector, it seems that we should set the equation equal to the desired directional derivative. < 3, -4 > ⋅ < a, b > = 4 which becomes 3a-4b=4 I thought of making a list of possible combinations...

The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is \begin{equation} T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}. \end{equation} Let contract...

Edit: I see this was discussed in the related thread sorry for a repost. If acceleration causes a change in velocity, and jerk causes a change in acceleration, snap causes a change in jerk, crackle causes a change in snap, pop causes a change in crackle, stop causes a change in pop, drop causes...

Red line being the function and blue an approximation of the derivative. Does it look right?

I'm studying how derivatives and partial derivatives transform under a Galilean transformation. On this page: http://www.physics.princeton.edu/~mcdonald/examples/wave_velocity.pdf Equation (16) relies on ##\frac{\partial t'}{\partial x}=0## but ##\frac{\partial x'}{\partial t}=-v## But this...

It seems that the way to combine the information given is z = f ( g ( (3r^3 - s^2), (re^s) ) ) we know that the multi-variable chain rule is (dz/dr) = (dz/dx)* dx/dr + (dz/dy)*dy/dr and (dz/ds) = (dz/dx)* dx/ds + (dz/dy)*dy/ds ---(Parentheses indicate partial derivative) other perhaps...

How do I interpret geometrically the partial derivative in respect to a constant of a function such as ##\frac{ \partial}{\partial c} (acos(x) + be^x + c)^2##?

Fμν = ∂μAν- ∂νAμ ∂μFμν = ∂2μAν - ∂ν(∂μAμ) = ∂2μAνWhy ∂ν(∂μAμ) and not ∂μ∂νAμ ? And why does ∂ν(∂μAμ) drop out? thank you

Hi, just a question regarding neumann conditions, I seem to have forgotten these things already. I think this question is answerable by a yes or a no. So given the 2D heat equation, If I assign a neumann condition at say, x = 0; Does it still follow that at the derivative of t, the...

Hello! Happy Sunday, I am trying to create a MATLAB GUI that can take an input ( i am starting out with just one variable) and take its derivative and display the result. I have attached pictures of what my GUI looks like, and also the code I wrote so far. Let me also describe my method: I have...

here I am trying to find ##\frac{d}{dt}dx## where ##x(t)## is the position vector Now ##\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x}{dt}dt=\frac{\partial v_x}{\partial t}dt+\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz## Now dividing by ##dx##...

I have been reading a book on classical theoretical physics and it claims: -------------- If a Lagrange function depends on a continuous parameter ##\lambda##, then also the generalized momentum ##p_i = \frac{\partial L}{\partial\dot{q}_i}## depends on ##\lambda##, also the velocity...

For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.

Consider the following Lagrangian density $$\mathscr{L}=\mathbf{E}\cdot\left(\nabla^{2}\mathbf{E}\right)$$ where $$E_{i}=\partial_{i}\phi\;(i=x,y,z )$$. In th is case the potential and its 3rd derivatives are the independent variables. Acording to Barut's classical theory of fields book, for...

Why ##\frac{\partial (0.5*m*\dot{x}^2-m*g*x)}{\partial x}=-mg##? why ##\frac{\partial \dot{x}}{\partial x}=0##? Why ##\frac{\partial (0.5*m*\dot{x}^2-m*g*x)}{\partial \dot{x}}=m*\dot{x}## ? why ##\frac{\partial x}{\partial \dot{x}}=0##? Does it assume that speed is same at every location? I...

I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement: $$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the...

205.8.9 Find the derivative of the function $y=\cos(\tan(5t-4))\\$ chain rule $u=\tan(5t-4)$ $\frac{d}{du}\cos{(u)} \frac{d}{dt}\tan{\left(5t-4\right)}\\$ then $-\sin{\left (u \right )}\cdot 5 \sec^{2}{\left (5 t - 4 \right )}\\$ replacing u with $\tan(5t-4)$ $-\sin{(\tan(5t-4))}\cdot 5...

In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))... ## \int_0^\phi \frac {d} {dx} f(x) dx =...

I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as ' a ' instead of ' a / b ' . ------ b

How/why does the first equal sign hold? Where does each derivative come from:

First I am sorry if this question is futile, or even stupid but I find it confusing, and I would like to clarify in my mind. Taking the derivative ##y=x^2## with respect to x by using limit is very easy stuff. But would you please illustrate how to apply this to the equation ##y^2=x^2## It...

Hello Physics Forums, I have a simple parametric surface in R3 <x,y,z(x,y)>. I've calculated the the usual mean curvature: H= ((1+hx^2)hyy-2hxhyhxy+(1+hy^2)hxx)/(1+hx^2+hy^2)^3/2 I needed to take the variational derivative of this expression. Since it has second order spatial derivatives the...

In Carrol's gr notes the covariant derivative of a vector is given as ∇μAϑ=∂μAϑ+ΓϑμλAλ...(1) For a geodesic in 2-D cartesian coordinates the tangent vector is V=##a\hat x+b\hat y##(a and b are constt.)where the tangent vector direction along the curve is ##\hat n=\frac{a\hat x+b\hat...

For the simple case of a 2-D curve in polar coordinated (r,θ) parametrised by λ (length along the curve). At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##. The non-zero christoffel symbol are Γ122 and Γ212. From covariant derivative...

Hi, I'm trying to understand the Euler-Poincare equations, which reduce the Euler-Lagrange equations for certain Lagrangians on a Lie group. I'm reading Darryl Holm's "Geometric mechanics and symmetry", where he suddenly uses what seems to be a variational derivative, which I'm having a hard...

Homework Statement [/B] From the Rodrigues’ formulae, I want to derive nature of the spherical Bessel and Neumann functions at small values of p. Homework Equations [/B] I'm going to post an image of the Bessel function where we're using a Taylor expansion, which I'm happy with and is as far...

<Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x\rightarrow 0} (x-tanx)/x^3$$ I solve it like this, $$\lim_{x\rightarrow 0}1/x^2 - tanx/x^3=\lim_{x\rightarrow 0}1/x^2 - tanx/x*1/x^2$$ Now using the property $$\lim_{x\rightarrow 0}tanx/x=1$$,we have ...

A balloon's volume is increasing at a rate of dV/dt. Express the rate of change of the circumference with respect to time (dc/dt) in terms of the volume and radius. Homework Equations Vsphere = (4/3)(π)(r^3) C = (2)(π)(r) The Attempt at a Solution [/B] My strategy was to come up with two...

I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ...I need help to understand some remarks by Fortney following Theorems 2.1 and 2.2 on the directional...

Why Lagrangian not depend of higher derivatives of generalised coordinates ?

Hi forum. I'm trying to prove a claim from Mathematical Analysis I - Zorich since some days, but I succeeded only in part. The complete claim is: $$\left\{\begin{matrix} f\in\mathcal{C}^{(n)}(-1,1) \\ \sup_{x\in (-1,1)}|f(x)|\leq 1 \\ |f'(0)|>\alpha _n \end{matrix}\right. \Rightarrow \exists...

I am relatively new to differential geometry. I am studying it from Fecko Textbook on differential geometry. As soon as he introduces the concept of lie derivative,he asks to do exercise 4.2.2 in picture. The question is,how do I apply ##\phi^*## to given function ##\psi## . I know that...

Hi guys, I am reading my lecture notes for Mechanics and Variations and I am trying to understand the maths here. From what I can see there we differentiated with respect to a derivative. Could you tell me how do we do that? Thanks

Homework Statement Without using vector identities, show that ##\nabla \cdot [\vec{A}(r) \times \vec{r}] = 0##. Homework Equations The definitions and elementary properties of the dot and cross products in terms of Levi-Civita symbols. The "standard" calculus III identities for the divergence...

I need a little help with understanding a differential relationship between functions. If g and f are vector fields and f(g(x,y),q(x,y))=∇2g(x,y) How could you, if possible, express ∂f/∂g explicitly? Please help a bit confused.

Homework Statement I'm given a gas equation, ##PV = -RT e^{x/VRT}##, where ##x## and ##R## are constants. I'm told to find ##\Big(\frac{\partial P}{\partial V}\Big)_T##. I'm not sure what that subscript ##T## means? Homework Equations ##PV = -RT e^{x/VRT}## Thanks a lot in advance.

This is a simple and maybe stupid question. Can you take a derivative of a vector component with respect to a vector? Or even more generally,can you take the derivative of a component of a tensor with respect to the whole tensor? For instance in the cauchy tensor could you take the xx component...

If we have two sets of coordinates such that x1,x2...xn And y1,y2,...ym And if any yi=f(x1...,xn)(mutually dependent). Then dyi=(∂yi/∂xj)dxj Again dyi/dxk=(∂2yi/∂xk∂xj)dxj+∂yi/∂xk Is it the contravariant derivative of a vector?? Or in general dAi/dxk≠∂Ai/∂xk

A different perspective to derivatives.

Hello all. I was playing around with the time dilation equation : √(1-v2/c2) Specifically, I decided to take the derivative(d/dv) of the equation. Following the rules of calculus, as little of them as I know, I got this: d/dv(√(1-v2/c2) = v / (c2√(1-v2/c2)). Now, this seems reasonable enough...

I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ... I need help with Question 2.4 (a) (i) concerned with computing a directional derivative ...

Homework Statement ##f(x) = (5x+6)^{10} , f'(x)=?## Homework Equations ##\frac{d}{dx}x^n = nx^{n-1}##? 3. The Attempt at a Solution [/B] I do know the solution ##f'(x) = 50(5x+6)^9##,but I don't know how this solution came to be.I downloaded this problem from the web and it only comes with...

I've heard the statement that by computing just the leading-order (tree level) diagrams of a process and then computing the derivative of this result with respect to the mass should correspond to the evaluation of the mass counter term diagrams. Can someone explain why this statement is...

I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Section 1.7: Derivatives in Several Variables as Linear Transformations ... I need some help in order to understand some...

Homework Statement What is the derivative of ##f(x)=\frac {2x^2} {cos x}##? Homework EquationsThe Attempt at a Solution ##F(x)=\frac {2x^2} {cos x}## So... ##f(x)=2x^2## and ##f'(x)=4x## ##g(x)=cosx## and ##g'(x)=-sinx## If I plug these into the quotient rule I thought that I would get...

I thought you guys might appreciate this. A lot of people notice that the derivative of area of a circle is the circle's circumference. This can be generalized to all regular polygons in a nice way.

Homework Statement Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$ where q is a constant vector. Homework EquationsThe Attempt at a Solution $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}}...