Derivatives Definition and 1000 Threads

In some books, when discussing the relation between partial/directional derivatives and tangent vectors, one makes a generalization called a "derivation". A derivation at ##\vec{a} \in \mathbb{R}^n## is defined as a linear map ##D: C^{\infty}(\mathbb{R}^n) \to \mathbb{R}## which for ##f,g \in...

Hello! :) Having the transformations: $$\xi=\xi(x,y), \eta=\eta(x,y)$$ I want to find the following partial derivatives: $$\frac{\partial}{\partial{x}}= \frac{\partial}{ \partial{\xi}} \frac{\partial{\xi}}{\partial{x}}+\frac{\partial}{\partial{\eta}}...

Problem: I did some of the problem on MatLab but I'm having a difficult time evaluating the derivatives at (0,0). Also, MatLab gave me the same answer for fxy and fyx, which according to the problem isn't correct. Any ideas? I used MatLab and computed: fx(x,y)=(2*x^2*y)/(x^2 + y^2) + (y*(x^2 -...

Homework Statement A biologist determines experimentally that the number of calories burned by a salmon swimming a distance d in miles upstream against a current v0 in miles per hour is given by Energy = kdv^5/v − v0 where v is the salmon’s swimming speed relative to the water it is in...

Homework Statement Homework Equations The Attempt at a Solution Umm can somebody explain to me what just happened. None of that makes any sense to me what so ever.

Homework Statement So I'm trying to find dy/dt. I used the chain rule to find dx/dt. I just don't understand how to put that answer back into an equation to find dy/dt Homework Equations 4x^3-6xy^2+3y^2=228 The Attempt at a Solution I found dx/dt=3 x=-3 and y=4 So how/what...

Homework Statement Suppose you are standing at the point (-100,-100,360) on a hill that has the shape of the graph of z=500-0.006x2-0.008y2. In what direction should you head to maintain a constant altitude? Homework Equations Duf = ∇f\bulletu formula for directional derivative...

Say you are given the equation of a surface f(x,y) and a point (x,y,z) on the surface. How would one find the gradient vector in which the directional derivative Duf is equal to zero.

Homework Statement If ##xs^2 + yt^2 = 1## (1) and ##x^2s + y^2t = xy - 4,## (2) find ##\frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \frac{\partial y}{\partial t}## at ##(x,y,s,t) = (1,-3,2,-1)##. Homework Equations Pretty much those just listed...

Homework Statement Let μ represent a positive constant. Find the derivatives of: (x^2)/(2μ) Please check work. I am confused about the "constant" part. Can't you just set μ = some positive number and find the derivative that way? Homework Equations (x^2)/(2μ) The Attempt at a Solution...

Homework Statement https://imagizer.imageshack.us/v2/622x210q90/833/sqaw.png I am having difficulty understanding the notation <h, f''(x0)h>

Find y' y=(x2+1)7(x9+2)5(x3+1)3(x8+7)3 Is there a shortcut to doing this problem? Or do I have to actually use the product rule more than 3 times?

Hello, For this question I managed to find an answer but I am not sure if what the question means is to plug in the x-value and find the slope, or first find the derivative of the function, and THEN solve for the slope using f'(x)=10x+5h-2 (what I got for the derivative of the function). If...

Has anyone tried to make physical theories where the derivatives do not commute? I mean there's a condition on the derivatives of every function for them to commute which is learned in first year calculus. I mean in QM and QFT we grew accustomed to operators that do not commute, so why not...

I have to integrate -partial^2f/partialx^2 -partial^2f/partialy^2 in the variable x-y How to do this?

Homework Statement Function f(x) = x^2/(x-1) should be expanded by Taylor method around point x=2 and 17th order derivative at that point should be calculated. Homework Equations Taylor formula: f(x)=f(x0)+f'(x0)*(x-x0)+f''(x0)*(x-x0)^2+... The Attempt at a Solution I...

Hello all, I am trying to calculate the second order of the partial derivative by x of the function: f(x,y)=(x^2)*tan(xy) In the attach images you can see my work. Both the answer in the book where it came from and maple say that the answer is almost correct, but not entirely. In the last...

Homework Statement The parametric equations of a curve are ##x=\frac{1}{2}(sint cost+ t), y=\frac{1}{2} t-\frac{1}{4} sin2t##, ##-\pi/2<t\leq0##. P is a point on the curve such that the gradient at P is 1. Find the equation of the normal at P. Hence, determine if the normal at P meets the...

Ok folks, I've taken a stab at the Latex thing (for the first time, so please bear with me). I've mentioned before that I'm teaching myself relativity and tensors, and I've come across a question. I have a few different books that I'm referencing, and I've seen them present the ordinary...

Hellow everybody! A form how various ODE are intercorrelated can be sinterized like this: ##t = t## ##y = y(t)## ##y' = y'(t,\;y)## ##y'' = y''(t,\;y,\;y')## ##y''' = y'''(t,\;y,\;y',\;y'')## Until here, no problems! But, how is such relationship wrt the PDE? Would be this...

Hey guys, I was wondering if anyone can post up the formulas for integration and derivatives for everything, or if you have a link you can send me to see them. Like taking the integral or derivative of e^x I'd like to know the process for how it works.

Here is the question: I have posted a link there to this thread so the OP can view my work.

Homework Statement Given a pair of parametric equations, ##x=f(t)## and ##y=g(t)## , The first derivative is given by ##\frac{dy}{dx}=\frac{g'(t)}{f'(t)}## and the second derivative is actually ##\frac{d}{dt}(\frac{dy}{dx})## But why we cannot find the second derivative of a parametric...

1. If the function f(x)=x3+a2+bx has the local minimum value at \frac{-2}{9}\sqrt{3}, what are the values of and a and b? Homework Equations $$f'(x)=0$$ The Attempt at a Solution I automatically took the derivative, getting $$f'(x)=3x^2+2ax+b$$ However, I have no idea where to go from...

1. Find a cubic function f(x)=ax3+bx2+cx+d that has a local maximum value of 3 at x=-2 and a local minimum value of 0 at x=1. Homework Equations $$f'(x)=0$$ The Attempt at a Solution The first thing I did was taking the derivative of f(x). $$f'(x)=3ax^2+2bx+c$$ I know that you can get the...

Homework Statement Hi, I'm new here, so pardon me for my mistakes. ;) I need someone good at maths to check my answer: Let y=x sinh x (a) Show that d^2 y/dx^2 = x sinh x+2 cosh x, and find d^4 y/dx^4 (b) Write down a conjecture for d^2n y/dx^2n. (c) Use induction to establish a...

A recent post of chisigma rings me the bell of an old problem I thought of posting in a forum (either here or MMF). Is there any particular approach to computing a closed form for derivatives of certain smooth and continuous functions of $\mathbb{R}$? For example, it is easy to find the $n$-th...

Hi Guys. I hope some of you are able to help me out. This is NOT a homework. I have to explain the difference between the three different rate of change (time derivatives) of a property (such as mass, momentum, internal energy, etc) in a continuum? 1 )The partial time derivative 2)...

Hi, I know this question may seem a little trivial, but is there any real difference between \left (\partial_{\mu} \phi \right)^{\dagger} and \partial_{\mu} \phi^{\dagger} and if so, could someone provide an explanation? Many thanks. (Sorry if this isn't quite in the right...

Homework Statement We have two coordinate functions of time, as follows: x(t) = 5 + 2t ; y(t) = -3+3t+2t2. Find velocity \vec{v}, acceleration \vec{a}, tangential acceleration \vec{a_t}, normal acceleration \vec{a_n} functions of time and their magnitude's functions of time. Homework...

Hi folks. Hope that you can help me. I have an equation, that has been rewritten, and i don't see how: has been rewritten to: Can someone explain me how? Or can someone just tell me if this is correct in tensor notation: σij,jζui = (σijζui),j really hope, that...

Homework Statement Show that the energy of a simple harmonic oscillator in the n = 2 state is 5Planck constantω/2 by substituting the wave function ψ2 = A(2αx2- 1)e-αx2/2 directly into the Schroedinger equation, as broken down in the following steps. First, calculate dψ2/dx, using A for A, x...

Hey everyone, I wasn't really sure where to post this, since it's kind of classical, kind of relativistic and kind of quantum field theoretical, but essentially mathematical. I'm reading and watching the lectures on Quantum Field Theory by Cambridge's David Tong (which you can find here), and...

1. A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is \frac{\pi}{3}, this angle is decreasing at a rate of -\frac{\pi}{3} rad/min. How fast is the plane traveling at that time? Homework Equations...

Question 1 When I take the derivatives of the Lagrangian, specifically of the form: \frac{\partial L}{ \partial q} I often find myself saying this: \frac{\partial \dot{q}}{ \partial q}=0 But why is it true? And is it always true?

Homework Statement Let h(u,v) = f(u+v, u-v). Show that f_{xx} - f_{yy} = h_{uv} and f_{xx} + f_{yy} = \frac12(h_{uu}+h_{vv}) . Homework Equations The Attempt at a Solution I'm always confused on how to tackle these types of questions because there isn't an actual function to...

1. Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is \pi/3. Homework Equations $$A=\frac{xysinθ}{2}$$ The...

1. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m^3/min, how fast is the water level rising when the water is 30...

Hello, I am not completely certain why in thermodynamics, it seems that everything is done as a partial derivative, and I am wondering why? My guess is because it seems like variables are always being held constant when taking derivatives of certain things, but it is still somewhat a mystery to...

1. Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high? Homework Equations...

Hi I have a question about partial derivatives? For example if I have a function x = r cos theta for all functions, not just for this function will dx/d theta be the inverse of dtheta/dx, so 1 divided by dx/d theta will be d theta/ dx? Please help on this partial derivative question...

1. A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 fts along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole? Homework Equations $$x^2+y^2=z^2$$ The Attempt at a Solution I've...

I've hit a snag in my studies, namely something my book labels "Corollary 10.1": [i]Are there any other functions with the same derivative as x^2+2=2x? You should quickly come up with several: x^2+3 and [itex x^2-4[/itex] for instance. In fact, d/dx[x^2+c]=2x for any constant c. Are there...

The problem I'm curious about is this: \frac{\partial}{\partial r}(\frac{\partial r}{\partial θ}) I found that the answer is zero using WolframAlpha, but obviously I won't have that on a future test xD. Can someone please explain to me how to think about the derivative above? How can I look...

Homework Statement let u=f(x,y) , x=x(s,t), y=y(s,t) and u,x,y##\in C^2## find: ##\frac{\partial^2u}{\partial s^2}, \frac{\partial^2u}{\partial t^2}, \frac{\partial^2u}{\partial t \partial s}## as a function of the partial derivatives of f. i'm not sure I'm using the chain rules...

Suppose we have a torsion free connection. Does anyone here know of a slick way to prove that covariant derivatives satisfy the Jacobi identity? I.e. that $$([\nabla_X,[\nabla_Y,\nabla_Z]] + [\nabla_Z,[\nabla_X,\nabla_Y]] +[\nabla_Y,[\nabla_Z,\nabla_X]])V = 0$$ without going into...

Se a function f(x(t, s), y(t, s)) have as derivative with respect to t: \frac{df}{dt}=\frac{df}{dx} \frac{dx}{dt}+\frac{df}{dy} \frac{dy}{dt} And, with respect to s: \frac{df}{ds}=\frac{df}{dx} \frac{dx}{ds}+\frac{df}{dy} \frac{dy}{ds} But, how will be the derivative with respect to...

Homework Statement This is an Optimization Problem, find the maximum value. P(R)=(E^2*R)/(R+r)^2 Homework Equations P'(R)=? The Attempt at a Solution I have the solution to this problem, and I can solve it, I just don't understand some parts. I tend to think that the...

First a little warm up problem. Suppose g:\mathbb{R}^N\to\mathbb{C} is some fixed function, and we want to find f:\mathbb{R}^N\to\mathbb{C} such that g(x) = u\cdot\nabla_x f(x) holds, where u\in\mathbb{R}^N is some constant. The problem is not extremely difficult, and after some work...

I determined the equilibrium point of a spring by setting the potential energy function U(r) equal to zero and solving for r. But I just looked at the guided solution, and they took the derivative of U(r) first, then solved for r. Is my approach correct? Can we solve for the equilibrium...