Derivatives Definition and 1000 Threads

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of Example 2.2.5 ... ... Duistermaat and Kolk's Example 2.2.5 read as follows: In the above text by D&K we...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of Example 2.2.5 ... ... Duistermaat and Kolk's Example 2.2.5 read as follows: In the above text by D&K we...

Existence of Partial Derivatives and Continuity ... Kantorovitz's Proposition pages 61-62 ... I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with another element of the proof of Kantorovitz's...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with another element of the proof of Kantorovitz's Proposition on pages 61-62 ... Kantorovitz's Proposition on pages 61-62 reads as follows: In the...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an element of the proof of Kantorovitz's Proposition on pages 61-62 ... Kantorovitz's Proposition on pages 61-62 reads as follows: I am...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an element of the proof of Kantorovitz's Proposition on pages 61-62 ... Kantorovitz's Proposition on pages 61-62 reads as...

It´s not a technical question, is about why the classic mechanics and even quantum mechanics equations are first or second order? ¿Exist any model with up order derivates?

I was trying to Extrapolate Eulers formula , after deriing the basic form I wanted to prove: ∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0 Here is my attempt but I get different answers: J(y) = ∫abF(x,yx,y,yxx)dx δ(ε) = J(y+εη(x)) y = yt+εη(x) ∂y/∂ε = η(x) ∂yx/∂ε = η⋅(x)...

Hi all. Suppose I have the ideal gas law $$P=\frac{RT}{v}$$If I'm asked about the partial derivative of P with respect to molar energy ##u##, I may think "derivative of P keeping other quantities (whatever those are) constant", so from the formula above I get $$\frac{\partial P}{\partial...

Could I please get help with the following question? f(x)=(2cos^2 x+3)^5/2 Any help would be very much appreciated:)

Hello! In my GR class we were introduced to the parallel transport as the way in which 2 tensors can be compared with each other at different points (and how one reaches the curvature tensor from here). I was wondering why can't one use Lie derivatives, instead of parallel transport. As far as I...

Hi, I am having some trouble with this problem. I have completed part a but I am stuck on part b and c. I used the quotient rule to try and find the first derivative, but I am unsure if I have done so correctly. This is my work for part b so far...

This is for research purposes. I am aware that first derivatives in thermodynamics always occur (a no-brainer). Do second derivatives occur in thermodynamics commonly as well?

Its a question about volume increase (in units cm^3) and height increase (cm) when pouring juice into a cup. Its stated that the volume of the juice in the cup increases at a constant rate, so I know the volume derivatives are zero. But the shape of the cup is inconsistent and there is a lot...

mod: moved from homework Does anyone know why and when this equation holds? I have searched online but cannot find the reason or the rules for the higher order derivatives.

Does anyone know why this is true?

So, I can't wrap around my head of why the Equation of the Tangent Line is: y = f(a) + f'(a)(x - a) I get it that it's the equation of a line, and so it should be something like y = mx + b. I also understand why f(a) = b (since it's a point in that line) and why f'(a) = m (since it's the slope)...

Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...

Hello, dear colleague. Now I'm dealing with issues of modeling processes of heat and mass transfer in frozen and thawed soils. I am solving this problems numerically using the finite volume method (do not confuse this method with the finite element method). I found your article: "Numerical...

Homework Statement Only 15 Homework Equations First derivative=maxima/minima/vertical tangent/rising/falling Second derivative=points of inflection/concave upward-downward The Attempt at a Solution $$x=y^3+3y^2+3y+2~\rightarrow~1=3(y^2+2y+1)y'$$ $$y'=\frac{1}{3(y+1)^2}>0,~y\neq...

Hi there, I have just read that the gauge field term Fμν is proportional to the commutator of covariant derivatives [Dμ,Dν]. However, when I try to calculate this commatator, taking the symmetry group to be U(1), I get the following: \left[ { D }_{ \mu },{ D }_{ \nu } \right] =\left( {...

Hi, I really wonder how these second derivatives can be written in terms of christofflel symbols. I have made so many search but could not find on internet What is the derivation of equations related to second derivatives in attachment?

Homework Statement Homework Equations Newton's binomial's: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## The Attempt at a Solution I use induction and i try to prove for n+1, whilst the formula for n is given: $$\frac{d^{n+1}(uv)}{dx^{n+1}}=\frac{d}{dx}\frac{d^{n}(uv)}{dx^n}=$$ The...

I thought Differentiation is all about understanding it in a graph. Every time I solve a question on differentiation I visualise it as a graph so it's more logical. After all, that IS what the whole topic is about, right? Or am I just wrong? But when you look at these questions...

Homework Statement Homework EquationsThe Attempt at a Solution I tried to find the slope of the tangent line, but this gave me 3.66 and the answer is 3.8 how do I find this?

Homework Statement If ## z=x^2+2y^2 ##, find the following partial derivative: \Big(\frac{∂z}{∂\theta}\Big)_x Homework Equations ## x=r cos(\theta), ~y=r sin(\theta),~r^2=x^2+y^2,~\theta=tan^{-1}\frac{y}{x} ## The Attempt at a Solution I've been using Boas for self-study and been working on...

How does one solve a problem like this? Suppose we have $$(e_\theta + f(\theta)e_\varphi) (e_\theta + f(\theta)e_\varphi)$$ What is the result of the above operation? As I remember it from the theory of covariant derivatives, the above relation would look like this $$e_\theta[e_\theta] +...

If F = Fxi + Fyj +Fzk is a force field, do the following derivatives have physical significance and are they related to the components of the stress tensor? I notice they have the same dimensions as stress. ∂2Fx / ∂x2 ∂2Fx / ∂y2 ∂2Fx / ∂z2 ∂2Fx / ∂z ∂y ∂2Fx / ∂y ∂z ∂2Fx / ∂z ∂x ∂2Fx / ∂x...

Hi. If I have a function f ( x , t ) = x - 6t with x ( t ) = t2 and I take the partial derivative of f with respect to x I get the answer 1 as t acts as a constant so its derivative is zero. But if I substitute t with x1/2 I get the answer 1 - 3x-1/2 which is obviously different and wrong , I...

Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...

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...

https://arxiv.org/pdf/1705.07188.pdf Equation 5 in this paper states that $$\frac{\partial F}{\partial p_i} = 2Re\left\lbrace\frac{\partial F}{\partial x}\frac{\partial x}{\partial p_i}\right\rbrace$$ Here, p_i stands for the i'th element of a vector of 'design parameters' \mathbf{p}. These...

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))...

Homework Statement To show that ##K=V^uK_u## is conserved along an affinely parameterised geodesic with ##V^u## the tangent vector to some affinely parameterised geodesic and ##K_u## a killing vector field satisfying ##\nabla_a K_b+\nabla_b K_a=0## Homework Equations see above The Attempt at...

Its about functions with two or more variables ? How do you keep this x and y constant , i don't understand .

For implicit differentiation, is dy/dx of x2+y2 = 50 the same as y2 = 50 - x2 ? From what I can take it, it'd be a no since. For x2+y2 = 50, d/dx (x2+y2) = d/dx (50) --- will eventually be ---> dy/dx = -x/y Where, y2 = 50 - x2 y = sqrt(50 - x2) dy/dx = .5(-x2+50)-.5*(-2x)

Just a quick question - Is it true that the domain of ##f'(x)## will always be less than or equal to the domain of the original function, for any function, ##f(x)##?

I am having some trouble solving the problem shown below. Can anyone point me in the right direction? or provide the location of a worked example? The volume V of a cone of height h and base radius r is given by V=1/3 πr^2 h. The rate of change of its volume V due to stress expansions with...

Homework Statement This is my 'carrying out a practical investigation' assignment for Maths. I've attached the coursework (what I've wrote up to now) and my main concern is whether I've got the right differential equation to find 3 new velocity values throughout the pendulum trajectory...

Hi. I don't understand the meaning of "up to total derivatives". It was used during a lecture on superfluid. It says as follows: --------------------------------------------------------------------- Lagrangian for complex scalar field ##\phi## is $$ \mathcal{L}=\frac12 (\partial_\mu \phi)^*...

Homework Statement Show that ##\frac{\partial U}{\partial T}|_{N} = \frac{\partial U}{\partial T}|_{\mu} + \frac{\partial U}{\partial \mu}|_{T} \frac{\partial \mu}{\partial T}|_{N} ## (Pathria, 3rd Edition, pg. 197) Homework Equations ##U=TS + \mu N - pV## The Attempt at a Solution I tried to...

Homework Statement Okay, I'm going to "cheat" a bit and add two programs here, but I don't want to clutter the board by making two threads. Anyways, here goes: (1) The value of the sine of an angle, measured in rads, can be found using the following formula: sin(x) = x - x3/3! + x5/5! - ...

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part V Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part IV Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part III Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part II Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part I Continue reading the Original PF Insights Post.

I've been thinking about it since yesterday and have noticed this pattern: We have, the first order derivative of a function ##f(x)## is: $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} ...(1)$$ The second order derivative of the same function is: $$f''(x)=\lim_{h\rightarrow...

Hi PF! Regarding derivatives, suppose we have some function ##f = y(t)x +x^2## where ##y## is an implicit function of ##t## and ##x## is independent of ##t##. Isn't the following true, regarding the difference between a partial and full derivative? $$ \frac{df}{dt} = \frac{\partial f}{\partial...

Derivatives in first year calculus Gateaux Derivatives Frechet Derivatives Covariant Derivatives Lie Derivatives Exterior Derivatives Material Derivatives So, I learn about Gateaux and Frechet when studying calculus of variations I learn about Covariant, Lie and Exterior when studying calculus...