Partial derivatives Definition and 417 Threads

Why do we treat velocity and coordinates as independent variables until the very end, where we then assume the dependence of velocity on coordinates via a time derivative? That is, let the Lagrangian of a given system be simply $$\mathcal L=\frac12mv^2$$ Now, plugging this into the...

The proof for the above ubiquitous formula (as in the summary) in "Chain rule for one independent variable" at the beginning of...

The interest is on number ##4##, In my working, ##f(x,y,z) = x+y^2+2z## and ##g(x,y,z) = 4x^2+9y^2-36z^2 = 36## ##f_x = 1, f_y=2y## and ##f_z = 2## and also ## g_x = 8λx, g_y = 18λy## and ##g_z = -72λz## using ##\nabla f (x,y,z) = λ\nabla f (x,y,z)## i shall have, ##1 = 8λx ## ##2y =...

##df=\frac {\partial f}{\partial r} dr+\frac {\partial f}{\partial \theta}d\theta\quad \nabla f=\frac{\partial f}{\partial r}\vec{e_r} +\frac{1}{r}\frac{\partial f}{\partial \theta }\vec{e_\theta }## On the other hand ## g_{rr}=1\:g_{r\theta}=0\:g_{\theta r}=0\;g_{\theta\theta}=r^2\;##So...

Note that ## \frac{\partial F}{\partial x}=\frac{2x}{2\sqrt{x^2+y'^2}}=\frac{x}{\sqrt{x^2+y'^2}}, \frac{\partial F}{\partial y}=0, \frac{\partial F}{\partial y'}=\frac{2y'}{2\sqrt{x^2+y'^2}}=\frac{y'}{\sqrt{x^2+y'^2}} ##. Now we have ## \frac{dF}{dx}=\frac{\partial F}{\partial x}+\frac{\partial...

My first point of reference is: https://math.stackexchange.com/questions/154968/is-there-really-no-way-to-integrate-e-x2 I have really taken time to understand how they arrived at ##dx dy=dA=r dθ dr## wow! I had earlier on gone round circles! ...i now get it that one is supposed to use partial...

Suppose Q=2x+t and x=t2, then ∂Q/∂t=1. But Q can also be written as Q=x+t2+t, then ∂Q/∂t=2t+1. We now have 2 different answers. But I think there can only be one correct answer. In reference to the equation in the image, no matter we write Q=2x+t or Q=x+t2+t, <Q> should be the same, so the LHS...

Hello, Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$ Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be...

Some may say that ##\frac{ \partial g }{ \partial t }## is correct because it is a term in a partial differential equation, but since ##g## is a one variable function with ##t## only, I think ##\frac{ dg }{ dt }## is correct according to the original usage of the derivative and partial...

Hello everyone, I seem to be majorly lacking in regards to intuition with partial derivatives. I was studying the Euler-Lagrange equations and realized that my normal intuition with derivatives seems to lead me to contradictory or non sensical interpretations when reading partial derivatives...

Can someone please help me to find out what happened here ?

Hi, Starting from dS in term of H and P, I'm trying to find ##(\frac{\partial H}{\partial P})_t## in term of ##P,V,T, \beta, \kappa, c_p##. Here what I did so far. ##ds = (\frac{\partial S}{\partial H})_p dH + (\frac{\partial S}{ \partial P})_H dP## ##ds = (\frac{\partial S}{\partial H})_p [...

He draws an n-manifold M, a coordinate chart φ : M → Rn, a curve γ : R → M, and a function f : M → R, and wants to specify ##\frac d {d\lambda}## in terms of ##\partial_\mu##. ##\lambda## is the parameter along ##\gamma##, and ##x^\mu## the co-ordinates in ##\text{R}^n##. His first equality is...

But, If I use chain rule than, I get that. ##\vec v_i = \frac{dr_i}{dt}=\sum_k \frac{\partial r_i}{\partial q_k} \cdot \frac{\partial q_k}{\partial t}## But, they found that?

Good day I just want to confirm if a function f(x,y) who has directional derivatives has automatically partial derivatives (even though the function itself is not necessarly differentiable)? Can we consider that partial derivatives are special cases of directional derivatives? Thank you in advance!

I've attached images showing my progress. I have used Maxwell relations and the definitions of ##\alpha##, ##\kappa## and ##c##, but I don't know how to continue. Can you help me?

It looks very easy at first glance. However, the variable S is a variable in the given expression. I have no clue to relate the partial derivatives to entropy and the number of particles.

a) ONLY The common way to solve this problem is minimizing the two-variable equation after using the substitution ##z^2=1/(xy)##. Yet I wondered if it is possible to optimize the distance equation with three varibles. So I wrote the following equations: Distance: $$f(x,y,z)=s^2=x^2+y^2+z^2$$...

I have no answer or solution to this. So I'm trying to seek a confirmation of whether this is correct or not: ##df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial t}dt ## ##\frac{df}{dt} = \frac{\partial f}{\partial x} \dot x + \frac{\partial f}{\partial t} ## Therefore, ##...

I research about coordinate systems and I found the following informations about transformation. Now, if I replace arctan (x/y) (according to the picture above) to φ, I think I can solve. But if I can do this, then what will be replaced to ψ? I mean, I know just taking partial derative about...

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I'm in a first-year grad course on statistical mechanics and something about multivariable functions that has confused me since undergrad keeps popping up, mostly in the context of thermodynamics. Any insight would be much appreciated! This is a general question, but as an example imagine...

So in my lecture notes on Differential Equations, it states that a first order ODE is exact if A(x,y)dx + B(x,y)dy = 0 and ∂A/∂y = ∂B/∂x. Okay I accept this definition. Then, there is a sentence like this: Our goal is to find the function V(x,y) satisfying Adx + Bdy = dV = ∂V/∂x(dx) +...

Hi, I just need some (hopefully) quick calculus help. I have the following: ##(y\frac {\partial } {\partial z}(z\frac{\partial f} {\partial x}))## After it is expanded this is the solution: ##(yz\frac {\partial^2 f} {\partial z \partial x} + y\frac{\partial f} {\partial x} \frac{\partial z}...

I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error: I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative \begin{align}...

Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...

I am given the following: $$u = (x,t)$$ $$\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0$$ and $$E = x + ct$$ $$n = x - ct$$ I need to solve for $$\frac{\partial^2 u}{\partial x^2}$$ and $$\frac{\partial^2 u}{\partial t^2}$$ using the chain rule.How would I even...

Let: ##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'## where ##V'## is a finite volume in space ##\mathbf{r}=(x,y,z)## are coordinates of all space ##\mathbf{r'}=(x',y',z')## are coordinates of ##V'## ##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...

Hi all, I have had the following question in my head for quite a while: Thermodynamic potentials written in differential form look like $$dU = TdS - PdV$$ and we can obtain equations for say, temperature by doing the following partial $$T = \frac {\partial U}{\partial S} |_V$$ Does this mean...

At the end of a long proof I came across something in tensor calculus that seems too good to be true. And if something seems too good to be true ... The something is that a second order partial derivative vanishes if one of the parts in the denominator is in the same reference frame as the...

Suppose we have to deal with the question : $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}=?\frac{\partial}{\partial y}\frac{\partial}{\partial x}$$ This seems true for independent variables. But if at the end x and y are linked in some way like $$x=f(t),y=g(t)$$ this is no more the...

Hello. Glad to meet you, everyone I am studying the [Mathematical Methods for Physicists; A Comprehensive Guide (7th ed.) - George B. Arfken, Hans J. Weber, Frank E. Harris] In Divergence of Vector Field, I do not understand that How to transform the equation in left side into that in right...

I have been studying multivariable calculus but I can't quite think visually how a function will be differentiable at a point. How can a function be differentiable if its partial derivatives are not continuous?

It is mentioned in Reif's book, statistical physics, that trough dimensional analysis it can be shown that: $$\frac{1}{\beta} = kT $$ where ##\beta## equals ##\frac{\partial \ln \Omega}{\partial E}## and k is the Boltzmann constant. I don't quite see how to reach this result, can anyone give me...

I am integrating the below: \begin{equation} \psi(r,v)=\int \left( \frac{\frac{\partial M(r,v)}{\partial r}}{r-2M(r,v)}\right)dr \end{equation} I am trying to write ψ in terms of M. Please, any assistance will be appreciated.

Homework Statement Hello I am given the equation: ut - 2uxx = u I was given other equations (boundary, eigenvalue equations) but i don't think I need that to solve this first part: The book says to get rid of the zeroth order term by substituting u = exp(t)V(x,t). I tried to but I can't find...

In Section 7.6 - Equivalence of Lagrange's and Newton's Equations in the Classical Dynamics of Particles and Systems book by Thornton and Marion, pages 255 and 256, introduces the following transformation from the xi-coordinates to the generalized coordinates qj in Equation (7.99): My...

So, I'm now studying thermodynamics and our teacher proved some time ago the following mathematical result: If f(x,y,z)=0, then (∂x/∂y)z=1/(∂y/∂x)z But today he used this relation for a function of four variables. Does this result still hold, because I'm not really sure how to prove it. If...

Hello everyone, I am trying to solve this wee problem regarding partial derivatives, and not sure how to do so. The following image shows level curves of some function \[z=f(x,y)\] : I need to determine whether the following partial derivatives are positive or negative at the point P...

I am trying to minimize the function below, ##R##, to find the optimum ##K##, ##V_d##, and ##V_m##. Currently I minimize ##V_d## and ##V_m## with gradient descent, and find the best K through a binary search, but, if possible, I would like to get rid of binary search and use only gradient...

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 D&K's notation for directional and partial derivatives ... ... D&K's definition of directional and...

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 D&K's notation for directional and partial derivatives ... ... D&K's definition of directional and...

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 another aspect of the proof of Proposition 2.3.2 ... ... Duistermaat and Kolk's Proposition 2.3.2 and its proof read...

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 another aspect of the proof of Proposition 2.3.2 ... ... Duistermaat and Kolk's Proposition 2.3.2 and its proof read...

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 the proof of Proposition 2.3.2 ... ... Duistermaat and Kolk's Proposition 2.3.2 and its proof read as...

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 the proof of Proposition 2.3.2 ... ... Duistermaat and Kolk's Proposition 2.3.2 and its proof read as...

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...

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...