Partial derivatives Definition and 435 Threads

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...

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 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...

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...

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

In 'Introduction to Electrodynamics' by Griffiths, in the section of explaining the Gradient operator, it is stated a theorem of partial derivatives is: $$ dT = (\delta T / \delta x) \delta x + (\delta T / \delta y) \delta y + (\delta T / \delta z) \delta z $$ Further he goes onto say: $$ dT =...

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...

I am reading a proof of why \left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z Given a wavefunction \psi, \hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...

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 Find the maximum and minimum value attained by f(x, y) = x2 + y2 - 2x over a triangular region R with vertices at (0, 0), (2, 0), and (0, 2) Homework Equations partial x = 0 and partial y = 0 at extrema The Attempt at a Solution partial x = 2x - 2 partial y = 2y 2x - 2 =...

Homework Statement I have the following data which I would like to model using an exponential function of the form y = A + Becx. Using wolfram mathematica, solving for these coefficients was computed easily using the findfit function. I was tasked however to implement this using java and have...

My physics book is showing an example of why it matters "what variable you hold fixed" when taking the partial derivative. So it asks to show that ##(\frac{\partial{w}}{\partial{x}})_{y} \neq (\frac{\partial{w}}{\partial{x}})_z## where ##w=xy## and ##x=yz## and the subscripts are what variable...

Homework Statement Sketch the surface of a paraboloid z=9-x2 -92 in 3-dimensional xyz-space Homework Equations I assume partial derivatives are involved in some manner The Attempt at a Solution [/B] I attempted to solve by making each variable equal to zero... That didn't work xD. I would...

Homework Statement So I know I have to take the derivative with respect to x, then respect to y, then respect to z, but I am not getting the right answer. I know that the answer is 0 and my professor did it with very few steps that I do not understand. Can someone please guide me through it?

Homework Statement a. Given u=F(x,y,z) and z=f(x,y) find { f }_{ xx } in terms of the partial derivatives of of F. b. Given { z }^{ 3 }+xyz=8 find { f }_{ x }(0,1)\quad { f }_{ y }(0,1)\quad { f }_{ xx }(0,1) Homework Equations Implicit function theorem, chain rule diagrams, Clairaut's...

Homework Statement Hi there, what is the difference between the partial derivative and the total derivative? how do we get the gradient "the actual gradient scalar value" at a point on a multivariable function? what does the total derivative tell us and what does the partial derivative tell...

Hi. I know how to calculate partial derivatives and mixed partial derivatives such as ∂2f/∂x∂y but I've now become confused about something. If I have a function of 3 variables eg. f(x,y,z) and I calculate ∂x then I am differentiating wrt x while holding y and z constant. Does that mean ∂x then...

A particle's position is given by $$r_i=r_i(q_1,q_2,...,q_n,t)$$ So velocity: $$v_i=\frac{dr_i}{dt} = \sum_k \frac{\partial r_i}{\partial q_k}\dot q_k + \frac{\partial r_i}{\partial t} $$ In my book it's given $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...

Homework Statement I have the function: f(x,y)=x-y+2x^3/(x^2+y^2) when (x,y) is not equal to (0,0). Otherwise, f(x,y)=0. I need to find the partial derivatives at (0,0). With the use of the definition of the partial derivative as a limit, I get df/dx(0,0)=3 and df/dy(0,0)=-1. However, my...

Homework Statement I'm reading through the derivations of the linear wave equation. I'm following everything, except the passage I highlighted in yellow in the below attachment: Homework Equations I'm not understanding why partials must be used because "we evaluate this tangent at a...

Homework Statement "Under mild continuity restrictions, it is true that if ##F(x)=\int_a^b g(t,x)dt##, then ##F'(x)=\int_a^b g_x(t,x)dt##. Using this fact and the Chain Rule, we can find the derivative of ##F(x)=\int_{a}^{f(x)} g(t,x)dt## by letting ##G(u,x)=\int_a^u g(t,x)dt##, where...

Hi all, It seems I haven't completely grasped the use of Partial Derivatives in general; I have seen many discussions here dealing broadly with the same topic, but can't find the answer to my doubt. So, any help would be most welcome: In Pathria's book (3rd ed.), equation (1.3.11) says: P =...

Homework Statement Show that (du/dv)T = T(dp/dt)v - p Homework Equations Using Tds = du + pdv and a Maxwell relation The Attempt at a Solution I've solved the problem, but I'm not entirely sure my method is correct. Tds = du + pdv ---> du = Tds - Pdv - Using dF=(dF/dx)ydx +(dF/dy)xdy...

I've been looking at the equation for r tilde prime in the image I attached below, but I cannot understand why it is that they say "Hence, the partial derivatives ru and rv at P are tangential to S at P". How does that equation imply that ru and rv are tangential to P?

Suppose you have a parameterized muli-varied function of the from ##F[x(t),y(t),\dot{x}(t),\dot{y}(t)]## and asked to find ##\frac{dF}{dt}##, is this the correct expression according to chain rule? I am confused because of the derivative terms involved. ##\frac{dF}{dt}=\frac{\partial...

Homework Statement Use the Chain Rule to find the 1. order partial derivatives of g(s,t)=f(s,u(s,t),v(s,t)) where u(s,t) = st & v(s,t)=s+t The answer should be expressed in terms of s & t only. I find the partial derivatives difficult enough and now there is no numbers in the problem, which...

Homework Statement Suppose ω = g(u,v) is a differentiable function of u = x/y and v = z/y. Using the chain rule evaluate $$x \frac{\partial ω}{\partial x} + y \frac {\partial ω}{\partial y} + z \frac {\partial ω}{\partial z}$$ Homework EquationsThe Attempt at a Solution u = f(x,y) v = h(y,z)...

Homework Statement \frac{d}{dt}\left(\frac{\dot{q}}{\sqrt{1+\left(\dot{q}\right)^{2}}}\right)=0\Rightarrow\frac{\dot{q}}{\sqrt{1+\left(\dot{q}\right)^{2}}}=const\Rightarrow\dot{q}=A\Rightarrow q=At+B Homework Equations Why it ok to say that...

Homework Statement v_{i}=\dot{x}_{i}=\dot{x}_{i}\left(q_{1},q_{2},..,q_{n},t\right) T \equiv \frac{1}{2}\cdot{\sum}m_{i}v_{i}^{2} \frac{\partial T}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial v_{i}}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial x_{i}}{\partial q_{k}}[/B]...

Homework Statement [/B] The area of a triangle is (1/2)absin(c) where a and b are the lengths of the two sides of the triangle and c is the angle between. In surveying some land, a, b, and c are measured to be 150ft, 200ft, and 60 degrees. By how much could your area calculation be in error if...

Homework Statement the equation is E= k((xy)x[hat] +(2yz)y[hat] +(3xz)z[hat]) Homework Equations partial of x with respect to y on the x component partial of y with respect to x on the y component The Attempt at a Solution my professor said during class that the partial of x with respect to y...

In my chemical thermodynamics class/notes (and other references I've used) it is stated throughout that internal energy U is a function of entropy and volume , i.e. it's "natural" variables are S and V: U = U(S,V) I suspect that I must take this "axiomatically" and move on. Since U is a state...

take the function f(x,y,z) s.t dF=(d'f/d'x)dx+(d'f/d'y)dy+(d'f/d'z)dz=0 where "d'" denotes a curly derivative arrow to show partial derivatives Mod note: Rewrote the equation above using LaTeX. $$df = (\frac{\partial f}{\partial x} ) dx + (\frac{\partial f}{\partial y} ) dy + (\frac{\partial...

I've just done a derivation and had to use the following u_{b}u^{c}\partial_{c}\rho = u_{b}\frac{dx^{c}}{d\tau}\frac{\partial\rho}{\partial x^{c}} = u_{b}\frac{d\rho}{d\tau} We've done this cancellation a lot during my GR course, but I'm not clear exactly when/why this is possible. EDIT: is...

F(r,s,t,v) = r^2 + sv + t^3, where: r = x^2 +y^2+z^2 /// s = xyz /// v = xe^y /// t = yz^2 find Fxx i have 2 solutions for this and i am not sure what is the right one the first solution finds Fx then uses formula : Fxx = Fxr.Rx + Fxs.Sx + Fxv.Vx+Fxt.Tx the 2nd solution find Fx then uses the...

Let ##f(x,y)## be a scalar function. Then $$\frac{∂f}{∂x} = \lim_{h \rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h} = f_x (x,y)$$ and $$\frac{∂}{∂y} \left (\frac{∂f}{∂x} \right ) = \lim_{k \rightarrow 0} \frac{f_x(x,y+k)-f_x(x,y)}{k} = \lim_{k \rightarrow 0} \left ( \frac{ \displaystyle \lim_{h...

I am working on implementing a PDE model that simulates a certain physical phenomenon on the surface of a 3D mesh. The model involves calculating mixed partial derivatives of a scalar function defined on the vertices of the mesh. What I tried so far (which is not giving good results), is this...

What is the difference between partial derivatives and gradients, if there is any? I'm asking because I need to derive a function " f (T,P) " for air convection; where T is temperature and P is pressure and both are variables in this case. Thanks

Homework Statement Let l, w, and h be the length, width and height of a rectangular box. The length l is increasing with time at at rate of 1 m/s, while the width and the height are decreasing at rates 2 m/s and 1m/s respectively. At a certain moment in time the dimensions of the box are l=5...

I have a multivariable function z = x2 + 2y2 such that x = rcos(t) and y = rsin(t). I was asked to find (I know the d's should technically be curly, but I am not the best at LaTeX). I thought this would just be a simple application of chain rule: ∂2/(∂y∂t) = (∂z/∂x)(ⅆx/ⅆt) + (∂z/∂y)(ⅆy/ⅆt)...

Hi, I'm a little confused about something. I have an object, and I want to take the partial derivative of its position wrt velocity and vice versa. I'm not sure how to begin solving this problem. Essentially, what I have is this: ## \frac{\partial x}{\partial \dot x} ## and ## \frac{\partial...

Homework Statement Find the equation of state given that k = aT^(3) / P^2 (compressibility) and B = bT^(2) / P (expansivity) and the ratio, a/b? Homework Equations B = 1/v (DV /DT)Pressure constant ; k = -1/v (DV /DP)Temperature constant D= Partial derivative dV = BVdT -kVdP (1) ANSWER is...

Homework Statement Find ∂2f ∂x2 , ∂2f ∂y2 , ∂2f ∂x∂y , and ∂2f ∂y∂x . f(x, y) = 1,000 + 4x − 5y Homework EquationsThe Attempt at a Solution Made somewhat of an attempt at the first one and got 0, however my teacher has poorly covered this in class, and I would value some further explanation.

Homework Statement Homework EquationsThe Attempt at a Solution I have the solution to this problem and the issue I'm having is that I don't understand this step: Maybe I'm overlooking something simple but, for the red circled part, it seems to say that ∂/∂x(∂z/∂u) =...

Hey! :o Let $w=f(x, y)$ a two variable function and $x=u+v$, $y=u-v$. Show that $$\frac{\partial^2{w}}{\partial{u}\partial{v}}=\frac{\partial^2{w}}{\partial{x^2}}-\frac{\partial^2{w}}{\partial{y^2}}$$ I have done the following: We have $w(x(u,v), y(u, v))$. From the chain rule we have...

I would like to define t^*= \phi(r, t) given dt^* = \left( 1-\frac{k}{r} \right) dt + 0dr where k is a constant. Perhaps it doesn't exist. It appears so simple, yet I've been running around in circles. Any hints?

I have completed the exercise, but I did something weird in one step to make it work, and I'd like to know more about what I did...or if what I did was at all valid. 1. Homework Statement Show that Laplace's equation \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial...

Find the partial derivatives of the following function: Q=(1/3)logeL+(2/3)logeK Any help would be much appreciated! Below is my working out so far: \frac{\partial Q}{\partial L}= \frac{\frac{1}{3}}{L} \frac{\partial Q}{\partial K}= \frac{\frac{2}{3}}{L} Are these correct?

In the proof, mean value theorem is used (in the equal signs following A). Hence, the conditions for the theorem to be true would be as follows: 1. ##\varphi(y)## is continuous in the domain ##[b, b+h]## and differentiable in the domain ##(b, b+h),## and hence ##f(x,y)## is continuous in the...