Partial derivatives Definition and 435 Threads

I'm reading through the book Quantum Mechanics (Second Edition) by David J. Griffiths and it got to the part about proving that if you normalise a wave function, it stays normalised (Page 13). That part that I don't get is how they say: ## \dfrac{i \hbar}{2m} \left( \Psi^* \dfrac{\partial^2...

I'm thinking in particular about Lenny Susskind's lectures, but I've seen other lecturers do it too. They'll be writing equation after equation using the partial derivative symbol: \frac{\partial f}{\partial a} And then at some point they'll realize that some problem they're currently...

Homework Statement The separation of layers is considered to occur at the thermocline, which is defined as the location of the steepest slope in the temperature gradient. Mathematically, this occurs at the inflection point – so the position of the thermocline can be found from the following...

Homework Statement If z=f(x,y) with u= x^2 -y^2 and v=xy , find the expression for (∂x/∂u). the (∂x/∂u) will be used to calsulate ∂z/∂u. my question is how to find (∂x/∂u). I don't know what to keep constant. Maybe the question has some problem. The answer is (∂x/∂u)=(x/2)/(x^2+y^2)...

Hello guys! Lately I've been studying some topics in Physics which require an extensive use vector calculus identities and, therefore, the manipulation of partial redivatives of vectors - in particular of the position and velocity vectors. However, I am not sure if my understanding of partial...

Homework Statement Where T(x,t)=T_{0}+T_{1}e^{-\lambda x}\sin(\omega t-\lambda x) \omega = \frac{\Pi}{365} and \lambda is a positive constant. Show that T satisfies T_{t}=kT_{xx} and determine \lambda in terms of \omega and k. I'm not to sure what is meant by the latter part of "determine...

f(z,t)=\frac{A}{b(z-vt)^{2}+1}... \frac{\partial^{2} f(z,t)v^{2} }{\partial z^2}=\frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}}=\frac{\partial^2 f}{\partial t^2} \frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}} this...

I have z=(e^y)φ*[y*e^(x^2/2y^2)].I have to prove that y*(dz/dx) -x*(dz/dy)=0.First of all what does φ mean there?

Hello PH, This is my first post. I came here while studying partial derivatives and after clicking here and there for over 4hrs for an answer. While practicing the derivatives rules i came across the hideous quotient rule. I've solved around 20 fractional problems trying to find a decision...

The idea of varying one thing but keeping others constant is central in scientific analysis. People want to know, other things constant, the effect of taking vitamins, smoking or drinking alcohol, just as examples. Is the idea of the partial derivative analogous to scientific empiricism's...

Homework Statement For the heat equation u_{t}=\alpha^{2}u_{xx} for x\in\mathbb{R} and t>0, show that if u(x,t) is a strong solution to the heat equation, then u_{t} and u_{x} are also solutions. Homework Equations u_{t}=\alpha^{2}u_{xx} The Attempt at a Solution I've considered...

Homework Statement general course question Homework Equations N/A The Attempt at a Solution fx is a first order partial derivative fxy is a second order partial derivative fxyz is a third order partial derivative I understand that Clairaut's Theorem applies to second order...

I am utilitizing rotation vectors (or SORA rotations if you care to call them that) as a means of splitting 3D rotations into three scalar orthogonal variables which are impervious to gimbal lock. (see SO(3)) These variables are exposed to a least-squares optimization algorithm which...

In a thermodynamics question, I was recently perplexed slightly by some partial derivative questions, both on notation and on physical meaning. I believe my questions are best posed as examples. Suppose we have an equation, (\frac{\partial x(t)}{\partial t}) = \frac{1}{y}, where y is a...

let f = x2 + 2y2 and x = rcos(\theta), y = rsin(\theta) . i have \frac{\partial f}{\partial y} (while holding x constant) = 4y . and \frac{\partial f}{\partial y} (while holding r constant) = 2y . i found these partial derivatives by expressing f in terms of only x and y, and then in...

Hi everyone, I am working on simplifying a differential equation, and I am trying to figure out if a simplification is valid. Specifically, I'm trying to determine if: \frac{\del^2 p(x)}{\del p(x) \del x} = \frac{\del^2 p(x)}{\del x \del p(x)} where p(x) is a function of x. Both p(x)...

Homework Statement Determine whether a function with partial derivatives f_x(x,y)=x+4y and f_y(x+y)=3x-y exist. The Attempt at a Solution The method I've seen is to integrate f_x with respect to x, differentiate with respect to y, set it equal to the given f_y and show that it can't be...

I'm working on a calculus project and I can't seem to work through this next part... I need to substitute equation (2) into equation (1): (1): r\frac{\partial}{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial ^{2}T}{\partial\Theta^{2}}=0 (2): \frac{T-T_{0}}{T_{0}}=A_{0}+\sum from n=1...

Suppose I have a transformation (x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2)) and I wish to find \partial x'_1\over \partial x'_2 how do I do it? If it is difficult to find a general expression for this, what if we suppose f,g are simply linear combinations of x_1,x_2 so something like ax_1+bx_2 where...

Could someone please explain how the formula at the bottom of the page is derived i.e. how is the Taylor theorem used to obtain it ?

Homework Statement Suppose f: R -> R is differentiable and let h(x,y) = f(√(x^2 + y^2)) for x ≠ 0. Letting r = √(x^2 + y^2), show that: x(dh/dx) + y(dh/dy) = rf'(r) Homework Equations The Attempt at a Solution I have begun by showing that rf'(r) = sqrt(x^2 + y^2) *...

what does d/ds (e^s cos(t)du/dx + e^s sin(t)du/dy) give, given that u = f(x,y) i don't know how to manipulate d/ds and how to derive using d/ds. i am trying to simplify the expression, but i don't know, i just get stuck in the middle of can't get farther than here...

Homework Statement I'm taking a fluid mechanics class and I'm having an issue with acceleration and background knowledge. I know this is ridiculous, but I was hoping someone might be able to explain it for me. Homework Equations I definitely understand: ##a=\frac{d\vec{V}}{dt}## And I...

Find the two first-order partial derivatives of z with respect to x and y when z = z(x, y) is defined implicitly by z*(e^xy+y)+z^3=1. I started by multiplying the brackets out to give; ze^xy + zy + z^3 - 1 = 0 i then differentiated each side implicitly and got; dz/dx = yze^xy and...

Homework Statement let u be a function of x and y.using x=rcosθ y=rsinθ,transform the following expressions in the terms of partial derivatives with respect to polar coordinates:(d^u/dx^2(double derivative of u with respect to x)+d^2u/dy^2(double derivative of u with respect to y)...

Homework Statement If z=\frac{1}{x}[f(x-y)+g(x+y)], prove that \frac{\partial }{\partial x}(x^2\frac{\partial z}{\partial x})=x^2\frac{\partial^2 z}{\partial y^2} Homework Equations The Attempt at a Solution I don't know how I'm supposed to find the partial derivative of z with respect to...

Homework Statement If f(x,y,z) = 0, then you can think of z as a function of x and y, or z(x,y). y can also be thought of as a function of x and z, or y(z,x) Therefore: dz= \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y} dy and dy= \frac{\partial y}{\partial x}dx +...

Homework Statement Taking k and ω to be constant, ∂z/∂θ and ∂z/∂ф in terms of x and t for the following function z = cos(kx-ωt), where θ=t2-x and ф = x2+t. Homework Equations The Attempt at a Solution I'm finding this difficult as t and x are not stated explicitly. I know how to...

Homework Statement Find all first and second partial derivatives of the following function: z = e^(-ET) where E and T are functions of z. I know how to do partial differentiation, but not when the variables are functions of z? I don't understand - is there some sort of implicit...

Homework Statement The surface z=f(x,y)=√(9-2x2-y2) and the plane y=1 intersect in a curve. Find parametric equations for the tangent line at (√(2),1,2).Homework Equations Partial derivativesThe Attempt at a Solution Okay, so I'm just trying to work through an example in my textbook, so...

Homework Statement f'_x = kx_k, k = 1, 2, ..., n The Attempt at a Solution The partial should be f(sub)x(sub)k, as in, the partial derivative of f with respect to x_k. I wasn't sure how to represent that using TeX. I'm honestly at a complete loss here, because I'm not entirely sure what the...

Calculus partial derivatives problem [y^(-3/2)arctan(x/y)] *urgent* Homework Statement f(x,y) = y^(-3/2)arctan(x/y)...find fx(x,y) and fy(x,y) [as in derivatives with respect to x and with respect to y]. Homework Equations The Attempt at a Solution mathematics is not my strong suit..i tried...

Homework Statement I have an expression for the partial derivative of u with respect to s, which is \frac{\partial\,u}{\partial\,s} = \frac{\partial\,u}{\partial\,x}x + \frac{\partial\,u}{\partial\,y}y How do I compute \frac{\partial^2u}{\partial\,s^2} from this?

$x = r\cos\theta$ and $y=r\sin\theta$ $$ \frac{\partial u}{\partial\theta} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta} = -r\sin\theta\frac{\partial u}{\partial x} + r\cos\theta\frac{\partial u}{\partial y} $$...

Homework Statement Please look at the attached pic. I don't know how to type all these symbols in. Homework Equations Im not sure how to start The Attempt at a Solution I tried using the cyclic rule but the problem just started getting messier.

say you have a function f(x,y) \nablaf= \partialf/\partialx + \partialf/\partialy however when y is a function of x the situation is more complicated first off \partialf/\partialx = \partialf/\partialx +(\partialf/\partialy) (\partialy/\partialx) ( i wrote partial of y to x in case y was...

Find the first order partial derivatives of the function x = f(x,y) at the point (4,3) where: f(x,y)=ln|(x+√(x^2+y^2))/(x-√(x^2+y^2))| I understand the method of partial derivatives and implementing the given point values once the partial derivatives are found, however I am having trouble...

A 3-dimensional graph has infinite number of derivatives (in different directions) at a single point. I've learned how to find the partial derivative with respect to x and y, simply taking y and x to be constant respectively. But what do I do if I want to take the partial derivative with respect...

Hi all, I have the following partial derivatives ∂f/∂x = cos(x)sin(x)-xy2 ∂f/∂y = y - yx2 I need to find the original function, f(x,y). I know that df = (∂f/∂x)dx + (∂f/∂y)dy and hence f(x,y) = ∫∂f/∂x dx + g(y) = -1/2(x2y2+cos2(x)) + g(y) Then to find g(y) I took the...

Homework Statement The problem is attached in the picture. The top part shows what is written in the book, but I am not sure how they got to (∂I/∂v)...The Attempt at a Solution It's pretty obvious in the final term that the integral is with respect to 't' while the differential is with...

Homework Statement The equations xu^2 + yv = 2, 2yv^2 + xu = 3 define u(x,y) and v(x,y) in terms of x and y near the point (x,y) = (1,1) and (u,v) = (1,1). Compute the following partial derivatives: (A) ∂u/∂x(1,1) (B) ∂u/∂y(1,1) (C) ∂v/∂x(1,1) (D) ∂v/∂y(1,1) The answers are: (A)...

Homework Statement 1. Is (∂P/∂x)(∂x/∂P) = 1? I realized that's not true, but I'm not sure why.2. Say we have an equation PV = T*exp(VT) The question wanted to find (∂P/∂V), (∂V/∂T) and (∂T/∂P) and show that product of all 3 = -1.The Attempt at a SolutionI tried moving the variables about...

I just got to a point in multivariable calculus where I realize I can solve problems in assignments and tests but have no actual idea of what I'm doing. So I started thinking about stuff and came up with a few questions: 1. Is picturing the derivative as the slope of the tangent line to a...

Homework Statement From step 1 to step 2, what do they mean by "Taking the weighted sum of the two squares " ? I tried and expanded everything in step 2 and it ends up as the same as step 1 (as expected), The Attempt at a Solution I tried looking up "weighted sum" and "...

Homework Statement Find a differential of second order of a function u=f(x,y) with continuous partial derivatives up to third order at least.Hint: Take a look at du as a function of the variables x, y, dx, dy: du= F(x,y,dx,dy)=u_xdx +u_ydy. Homework Equations The Attempt at a Solution I'll be...

Homework Statement I have got a question concerning the following function: f(x,y)=\log\left(x^2+y^2\right) Partial derivatives are: \frac{\partial^2f}{\partial x^2}=\frac{y^2-x^2}{\left(x^2+y^2\right)^2} and \frac{\partial^2f}{\partial y^2}=\frac{x^2-y^2}{\left(x^2+y^2\right)^2} The...

Hello Everyone! This has been confusing me a lot: consider a function $f(x) = x^2 + 2x + 3$. Now, $\frac{\partial f}{\partial x} = 2x + 2$. Now, someone tells me that $y = x^2$. What is $\frac{\partial f}{\partial x}$ now?

Why is partial derivative with respect to time used in the continuity equation, \frac{\partial \rho}{\partial t} = - \nabla \vec{j} If this equation is really derived from the equation, \frac{dq}{dt} = - \int\int \vec{j} \cdot d\vec{a} Then should it be a total derivative with...

In the ordinary least squares procedure I have obtained an expression for the sum of squared residuals, S, and then took the partial derivatives of it wrt β0 and β1. Help me to condense it into the matrix, -2X'y + 2X'Xb. ∂S/∂β0 = -2y1x11 + 2x11(β0x11 + β1x12) + ... + -2ynxn1 + 2xn1(β0xn1 +...

Dear all, I have a confusion about partial derivatives. Say I have a function as y=f(x,t) and we know that x=g(t) 1. Does it make sense to talk about partial derivatives like \frac{\partial y}{\partial x} and \frac{\partial y}{\partial t} ? I doubt, because the definition of...