Gradient Definition and 723 Threads

There have been some discussions here as to what type of processes create entropy rather than just move it around. It is established that a gradient of temperature can create entropy. However, the issue moved to partial pressure, and then even away from that. The previous discussion...

I am having difficulties to understand why in mathematics when calculating line integrals using gradient theorem we use F=grad(U), and in physics it is always F=-grad(U)? It seems important to me, because I may end up getting answer with opposite sign. Is it somehow related to Newton's third law?

Homework Statement The temperature ##T## in a region of Cartesian ##(x,y,z)-## space is given by $$T(x,y,z) = (4 + 3x^2 + 2y^2 + z^2)^{10},$$ and a fly is intially at the point ##(-5,6,7)##. Find a vector parametric representation for the curve which the fly should move in order to ensure...

Hi, I have a quick question about making quantum mechanics relativistic by simply replacing the hamiltonian by a relativistic hamiltonian. If we write the hamiltonian operator as: H = \sqrt{P2c2 + m2c4}, schrodinger's equation in position basis becomes: i\hbar\dot{\psi} =...

Why does the formula for the gradient - that is (for functions of 2 variables), the partial with respect to x plus the partial with respect to y give the direction of greatest increase? i.e. the direction of maximum at some point on a surface is given by f_xi+f_yj And why, when you times...

Hi all, I've to recover a function ∅ such that F=∇∅,F=3x^2y i +(x^3+2yz)j+y^2k. So, ∂∅/∂x=3x^2y, Integrating w.r.t. x ∅=x^3y+f(y,z) ,Assuming there may be function of y and z.----------1 ∴∂∅/∂y=x^3+f'(y,z) now, ∂∅/∂y=(x^3+2yz)=x^3+f'(y,z) ∴f'(y,z)=2yz To find ∅ from 1 I've to find...

Homework Statement So I'm trying to calculate the error on the gradient I've obtained for my lab work. The line of best fit is too precise to use the parallelogram method and I'm still at the stage of my course where calculations of the gradient and such must be done by hand and not using a...

Homework Statement Find \nabla\left( \dfrac{1}{\left| \vec{r}-\vec{r'}\right| }\right) Homework Equations The Attempt at a Solution \left| \vec{r}-\vec{r'}\right| =\sqrt{(x-x^\prime)^2 + (y-y^\prime)^2 + (z-z^\prime)^2} and so therefore the derivative of the scalar would be 0. Of...

Homework Statement Suppose f:R^2 - {0} → R is a differentiable function whose gradient is nowhere 0 and that satisfies -y(df/dx) + x(df/dy) = 0 everywhere. a) find the level curves of f b) Show that there is a differentiable function F defined on the set of positive real numbers so that...

Homework Statement Find the equations of both the straight lines that are inclined at an angle of 45 ° with straight line 2x + y - 3 = 0 and passing through the point (-1 , 4) Homework Equations tan θ = (m1 - m2)/(1+ m1m2) The Attempt at a Solution If I were to use the equation above, how...

Homework Statement Show that the operation of taking the gradient of a function has the given property. Assume u and v are differentiable functions of x and y and that a and b are constants. Operation: (∇(u))n = n*un-1*∇u Homework Equations The gradient vector of f is <∂f/∂x,∂f/∂y>, where f...

Hi all, I'm not quite sure if this is the right place to post my question, so forgive me if its not... I've written a program in Python that analyses data that I got from a compression experiment (mechanical testing of rocks and such), and I've written a piece of code that estimates the...

Let's say we have a scalar function U in terms of r,theta and phi. why cannot this be the gradient at any point P(r,theta,phi)- partial of U wrt. r in the direction of r+partial of U wrt. theta in direction of (theta)+partial of U wrt. phi in the direction of (phi)?

Homework Statement A set of narrow vertical slits is located a distance D from a screen. The slits are equally spaced and have the same width. The intensity pattern in the figure is observed when light from a laser passes through the slits, illuminating them uniformly. The screen is...

Suppose that an object is moving in a space V, so that its position at time t is given by r=(x,y,z)= (3sin πt, t^2, 1+t) How to find the direction of the vector along which the cat is moving at t = 1? I have no idea where to find out the direction of the vector along which the object is...

Homework Statement What is the gradient of the chord of the curve y = 2x^2 between the points x = 1 and x = 1+ h? Homework Equations differentiation by first principles dy/dx = f(x+h) - f(x)/hThe Attempt at a Solution use of the formula to receive 4x +2h

Homework Statement Hi. I have an assignment to find the specific heat capacity of water. We did an experiment in class where we used an electric kettle with power output of 1850W-2200W to heat up 1,400g of water (we actually used 1,400 mL of tap water but we were told to assume that the tap...

Hello everyone, So I am an Aero-Thermo Intern at Pratt and Whitney, and my supervisor gave me the following problem to set up a mathematical Excel model for the temperature gradient inside an enclosed box with a heat source underneath it to be used in thermal analysis of an engine component...

Calculate gradient of f f(x,y)=x^3+2y^3 at point P (1,1) and the directional derivative at P in the direction u of the given vector A -> i-j I tried to attempt this but i honestly don't know where to start. I began to take the partial derivatives of f. I got f'=3x^2dx+6y^2dy, however that...

Using tensors, I'm supposed to find the usual formula for the gradient in the covariant basis and in polar coordinates. The formula is \vec{grad}=[\frac{\partial}{\partial r}]\vec{e_{r}}+\frac{1}{r}[\frac{\partial}{\partial \vartheta}]\vec{e_{\vartheta}} where \vec{e_{r}} and...

Homework Statement Consider an isothermal atmosphere (T = const.) over a sufficiently small range of radii, so that you can assume that the gravitation acceleration g is constant. Use the equation for the gas pressure gradient in hydrostatic equilibrium to show that the gas pressure decreases...

Homework Statement Consider f(\vec{x}) = |\vec{x}|^r, where \vec{x} \in ℝ^n and r \in ℝ. Find \vec{∇}f The Attempt at a Solution I know a vector function maps real numbers to a set of vectors, but here I believe we have the opposite. (inverse of a vector function, assuming inverse...

Homework Statement The temperature T of a plate lying in the (x,y) plane is given by T(x,y) = 50 - x^2 - 2y^2. A bug on the plate is intially at the point (2,1). What is the equation of the curve the bug should follow so as to ensure that the temperature decreases as rapidly as possible...

Homework Statement Determine the field gradient of a 50-cm-long Stern-Gerlach magnet (d1) that would produce a 1-mm separation at the detector between spin-up and spin-down silver atoms that are emitted from an oven at T=1500. Assume the detector is located 50 cm from the magnet (d2). Note...

How do you solve ((grad(f(x,y,z))))^2?

Homework Statement Determine Planck's Constant from gradient Homework Equations The Attempt at a Solution I have a graph entitled "Photoelectric effect: stopping voltage as a function of light frequency" The y-axis is the Stopping Voltage (V), the x-axis the Frequency (Hz)...

Homework Statement Find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. g(x,y) = x2/2 - y2/2; (√2, 1)Homework Equations ∇f = (∂f/∂x)i + (∂f/∂y)jThe Attempt at a Solution ∇g = <x, -y> ∇g(√2, 1) = <√2, -1>...

Let's say we have a function f(x,y,z)=k which is a level surface for a function of 3 variables. Now say at some point P we want to find the derivative in the direction of some vector, u. (the change in z in the direction of u at point P). We can easily find this direction derivative using...

I'm looking for a physical proof, something I can understand easily, though a mathematical proof might help too. Apologies if its the wrong section, encountered this while studying mechanics :|

Hey guys, this is for my classical E&M class but it's more of a math problem. Homework Statement Show: ∇(\vec{A} . \vec{B}) = \vec{B} \times (∇ \times \vec{A}) + (\vec{B} \times ∇)\vec{A} + \vec{A} \times (∇ \times \vec{B}) + (\vec{A} \times ∇)\vec{B} Homework Equations I tried...

Gradient as many knows is rise/run or the ratio between the change of Y over change of X However, what's the meaning then? It has no significant true meaning to a linear slope, the only true meaning is - when i inverse tangent O/A(the gradient) ( right hand side of the picture ) i get...

Can anyone tell me what a high gradient RF structure is? I think it is referring to the strength of say an electric field in a structure oscillating at radio frequency but I am not sure. I would appreciate any explanation.

Hello, The value of electric potential(∅) is known at every node in a 3d finite element mesh. The relation between electric current density(i) and electric potential(∅) is i=k.∇∅, I am writing a code in c, I want to know how to find the gradient of electric potential(∇∅) at every node so...

Hello, I know the value of a field variable in a 3d mapped finite element mesh. Can anyone suggest an effective method/methods to find its gradient throughout the mesh.

http://img580.imageshack.us/img580/8859/proofea.jpg i am having trouble figuring out how to proceed without knowing anything about the function ψ=ψ(x,y,z). all i know is that it's a function of x,y,z. but how can i figure out how the surface changes w.r.t the tangent vector if i don't even...

Howdy, I guess I need to explain this situation a little bit.. I am doing a project about a guy buying a house, but using a gradient series approach to do the payments, for example, say the original monthly payment for a $225,000 15 year loan is ~$1600. The guy will pay that the first month, and...

Homework Statement The statement that the simple surface F has gradient at the ordered pair ((x,y),F(x,y)) of F means that there exists only one ordered number pair (p,q) such that if c is a positive number then there exists a rectangular segment , S, containing (x,y) such that, if (u,v) is in...

Given certain function f(x), a standard way to minimize it is to set its derivative to zero, and solve for x. However, in certain cases the method of gradient descent is used; compared to the previous method (call it 'method I')that simply sets the derivative to zero and solves for x, the...

Hi, I need to plot a vector field 2x,2y,0 which is the gradient of the function x^2+y^2. I need to plot it only in some interesting points (points on the paraboloid x^2+y^2). So I tried something like a=Table[{{x,y,x^2+y^2},{2x,2y,0}},{x,-2,2,0.5},{y,-2,2,0.5}] ListVectorPlot3D[a] It...

Hi everyone, I need help with a derivation I'm working on, it is the differentiation of the norm of the gradient of function F(x,y,z): \frac{∂}{∂F}(|∇F|^{α}) The part of \frac{∂}{∂F}(\frac{∂F}{∂x}) is bit confusing. (The answer with α=1 is div(\frac{∇F}{|∇F|}), where div stands for...

I have been wondering if there is an explanation to why diffusion always goes down the concentration gradient? If it is a random molecular movement than why do we always end up with a uniform distribution of molecules?

Homework Statement I need to show that ##\displaystyle\int_\Omega (\nabla G)w dxdy=-\int_\Omega (\nabla w) G dxdy+\int_\Gamma \hat{n} w G ds## given ##\displaystyle \int_\Omega \nabla F dxdy=\oint_\Gamma \hat{n} F ds## where ##\Omega## and ##\Gamma## are the domain and boundary respectively...

This is not a problem statement this is not homework this is not a textbook exercise. This is my own question about a formula in a textbook. Homework Statement I am trying to understand the way that equation 4.2.2 is rewritten as equation 4.2.3 Homework Equations Source: Stanley...

Homework Statement Consider the surface and point given below:- Surface: f(x,y)= 4-x2-2y2 Point: P(1,1,1) a) Find the gradient of f. b) Let C' be the path of steepest descent on the surface beginning at P and let C be the projection of C' on the xy-plane. Find an equation of C in the...

Homework Statement Find the gradient vector of surface z=(x^2) * (y^3) at A(1,1). Homework Equations The Attempt at a Solution I am confused with book's solution. Books solution is : f(x,y,z) = (x^2) * (y^3) - z grad(f) = < 2x(y^2) , 3(y^2)(x^2) , -1 > at A(1,1) =...

Hello, I am a bit trumped. I know how to calculate the voltage of a single uniform cylinder: V(r) = (-q/2 Pi ep0) Ln(r/R0) + V0 q: Charge density r: radius from outside of the cylinder. r >= R0 R0: radius of the uniform cylinder V0: applied voltage to the cylinder Here is my...

Homework Statement https://dl.dropbox.com/u/64325990/Capture.PNG I'm not even sure where to start :O

Homework Statement A wildebeest is charging across a plain. His path takes him to location (x,y) where x is his distance (in miles) east of his starting point and y is his distance in miles north of his starting point at time t. So x and y are functions of t. The air temperature is a...

Homework Statement I have been able to solve all the gradient problems when you are given the starting point, and the actual function. But I am getting caught up on this one which goes in reverseSuppose that the maximum rate of change of f at (1,-1) is 25 and it occurs in the direction of 3i...

I have the function: f(x,y)= x*(y^2)*e^-((x^2+y^2)/4) I am not sure how to find the point where the gradient is the greatest. The gradient I found after taking the partials is: partial with respect to x: e^(-(x^2+y^2)/4)*((y^2)-.5(x^2)(y^2)) partial with respect to...