Gradient Definition and 698 Threads

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...

How do we prove that the gradient points in the direction of the maximum increase? Would it be enough to simply state that the gradient is just the derivates of a function w.r.t all the variables a function depends upon. Since the derivative of a term w.r.t a certain variable gives the maximum...

Homework Statement hi i have the folowing data i would like to plot in matlab plotERRLW = 0.0466 0.0111 0.0074 0.0046 NX = 50 500 1000 2000 i am using a logarithmic graph to gain a straight line, if i wished to find the gradient of the line...

The first order KKT condition for constrained optimization requires that the gradient for the Lagrangian is equal to zero, however this does not necessarily imply that the objective function's gradient is equal to zero. Is it absurd to include in one's Lagrangian the requirement that the entry...

I am trying to understand how Hamiltonian gradient works. [SIZE="4"]H(q,p)=U(q)+K(p) U(q): potential energy K(p): kinetic energy q: position vector p: momentum vector both p and q are functions of time H(q,p): total energy [SIZE="4"]\frac{d{{q}_{i}}}{dt}=\frac{\partial H}{\partial {{p}_{i}}}...

Hi Folks, After a general search online, I have not yet found a simple description of refraction in a medium with an inhomogeneous refractive index. For example: if we have a block of glass with a beam of light shining through it, and the block has a gradient in the real part of the...

Under what circumstances can a conservative field NOT be irrotational?

I am doing my research in probability. I have found some probability distribution of a random variable X on the n dimensional unit sphere. Let b be a smooth and lipschitz vector field mapping X to R^n. I have also found that for all continuous differentiable function f mapping X to R, the...

Homework Statement Is F = (2ye^x)i + x(sin2y)j + 18k a gradient vector field? The Attempt at a Solution Yeah I just don't know...I started to find some partial derivatives but I really don't know what to do here. Please help!

Homework Statement Given: Concentration of Fluid = F(x,y,z) = 2x^2 + 4y^4 + 2*x^2*z^2 at point (-1,1,1) Found Grad(F(x,y,z)) = <-8,16,4> ----If you start to move in the direction of Grad(F) at a speed of 8, how fast is the concentration changing?Homework Equations Already found the gradient...

Hi all, I am struggling to find any elementary material on the "gradient flow of a functional" concept. From introductions in advanced papers I seem to have understood that, assigned a functional F (u), the gradient flow is charactwerized by an equation of the type Du / Dt = P u , where P...

Homework Statement The evaluated partial derivative of f(x,y) with respect to x is -16 and 6 with respect to y at some point (x0,y0). What is the vector specifying the direction of maximum increase of f? Homework Equations The direction of maximum increase of f is given by...

Homework Statement An artificial hill has altitude given by the function A(x,y)=300e^-(x^2 +y^2)/100 where the positive y-axis points north and the positive x-axis points east. a.)What would be the instantaneous rate of change of her altitude if she walks precisely northwest, starting from...

If given only: f(5,2) = 80 fx(5,2) = 8 fy(5,2) = -6 Suppose 80 is measured in degrees Fahrenheit. Find the direction where the temperature would get cooler. I just did 8a - 6b = 0 (since using the dot product, <8,-6> * <a,b> = 0. Then I solved for a, b, and this was the vector equation. Then...

Hi I am having trouble getting my head around the definition of a gradient. I know a gradient tells us the direction of steepest slope that one must follow to arrive at a maximum and I know it is defined as: However I haven't got a gutt feeling for it, I need these questions answering...