Gradient Definition and 698 Threads

Homework Statement can anyone explain/prove why the gradient vector is perpendicular to level curves? Homework Equations The Attempt at a Solution

Homework Statement I need some help regarding the gradient operator. I recently came across this statement while reading Griffith's Electrodynamics "The gradient ∇T points in the direction of maximum increase of the function T." Wolfram Alpha also states that "The direction of ∇f is the...

I am a student of 11th standard and being introduced to Bernoulli's principle made me wonder , how does flow takes place in positive pressure gradient (i.e. from low pressure region to high pressure region), in a diffuser or a diverging part of a venturi meter , since we know that flow always...

[SIZE="4"]Definition/Summary The gradient is a vector operator denoted by the symbol \mathbf\nabla or grad. The gradient of a differentiable scalar function f\left({\mathbf x}\right) of a vector \mathbf{x}=\left(x_1,x_2,\ldots,x_n\right) is a vector field whose components are the partial...

There is a paper in chemical physics by Overbeek in which he describes the electrostatic energy of a double layer as the "energy of the surface charges and bulk charges in a potential field"; the transformation that he provides appears to be a variant of the divergence theorem in which he...

I'm supposed to find the gradient vector of the function below at (0,0), and then use the dot product with the unit vector to find the directional derivative. Then find the directional derivative using the limit definition of a directional derivative, and explain why I get two different...

Homework Statement We take a sphere (1mm) which has a parabolically changing refractive index, which is given in a function. Homework Equations Depending on the gradient of the refractive in the sphere, how does it correlates with the focal length. The Attempt at a Solution I...

Hi. This is driving me mad: \hat{\vec{\nabla}}(\hat{\vec{A}})f=(\vec{\nabla}\cdot\vec{A})f + \vec{A}\cdot(\vec{\nabla}f) for an arbitrary vector operator ##\hat{\vec{A}}## So if we set ##\vec{A} = \vec{\nabla}## this should be correct...

I'm developing a classical model of a dipolar ion in an external electric field. It consists of two charges ##\delta_+## and ##\delta_-##, located at a fixed distance from each other. For the special case I'm considering, I end up with the potential energy $$ (\delta_+ + \delta_-)...

. I need to calculate the maximum gradient that a specific vehicle will be able to climb when subjected to known load conditions. There are 6 calculations required for 4 sets of conditions. Values common to all calculations: - Weight of vehicle : 1800kg Weight of payload : 500kg Max gross...

Hi All, I think I have confused myself ... perhaps you can tell me where my reasoning is wrong. The idea is that in general coordinates the partial derivative of a vector, \frac{\partial A^i}{\partial x^j}, is not a tensor because an additional term arises (which is the motivation for...

Hi, I was wandering, sometimes in physics, to get acceleration from a velocity time graph, you would have to find the gradient of the tangent of the curve. But in other graphs like say Voltage current graph, if you want to find the resistance at any point (Which is V/I) you simply take the...

*disclaimer I am not a physicist Had a weird thought the other day - when you focus light with a lens, for example a magnifying glass, you basically increase the 'concentration of photons' at a certain point, right? But then energy is conserved... so wouldn't focusing some of the light on...

Assume that f:\mathbb{R}^N\to\mathbb{R} is a differentiable function and that x_0\in\mathbb{R}^N is a local minimum of f. Also assume that N\geq 2 and that the gradient of f has no other zeros than the x_0. In other words \nabla f(x)=0\quad\implies\quad x=x_0 Is the x_0 a global minimum?

If the direction of the gradient of f in a point P is the direction of most/minor gradient, so a direction of the curl of f in a point P is the direction of most/minor curl too, correct? Also, if the gradient of f in the direction t is given by equation: ∇f·t, so the curl of f in the...

hey pf! i have a few question about the physical intuition for divergence, gradient, and curl. before asking, i'll define these as i have seen them (an intuitive definition). \text{Divergence} \:\: \nabla \cdot \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \cdot \vec{v} da...

Homework Statement complete problem attached Homework Equations The Attempt at a Solution part I in this question was a bit tricky but i managed to solve it , when i read part II i understood nothing , he usually asks about the tangent not the normal , he asks about the point...

From the attached image problem: When deriving the third term in the Lagrangian: \lambda_{2}(w^{T}∑w - \sigma^{2}_{\rho}) with respect to w, are w^{T} and w used like a w^{2} to arrive at the gradient or am I oversimplifying and it just happens to work out on certain problems like this? (∑...

I'm watching a lecture on Newton's method with n-dimensions but I am kind of hung up on why the professor did not use the negative sign while taking the first gradient? Is there a rule that explains this or something that I'm forgetting? The rest makes sense but highlighted in red is the part I...

I'm watching a lecture on Newton's method with n-dimensions but I am kind of hung up on why the professor did not use the negative sign while taking the first gradient? Is there a rule that explains this or something that I'm forgetting? The rest makes sense but highlighted in red is the part I...

Homework Statement Match the function with the description of its gradient. Homework Equations f(x,y,z)=√(x^2+y^2+z^2) 1. constant, parallel to xy plane 2. constant, parallel to xz plane 3. constant, parallel to yz plane 4. radial, increasing in magnitude away from the origin 5. radial...

Homework Statement Suppose I wish to fit a plane z = w_1 + w_2x +w_3y to a data set (x_1,y_1,z_1), ... ,(x_n,y_n,z_n) Using gradient descent Homework Equations http://en.wikipedia.org/wiki/Stochastic_gradient_descent The Attempt at a Solution I'm basically trying to figure out the...

Homework Statement Homework Equations The Attempt at a Solution Ok so I think I know how to get the direction. It's going to be perpendicular to the tangent of the level curve and pointing in the direction where f(x,y) is increasing. So on the graph that was provided it will...

According to the theory, E= -dv/dx or E.dx = -dv So if both are positive, the potential drop should increase. But as we know, if a positive charge is placed, as the distance from it keeps on increasing, field strength starts decreasing and potential drop should increase But this is...

Homework Statement Consider the function f(x,y) = cos(x^2+3y). Write down the gradient of f. Then find the lines in the x-y plabe where ∇f = 0 Homework Equations ∇f = (∂f/∂x,∂f/∂y) The Attempt at a Solution -2xsin(x^2+3y) = 0 sin(x^2+3y) = 0 y = -(1/3)x^2 and...

is electric field strength always equal to negative potential gradient or can it be equal to positive potential gradient sometimes?

I am trying to understand stresses that are induced by thermal gradients. Now, I can think of a hundred different questions to ask, but I want to take baby steps to get there. Let's just talk about a simple cantilever beam in the x-y plane where the x-axis is the beam's longitudinal axis and...

Level function [L(x,y,z)] = (1/r^2) (x^2 + y^2 + z^2) = 1 Vector [N([x(h,g)], [y(h,g)], [z(g)])] = parametric equation to sphere Level function [L(x,y,z)] The parametric equations have 2 parameters, h and g [x(h,g)] = (r [sin (a + gv)]) [cos (b + hw)] [y(h,g)] = (r [sin (a + gv)]) [sin (b +...

Homework Statement Find the derivative of \frac{Q}{4\pi \epsilon_0 r} Homework Equations \frac{d}{dx} \frac{1}{x}=\ln x The Attempt at a Solution Assuming Q and the rest of the variables under it are constant, \frac{Q}{4\pi \epsilon_0}\frac{1}{r} then the derivative should be \ln...

Homework Statement The magnetic field of the Earth it's approximately B=3\times10^{-5}T at the equator and diminishes with the distance from the center of the Earth as 1/r^3, as a dipole. Consider a population of electrons on the equatorial plane with energy 30keV, at 5 Earth radius from the...

Homework Statement I don't seem to understand the proper intuition behind the electric field intensity potential difference relation? please can anyone explain it with solid intuition and maybe a good analogy...and can anyone give a short analogy about the concept of electric field...

Hi, Short question: If you take the inner product of two arbitrary wave functions, and then the gradient of that, the result should be zero, right? (Since the product is just a complex number.) Am I missing something? ∇∫dΩψ_{1}*ψ_{2} = 0

Hello, I have been playing around with a simulation software package and ran a heat transfer simulation however, to me the results did not seem intuitive. From my understanding the distance heat travels in a specific amount of time is proportional to the temperature difference, for example...

Is there any data about such measurements? I think, there has to be a pressure gradient in a floating liquid sphere due to surface tension.

Is it just ∇-1 with the vector hat?

Dear, I have a task to model the behaviour of certain interphase material. Let's say that functions which describe the change of material parameters are known. i.g. change of the Young's modulus as function of distance from neighbouring material (or (0,0) origin) - PAR=PAR(x)...

\partialThis is my first post, so I apologize for all my mistakes. Thank you for the help, in advance. These are test review questions for Multi Variable Calculus. Homework Statement Let f(x,y) = tan-1(y2 / x) a) Find fx(\sqrt{5}, -2) and fy(\sqrt{5}, -2). b) Find the rate of change...

Hey! I've been working on this problem - I think it would be easy for a chemist to answer. If anyone can help me out, I'd appreciate it! Suppose I want to create a pH gradient from 3 to 9, in increments of 0.5: pH = (3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9) Each pH is in a different...

Homework Statement By use of the graph in part (a), calculate the gravitational field strength at a distance 2R from the centre of the Earth. Homework Equations g = delta V / Delta r filed strength is the potential gradient The Attempt at a Solution I have drawn the...

Hi all I was hoping someone could help solve a gradient problem, I am more concerned about understanding what the question is asking me. Homework Statement I have two straight lines which represents the vertical profile of a road. Line AB has a gradient of 1 in 169 (for every 1...

"What is the acceleration of a car of mass 1200kg, when the driving force on it is 4000N and the total frictional force on it is 900N? What is the acceleration of this car climbing a hill with a gradient of 8 degrees?" I got the first part, as obviously a= f/m, resultant force is 3100, and...

i am not sure if this post should be under calculus or not, but i think i'll get a more "complete" answer here. at any rate, I'm wondering if anyone can clarify the intuition behind the gradient theorem: \iiint\limits_V \nabla \psi dV=\iint\limits_S \psi \vec{dS} by intuition, i refer to a...

Hi, I am trying find the simplified expression of this: ∇(E \cdot E) Where E is the electric field that can written as E_{0}(exp(i(kx-ωt)) I know that since the two vectors are the same => E \cdot E = ||E||^{2} Do I take the gradient of the magnitude then? It just doesn't feel...

Hi guys, I'm trying to take the gradient of the potential function, and know the answer, but am not sure how to go about it. Can someone help me step by step as to how to do this. So the potential function is: \begin{equation} U = \frac{1}{2} G \sum^{N}_{i=1} \sum^{N}_{j=1,j \neq i}...

Problem: Starting from the gradient of a scalar function T(x,y,z) in cartesian coordinates find the formula for the gradient of T(s,ϕ,z) in cylindrical coordinates. Solution (so far): I know that the gradient is given by \nabla T = \frac{\partial T}{\partial x}\hat{x}+\frac{\partial...

Hello, I am new here and took a look through the forms, if there is a better place for this question, feel free to move it there. I have been busting my noggin trying to google the answer to this problem I have. (Which isn't school/homework related) I am looking to determine an approximate...

Homework Statement Attached. Homework Equations E=-∇V The Attempt at a Solution I think that the answer is C because it goes in the direction opposite the electric field and crosses through the most equipotential surfaces. Any confirmation or denial would be great. Thank you.

Homework Statement Calculate the gradient of: (a) V1=6xy-2xz+z (b) V2=10ρcos(phi)-ρz (c) V3=(2/r)cos(phi) Homework Equations The Attempt at a Solution Upload

Let ##v(x,y)## be function of (x,y) and not z. \nabla v=\hat x \frac{\partial v}{\partial x}+\hat y \frac{\partial v}{\partial y} \nabla \times \nabla v=\left|\begin{array} \;\hat x & \hat y & \hat z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\...

If ##\vec F(x'y'z')## is function of ##(x'y'z')##. ##\nabla## is operator on ##(x,y,z)##. So: \nabla\left[\vec F(x'y'z') g(x,y,z)\right]=(\vec F(x'y'z') \nabla g(x,y,z) or \nabla(\vec F g)=\vec F \nabla g Am I correct?