Gradient Definition and 723 Threads

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.

I am trying to determine my scalar function \(f(u_1, u_2, u_3)\) of elliptical cylindrical coordinates. \begin{align*} x &= a\cosh(u)\cos(v)\\ y &=...

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?

http://en.wikipedia.org/wiki/Gradient#Cylindrical_and_spherical_coordinates which formula do we apply to get the gradient in spherical coordinates?

in a text a read that " \oint \nabla A \cdot dl = 2 \pi n wich implies that the gradient of A has a pole singularity" why there is a singularity? I thing that this is a contidion to integral is nonzero but ¿what is the theorem used?

I've always thought of the gradient of a scalar function (id est, ##\nabla\varphi##) as being a vector field. However, I started thinking about it just now in terms of transformation with respect to coordinate changes, and I noticed that the gradient transforms covariantly. Thus, shouldn't the...

Hi there, I just started to learn about gradients. I can calculate them with ease; but I don't think I really understand them conceptually. I understand the usual example of the temperature scalar field where the temperature in a room is a function of your position T(x, y, z). But when it comes...

I am interested in the math involved to calculate the ideal natural teardrop shape for a hot air balloon. I want to learn the details of what is involved to calculate this accurately. I read this https://www.physicsforums.com/showthread.php?t=658802 which was a really nice start, but it...

Hello, would someone know what is the smallest radius of curvature achievable with current gradient index optics (GRIN) technology? I mean, how much could one "curve" a ray of light? Many thanks! :smile:

I was doing some simple physics with a ball resting on a table and I made this curve (0,0) (25, 6.8) (50, 27.51) (75, 63.4) (100, 112.34) (125, 175.7) (150, 253.3) (175, 345.4) I was wondering if anyone could identify what sort of curve it is? I am just curious. This is not a homework...

how come there is an electrical gradient in lasers? i means lasers are just monochromatic photons so how come a particle feels an electrical force there

Let's say we have some time-independent scalar field \phi. Obviously \phi\left(\mathbf{q}\right)-\phi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\,\mathbf{q}]} \nabla\phi(\mathbf{x})\cdot d\mathbf{x}. This is of course still true if the path \gamma is the trajectory of a particle moving...

Homework Statement The velocity of a body traveling in a circular orbit around another body situated at the centre of the circle, is given by v = √(GM/r) where G is the Universal Gravitational constant, M is the mass of the central body and r is the radius of the orbit. By taking natural...

I am trying to understand MRI scanners. I know that MRIs work by aligning the protons in the direction of the large magnetic field and the radio frequency sets the frequency of the oscillations to the lamour frequency - also raising its energy level. Then when the RF is switched off, the...

Hello. I can't seem to wrap my head around the geometry of the gradient vector in ℝ3 So for F=f(x(t),y(t)), \frac{dF}{dt}=\frac{dF}{dx}\frac{dx}{dt}+\frac{dF}{dy}\frac{dy}{dt} This just boils down to \frac{dF}{dt}=∇F \cdot v Along a level set, the dot product of the gradient vector and...

I do not understand the following from Griffiths’ Electrodynamics – page 424 Equation 10.21. \nabla p = \dot{p} \nabla {tr} = … I’m not sure how much of this applies (I think my question is on the math) but p is the charge distribution, tr is the retarded time. Is this an...

Hi, Homework Statement The gradient ∇3 can be generalized for spacetime as: ∇4 =(∇3 ,d/dct)=(d/dx,d/dy,d/dz,d/dct) Show that ∇4 is a four-vector. Homework Equations The Attempt at a Solution I just have to write that : d/dx'=γ(d/dx-βd/dct) d/dy'=d/dy d/dz'=d/dz...

Hey guys, I'm not sure how to interpret euler's fluid equations \rho (\partial / \partial t + {\bf U} \cdot ∇) {\bf U} + ∇p = 0 I'm not sure what the meaning of {\bf U} \cdot ∇ {\bf U} is. am I able to simply evaulate the dot product as U_{x}\partial_{x} + U_{y}\partial_{y}+...

Hi to all Homework Statement ∫∫∫∇ψdv = ∫∫ψ ds over R over S R is the region closed by a surface S here dv and ψ are given as scalars and ds is given as a vector quantitiy. and questions asks for establishing the gradient theorem by appliying the divergence theorem to each component...

Homework Statement T(x,y,z)=8x^2-7xy+7xyz a. find the rate of change of t at point p(-1,1,-1) in the direction u=<8,10,-8> b. which direction does the temperature increase fastest c. find the maximum rate of increase at P. Homework Equations gradient of T=(16x-7y+7yz, -7x+7xz, 7xy)...

Homework Statement if f(x,y,z) indicates electrical charge in the water at position(x,y,z) and the gradient is <12,-20,5>, in which direction should the shark swim to find its prey? Homework Equations The Attempt at a Solution is the answer in the direction of <12,-20,5> im...

Why do gradient show rate of maximum increase not decrease always?

Homework Statement Show that ∇_{x}|x-y|-3= -(x-y)|x-y|-3 x and y are vectors.Homework Equations The Attempt at a Solution When dealing with just a straight up absolute value I know that a solution can be found by using a piece wise approach, but I don't think that's what I should be using...

Hi guys First things first, I'll lay out the problem. I have a box of volume V containing a constant sink of oxygen (e.g. a candle or an animal); this box is sealed except for a smallish aperture of area, A and depth, L (the L meaning the walls of the box have finite thickness). After a...

So I have a function \(f(x,y)=\sqrt{2x+3y}\) and need to find the gradient at the point (-1,2). I got this part already, its \(\frac{1}{2}\hat{i}+\frac{3}{4}\hat{j}\). The part I'm having trouble with is when it asks me to sketch the gradient with the level curve that passes through (-1,2). The...

Homework Statement I recently conducted an experiment to determine the moment of inertia of a disc using a tachometer attached to a disc marked with reflected strips, a weight, and an oscilloscope. The resulting oscilloscope data was plugged into fitplot to generate a graph of voltage...

Homework Statement Does x' = xex2tanh(x+y) + (1/2)ex2sech2(x+y) y' = (1/2)ex2sech2(x+y) contain a limit cycle? Possibly-relevant theorem below. Homework Equations Theorem. Closed orbits are impossible in gradient systems. Definition. If x' = -ΔV for some cont. diff'ble, single-valued...

Hi everyone, I have a plot of some data points that have error bars on the y axis. A bit of software I am using gives me the best fit gradient and a "Standard Error", but it doesn't take the size of error bars into consideration. I'm assuming that it just looks at how well the gradient...

I'm a bit confused here. If I have Y(x2,x3,x4)=(sqrt(1-x2^2-x3^2-x4^2),x2,x3,x4), how do I find the magnitude of the gradient? I know that for Y(s)=(sqrt(1-s^2),s) the gradient is (-s/sqrt(1-s^2),s) and the magnitude of the gradient is 1/sqrt(1-s^2), and I'm supposed to get an expression similar...

According to Carroll, \nabla \phi is covariant under rotations. This really confuses me. For example, how could equations like \vec{F}=-\nabla V be rotationally covariant if force is a contravariant vector? I know this is strictly speaking more of a mathy question, but I still figured this...

Can we approach spin by gradient. For example, spin 1/2 can be written as 180 degree turning in 360 degree space while spin 2 is 720 degree turning in 360 degree space? If I have a ball spinning with angular momentum perpendicular to rotation plane, what is the spin value of the ball? Can some...

All the demonstrations on Jeans instability start with: \rho g \ge \nabla P Then they substitute \nabla P with nkT/R. But from ideal gas law: PV = nkT, so P = nkT/V = nkT/R^3 (I'm not interested in proportionality factors, so let's not bother about 4/3 \pi, ecc...) Now, gradient of P, to...

Gradient of a dot product identity proof? Homework Statement I have been given a E&M homework assignment to prove all the vector identities in the front cover of Griffith's E&M textbook. I have trouble proving: (1) ∇(A\bulletB) = A×(∇×B)+B×(∇×A)+(A\bullet∇)B+(B\bullet∇)A Homework...

Homework Statement I know you can find the gradient of a scalar using partial derivatives. Does it make sense to find the gradient of a vector, however? A homework problem of mine asks to find the gradient of a vector. I'm starting to think it's a trick question... Homework Equations ∇ dot...

Suppose F(x,y,z) = 0 grad (F) = 0 ? e.g. F = x + y + z grad (F) = <1,1,1> =/= <0,0,0> ?? I don't know why I get an opposite result

Homework Statement For a hill the elevation in meters is given by z=10 + .5x +.25y + .5xy - .25x^2 -.5y^2, where x is the distance east and y is the distance north of the origin. a.) How steep is the hill at x=y=1 i.e. what is the angle between a vector perpendicular to the hill and the z...

Hello, I've been reading up on Smoothed Particle Hydrodynamics. While reading some papers I found some math that I do not know how to do because I never took multi variable calculus. I need the gradient and laplacian of all three of the following functions ( h is a constant )...

Homework Statement compute the gradient: ln(z / (sqrt(x^2-y^2)) Homework Equations ∇=(∂/(∂x)) + ... for y and z I just want to know how to do the first term with respect to x The Attempt at a Solution I am so rusty I don't know where to begin.

I have $$ u(r,\theta) = r\cos(\theta)\left[1 - \left(\frac{1}{r}\right)^2\right] $$ and the gradient is $$ 1 + \frac{2 x^2}{(x^2 + y^2)^2} - \frac{1}{x^2 + y^2}, \frac{2 x y}{(x^2 + y^2)^2} $$ How was this obtained?

Hi all, I am trying to self-learn continuum mechanics, and I have a question regarding the development of the deformation gradient (which ultimately leads to green's deformation tensor). I have attached the specifics of the question in a attached photo. Ultimately, there comes a point...