Gradient Definition and 698 Threads

Homework Statement Find the gradient of 3r^2 in spherical coordinates, then do it in Cartesian coordinates Homework Equations \nabla f=\hat r \frac{\partial f}{\partial r} + \hat \theta \frac{1}{r} \frac{\partial f}{\partial \theta}+ \hat \phi \frac{1}{r\sin \theta}\frac{\partial...

Homework Statement Suppose you are climbing a hill whose shape is given by the equation below, where x, y, and z are measured in meters, and you are standing at a point with coordinates (120, 80, 1064). The positive x-axis points east and the positive y-axis points north. z = 1200 - 0.005x2...

This is doing my nut in. I'm looking at causes of errors in a rotating gradiometer. It uses a loop of superconductor formed so that the current in the loop is proportional to the gradient of the magnetic field threading the loop. I think that an error current will arise due to the Lorentz...

Hi, Say I have a sphere of radius r that has a constant surface temperature of T_s. The sphere is surrounded by air at a constant temperature T_amb. I am interested in the temperature gradient surrounding the sphere. From the little I know, I think i have to look at the natural...

There is an equation for the electric field E=V/d. This tells me the change in voltage per distance. Lets say I have a 1-meter wire and a 1-volt battery, so the electric field would be 1V/m. What is the significance of this in the circuit?

Homework Statement I have a derivation for an equation here: https://www.physicsforums.com/showthread.php?t=334692 Basically, I need to invert the gradient operator, so I have: \nabla B = k_z k is known and I want to solve for B numerically. How do I get rid of the gradient...

Homework Statement From Townsend "Modern Approach to Quantum Mechanics", problem 1.1: "Determine the field gradient of a 50-cm long Stern-Gerlach magnet 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=1500K...

Homework Statement Calculate the gradient of the scalar field f(x,y) = x^{2} - y^{2} . Sketch the gradient for a few point on two straight lines y = x and y = -x on the plane and comment on the properties of the sketch. Homework Equations The Attempt at a Solution So I worked...

Homework Statement Check the gradient theorem for the scalar field T= x^2 + 4xy + 2yz^3 and the paths a) (0,0,0) to (1,0,0) to (1,1,0) to (1,1,1) Homework Equations Equations = none well maybe divergence of a vector field= (df/dx)*x + (df/dy)*y + (df/dz)* z where x,y and z are...

Gradient and velocity Just curious Let's say I have a plane with the equation 4x + 5y + 6z = 45 If I find \nablaF(x,y,z) and then find it's magnitude, I get the direction of steepest descent/ascent in the direction of <\partialF(x,y,z)/\partialx,\partialF(x,y,z)/\partialy...

Okay, I have two points on my graph - (-2,72) and (0,64). Here is the question on my assignment - "Find another two points on your curve which have the same gradient as those in Parts 3 and 4 and find the equations of the tangents to the curve at these points." The problem I have is I don't...

Homework Statement Find the gradient of y with respect to x: y=\frac{3\sqrt{\theta^{2}+1}}{\frac{1}{2}cos(x^{2}+2\theta)} Well, I am at a complete loss where to start with this. The learning package I have in all its examples and text has no similar worded examples or utilises 2...

Hi all, just an very elementary question, arising from the first study of generating harmonic solutions. How to get the gradient twice for 1/r in spherical coordinates?

Can someone remind me why the electric field is defined as the negative gradient times the electric potential, rather than the gradient times the electric potential? Thanks, JL

Hello everyone, I have 2 components of a gradient, for example, the dz/dx and the dz/dy, I want to find the overall gradient it forms, how would I do that? Is it simply by combining the two gradients like this: overall gradient = ((dz/dx)^2 + (dz/dy)^2)^(1/2) I don't need the direction, I...

Hi All, A potassium channel is designed to filter out potassium ions at a maximum rate of 100,000,000 ions s-1 A common concentration (cell's interior) of potassium is 100 mM and 5 mM outside. This means there is 553 water molecules for 1 potassium ion inside (55.3/0.100) and 11,060 water...

Physical interpretation of gradient says that its a vector normal to equipotential (or level) surface \phi(x,y,z) = 0 but what about other surfaces, say the surface which are not equipotential? This is my first question. ok, now as grad \phi is a vector normal to surface it can't be 0...

The equation widely used to calculate the force on a projectile in an electromagnetic launcher, more specifically a railgun, is: F = 0.5 * L' * I^2 where: --> F is the force in Newtons --> L-prime is the inductance gradient of the rails in henries/meter (H/m) --> and I is the current...

Homework Statement Find the gradient vector of: g(r, \theta) = e^{-r} sin \theta Homework Equations The Attempt at a Solution I know how to get gradients for Cartesian - partially derive the equation of the surface wrt each variable. But I have no idea how to do it for...

Homework Statement The temperature in a room is given by T(x,y,z)=2x^2+3y^2 - 4z. A flying bug located at P(1,1,2) in the room desires to fly in such a direction that it will warm as soon as possible. In what direction must the bug fly? Taking the gradient of T: 4x(x hat)+6y(y hat)-4(z...

Homework Statement Lord Kelvin used the heat flow at the surface of the Earth to argue that the Earth was 100 million years old withing a factor of 4 error. a.) Reproduce his logic by deriving the temperature gradient at the surface of the Earth for a planet that is cooling by conduction...

grad(A*B)=(A*grad)B + (B*grad)A + A curl B + B curl A i'm not sure how to read the RHS to begin to work out the index definition.i'm thinking if add and subtract terms this will work out. i think i can see the first two terms, but the last two maybe "A cross nabla" is what they mean acting on...

In physics texts, its customary to write (and even to define the gradient as) the following: dT = (\nabla T) \cdot dl Working in Cartesian coordinates, we can expand this as follows: dT = \frac{\partial T}{\partial x} dx + \frac{\partial T}{\partial y} dy + \frac{\partial T}{\partial...

Homework Statement http://img245.imageshack.us/img245/2353/87006064.th.jpg I need to find the unit vector in the direction of \vec{F} at the point (1, 2, -2). Homework Equations The Attempt at a Solution well first of all I need to find what F is right, which is gradf.. how can I get...

Homework Statement h(sph)=exp(r2sin2(\theta)sin2(\phi)+r2cos2(\theta)) need to find gradient of this function, i have er and etheta... but can someone please tel me why when maple differentiates with respect to phi, why does it say it equals zero? coz i get...

Homework Statement Suppose that the function f: Rn --> R has first-order partial derivatives and that the point x in Rn is a local minimizer for f: Rn --> R, meaning that there is a positive number r such that f(x+h) > f(x) if dist(x,x+h) < r. Prove that Df(x)=0. Homework Equations...

hi all, do you know what is the gradient of a tensor looks like? I mean the del operator on a second order tensor, not the divergence of the tensor. And actually I need them in polar coordinates.. I have been searching so hard in web, but I can't find anything useful. Please help.

Homework Statement Calculate the gradient vector at the point S for the function, f(x,y,z)=x-\sqrt{z^2 - y^2}; S(x,y,z)=(4, 8, -6). 2. The attempt at a solution \frac{\partial f}{\partial x} = 1 \frac{\partial f}{\partial y} = \frac{y}{\sqrt{z^2-y^2}} \frac{\partial f}{\partial z} =...

hi. the is it true that if \vec{m} is a constant vector, then \nabla \cdot \vec{m}=0?

Wave speed changes only when medium changes. But so far, all I've seen is a definite boundary behavior where one medium abruptly ends and another one begins. What happens if there is a gradient. For example, what happens when a wave is passed through a rope with a density gradient. It is very...

Does anybody know, or know where to find, the expressions for the gradient and/or divergence in hyperspherical coordinates. Specifically, I'd like to know \nabla \cdot \hat{r} in dimensions higher than 3.

Homework Statement Let f(x,y,z)= |r|-n where r = x\hat{i} + y\hat{j} + z\hat{k} Show that \nabla f = -nr / |r|n+2 2. The attempt at a solution Ok, I don't care about the absolute value (yet at least). I take partial derivatives of (xi + yj + zk)^-n and get \nabla f =...

1. A rod carrying a uniform charge distribution is bent into a semi circle with the center on the orgin and a radius R. Calcualte the Electric field at the center of the semi circle using the electric potential expression found in part a 2. E = -(gradient)V 3. The electric...

Homework Statement The elevation of a mountain above sea level at (x,y) is 3000e^\frac{-x^2-2y^2}{100} meters. The positive x-axis points east and the positive y-axis points north. A climber is directly above (10,10). If the climber moves northwest, will she ascend or descend and at what...

Homework Statement Consider the function f (x,y). if you start at the point (4,5) and move to the point (5,6) . the directional derivative is 2. Starting at the point (4,5) and moving toward the point (6,6)gives a directional derivative of 3.Find grad f at the point (4,5) . Homework...

Homework Statement View the curve below as a contour of f(x,y). (y-x)^2 + 2 = xy - 3 Use gradf (2,3) to find a vector normal to the curve at (2,3). Homework Equations The Attempt at a Solution I am not sure how do I get the vector normal to the curve, is it using a cross...

Homework Statement First problem: Let f(x,y) = x-y and u = vi+wj. In which direction does the function decrease and increase the most? And what u (all of them) satisfies Duf = 0 Second problem: Let z = f(x,y), where x = 2s+3t and y = 3s-2t. Determine \partial{z^2}/\partial{s^2}...

The temperature in an auditorium is given by T = x2 + y2 - z. A mosquito located at (1,1,2) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. In what direction must it fly? I know that the gradient of T will point to the direction where the...

Homework Statement 1. Calculate the gradient of the curve y = 2x3 - 5x2 + 46x + 87 at the point where it crosses the x-axix. 2. Show by differentiation and solving a quadratic equation, that there are no points on the above curve where the gradient is zero.Homework Equations y = 2x3 - 5x2 +...

Define f: R^{2} \rightarrow R , by f(x,y) = \int^{sin(x sin(y sin z))}_{a} g(s) ds where g:R -> R is continuous. Find the gradient of f. I tried using the FTC, and differentiating under the integral, but did not get anywhere, thanks for any suggestions.

I am given some function f(x,y) and I am asked to find what the maximal rate of change is at some point (x0y0) and the direction in which it occurs. Is this correct: Maximal rate of change=|\nabla{f}(x_0,y_0)| And for the direction, if \nabla{f}(x_0,y_0)=<a\, ,b\,> then the direction is...

Earlier today, I came up with an explanation of why 0^0 is undefined in terms of properties of exponentiation. In it, I was treating exponentiation as a function from R^2 to R. Then, it occurred to me that the gradient of a function f(x,y) = x^y would be a horrible nightmare. Perhaps something...

I'm doing a selfstudy on relativistic electrodynamics and stumbled over a problem (which i find rather important) i can't solve. It's concerning problem 12.55 in Griffiths introduction to electrodynamics. One needs show that the four gradient: \frac{\partial}{\partial x ^\mu} functions as a...

Greetings, I'm having trouble deciding what to do, and in what order for this question: Suppose F = F( x, y, z ) is a gradient field with F = \nablaf, S is a level surface of f, and C is a curve on S. What is the value of the line integral (over C) of F.dr ? I think I'm a little confused...

Hello, I was messing around with subscript summation notation problems, and I ended up trying to determine a vector identity for the following expresion: \overline{\nabla}(\overline{A}\cdot\overline{B}) Here are my steps for as far as I got: \hat{e}_{i}\frac{\partial}{\partial...

Hi, May i know the physical meaning of the following: (1) Curl of a vector field A(x,y,z) (2) divergence of a vector field A(x,y,z) (3) directional deriative of G(x,y,z) (4) gradient of a scalar field G(x,y,z)

The length contraction in special relativity says that a rod moving along its axis will appear shorter by γ to a stationary observer. I think, however, not only the rod will appear shorter, but also each small segment of the rod will show its snapshot of different time as in the moving frame, in...

What is the difference between strain rate and velocity gradient of a Newtonian fluid?

Hi. Is it possible for two separate points on an equipotential surface to have two different values for the force field? eg, point A and point B lie on an equipotential surface, but the equipotential surface spacing is much denser at A than at B - so the force field at A as the gradient...

Does anyone know how to draw a gradient field? For example, how do you draw one of f(x,y)=xy^2