Gradient Definition and 698 Threads

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...

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...

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...