Gradient Definition and 723 Threads

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 the Celsius temperature in a region in space is given by T(x,y,z)=2x^2-xyz. a particle is moving in this region and its position at time t is given by x=2t^2, y=3t, z=-t^2, where time is measured in seconds and distance in meters a) how fast is the temperature experienced...

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. 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 \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...

Homework Statement Suppose a rod measured in the S frame has a gradient of M. The S' frame travels at v (along x-axis) relative to S. What is the gradient of the rod in the S' frame? Homework Equations Lorentz tranformations The Attempt at a Solution y'=y x' =...

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...

Homework Statement Prove the following or disprove with a counterexample: Let f be a differentiable function in an open set U in R^3 and (a, b, c) be a point in U where the gradient of the function f isn't zero. If r: I -> U is a regular curve with a regular derivative on an open interval I...

Homework Statement http://www.physics.ox.ac.uk/olympiad/Downloads/PastPapers/BPhO_PC_2006_QP.pdf Question 11 Homework Equations h=v^2/g The Attempt at a Solution convert 36km/h into 10m/s 10 is final velocity so average should be 5m/s h=v^2/g=5^2/10=2.5m Using pythagoras...

Please check my answers. I tend to over-think and get simple questions wrong. A hypothetical cell membrane is positively charged on the intracellular side and negatively charged on the extracellular side. In this cell, the concentration of ion X+ in the intracellular space is high and in the...

Homework Statement My textbook never explains well so I have to figure out how to do problems by reverse engineering using the solution manual. However, here is one operation that I simply cannot reverse engineer. I do not see a common pattern in these four problems. I can't figure out what...

On a quiz, a true/false statement was given along the lines of: "The gradient is a specific example of a directional derivative." I marked "true" and got it wrong. I see why, I think, since the gradient is an actual "guide," a vector, towards the max rate of change, while the directional...

Is it possible to nontrivially represent the cross product of a vector field \vec{f}(x,y,z) with its conjugate as the gradient of some scalar field \phi(x,y,z)? In other words, can the PDE \vec{\nabla}\phi(x,y,z) = \vec{f}(x,y,z)\times\vec{f}^\ast(x,y,z) be nontrivially (no constant...

My textbook (Taylor, Classical Mechanics) and professor introduced the concept of \nabla_{1} to mean "the gradient of the function (potential energy) with respect to the position (x_{1},y_{1},z_{1}) of particle 1. I do not understand this. I am familiar with partial derivatives and...

Hi folks, I have a basic question I would like to ask. I ll start from the Euclidean analogue to try to explain what I want. Suppose we have a smooth function (real valued scalar field) F(x,y)=x^2+y^2, with x,y \in ℝ. We also have the gradient \nabla F=\left( \frac{\partial F}{\partial...

Homework Statement f is a function of two variables: y, z. I want to show that the gradient: \nabla f=\frac{\partial f}{\partial y}\hat y + \frac{\partial f}{\partial z}\hat z Transforms as a vector under rotation of axes. Homework Equations The rotation of axes: A...

Hello , i would like to know how do you calculate the error in the gradient of a graph when all the points fall on the line or is so close to the line to draw the maximum and minimum slope and using it in the general formula is not applicable. error in gradient = ±(max.slope- min slope) /2√N...

Homework Statement Consider the scalar field V = r^n , n ≠ 0 expressed in spherical coordinates. Find it's gradient \nabla V in a.) cartesian coordinates b.) spherical coordinates Homework Equations cartesian version: \nabla V = \frac{\partial V}{\partial x}\hat{x} +...

Hi there, To find the gradient of a curve, we draw a chord on the curve and then makes the 2nd coordinates ( B ) tends to A ( 1st coordinates ). To find the gradient of the chord, i.e, ΔY/ΔX, we replace the two coordinates into the equation of the curve. But my question is why do we...

Homework Statement For the surface z=2x^2+3y^2, find (i) the gradient at the point P (2,1,11) in the direction making an angle a with the x-axis; (ii) the maximum gradient at P and the value of a for which it occurs. Homework Equations...

Hi, Suppose a copper block is heated on one side so that one end is at 800K. Given the dimensions of the copper block, is there a way of calculating the temperature of a point in the block distance x from the heated end after a given time? With many thanks, Froskoy.

I'm reading about gas chromatography at the moment and the notes I'm reading mentioned a "generic scouting gradient" but didn't explain what it is. I've been googling it and found a few HPLC tutorials (in GC its temperature gradient whereas in the HPLC tutorials they're talking about mobile...

(1) Let f(x)=x^3+y^3+z^3-3xyz, Find grad(f). grad(f)=(3x^2-3yz, 3y^2-3xz, 3z^2-3xy). (2) Identify the points at which grad(f) is a) orthogonal to the z-axis b) parallel to the x-axis c) zero.I have managed to solve for (1), but don't have a clue how to solve for the second part. I have not...

Homework Statement Show that the gradient of the curve \frac{a}{x}+\frac{b}{y}=1 is -\frac{ay^2}{bx^2}. The point (p,q) lies on both the straight line [itex]ax+by=1[/tex] and \frac{a}{x}+\frac{b}{y}=1 where ab =/= 0. Given that, at this point, the line and the curve have the same gradient...

Hello, My question is concerning how to compute the complex gradient of the following cost functional with respect to W: F=Ʃ_i=1:M ||y_i-Go*W_i||^2 + Ʃ_i=1:M ||W_i - X*(E_i - Gc*W_i)||^2 Where the summations go from i=1 to i=M and the dimensions of the diferent elements are: y: Nx1 Go: Nxn W...

Why is the function decreasing the fastest in the direction of the negative of the gradient? Just because it increases the fastest in the direction of the positive of the gradient why does this have to mean it has to decrease the fstest in the negative of the gradient? If you stand facing a...

Hi there - I'm looking for a clear and intuitive explanation of how one obtains the gradient in polar / cylindrical / curvilinear coords. I do a lot of tutoring, but am finding that the method I've been using (basically chain rule + nature of directional derivative) just doesn't roll with...

Homework Statement I have an issue with Straight Line graphs, I have never done them before (I touched on them in Seconday School, y=mx+c that sort of stuff) Now I've been faced with a problem that I need to learn. It's not homework it's revision but I thought it was more relevant to post here...

Anyone know how to use the temperature gradient in a thick-walled tube to calculate the stress seen throughout the wall (radial stress gradient)? I've been scouring the internet for a good explanation but haven't found one.

Homework Statement What is the direction and rate of maximum increase? f(x,y) = x^2 + y^3, v = <4,3>, P = (1,2) Homework Equations The Attempt at a Solution The direction should be as same as the gradient so < 2,12> The rate of maximum increase is magnitude of the gradient so...

As we know grad F (F surface) is in normal direction. But we also have (grad F(r)) x r = F'(r) (r) x r = 0 this implies grad F is in direction of r i.e., radial direction. Radial and normal directions need not be same. Can any öne clarify THE DIRECTION OF GRAD VECTOR? Thanks,

As we know grad F (F surface) is in normal direction. But we also have (grad F(r)) x r = F'(r) (r) x r = 0 this implies grad F is in direction of r i.e., radial direction. Radial and normal directions need not be same. Can any öne clarify THE DIRECTION OF GRAD VECTOR?

Homework Statement f(x)= 3+6x-2x^3 (a) Determine the values of x for which the graph of f has positive gradient (b) Find the values of x for which the graph of f has increasing gradient Homework Equations I had originally thought the two terms meant the same thing, but when I checked the...

Can someone help me intuitively understand why if a field has zero curl then it must be the gradient of a scalar potential? Thanks!

Homework Statement Hello all, I encountered this practice problem for my midterm tomorrow involving the gradient operation. Let (r, θ) denote the polar coordinates and (x, y) denote the cartesian coordinates of a point P in the plane. A function is defined via f(P)=xsinθ away from the origin...

Homework Statement I've been given the task of analysing data from an experiment where an object was dropped with an initial velocity of 0.I've calculated the time in s2 from the original milisec. Distance Time s2 0.1 0 0.2 0.040804 0.4...

For a solenoidal velocity field [ tex ] \nabla \cdot \mathbf{u} [ /tex ] which means that [ tex ] \nabla [/tex ] is perpendicular to [ tex ] \mathbf{u} [ /tex ]. Similarly, for an irrotational velocity field [ tex ] \nabla \times \mathbf{u} [ /tex ] which means that [ tex ] \nabla [/tex ] is...

The path of a particle is given by r(t) = tsin(t) * i - tcos(t) * j where t≥0. The particle leaves the origin at t = 0 and then spirals outwards. Let θ be the acute angle at which the path of the particle crosses the x-axis. Find tan(θ) when t = 3pi/2. I was able to figure out a...

Homework Statement Find the set of points where the gradient of f is parallel to u =(1,1) for f(x,y)=x2 + y3 + 2xyHomework Equations the gradient of f(x,y)=((partial derivative of f wrt x), (partial derivative of f wrt y)) u=grad f The Attempt at a Solution fx = 2x+2y fy = 3y+2x 1=2x+2y...

Assuming that the accretion disk has been totally consumed by the black hole, does the temperature of the black hole due to Hawking radiation vary with respect with proximity with the black hole? For example, if I were next to the black hole, would this radiation would have a higher temperature...

Homework Statement "A gradient of a vector field is symmetric if and only if this vector field is a gradient of a function" Pure Strain Deformations of Surfaces Marek L. Szwabowicz J Elasticity (2008) 92:255–275 DOI 10.1007/s10659-008-9161-5 f=5x^3+3xy-15y^3 So the gradient of this function...