Gradient Definition and 698 Threads

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...

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...

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...

[b]1. For two differentiable vector functions E and H, prove that (Delta (dot) (e X h) = H (dot) (delta X e) - e (dot) (Delta X h) [b]2. Cross product and dot product. The Attempt at a Solution First I took did the left side of the equation, I took the cross product of vectors e and...

Hello all, I understand that the taylor expansion for a multidimensional function can be written as f(\overline{X} + \overline{P}) = f(\overline{X}) + \nabla f(\overline{X}+t\overline{P})(\overline{P}) where t is on (0,1). Although I haven't seen that form before, it makes sense...

In functions involving only two variables the gradient is supposed to be the instantaneous rate of change of one variable with respect to the other and this is usually TANGENT to the curve. So then why is the gradient NORMAL to the curve at that point, since it is supposed to represent the...

Homework Statement If z = f(x,y) such that x = r + t and y = e^{rt}, then determine \nabla f(r,t) Homework Equations \nabla f(x,y) = <f_x,f_y> The Attempt at a Solution Now if i follow this the way i think it should be done then i find the partials of f wrt x and y and then...

Hello, thanks for reading! I am slightly confused. According to the definition of the directional derivative, calculated at the point x in the direction y, f'(x;y) = lim\frac{f(\vec{x} + h\vec{y})-f(\vec{x})}{h}, h-->0 According to this definition, the directional derivative seems to not...

Homework Statement 1. for the surface z=9 - x^2 - y^2 find, i) the gradient at (1,1,7) in the direction making an angle alpha with the x-axis ii) the max gradient at the point (1,1,7) and the value of alpha for which it occurs 2. find the stationary point of z=x^2 +2x +3y^2 -3xy + 5...

Homework Statement there is a surface xy3z2=4. What is the unit normal to this surface at a pt in the surface (-1,-1,2)?? Homework Equations what is a unit normal to a scalar region? how can it be calculated? The Attempt at a Solution i calculated the gradient (del operator) of...

Hi z(x,y,t)=a sin(ωt) sin(k/Lx*pi*x) sin(l/Ly*pi*y) a = Amplitude ω = Frequency k and l are constants Lx = Length in x direction Ly = Length in y direction How can I find [using an equation] the slope of the surface [ie the gradient] at any given point on the surface? I know...

Hi all, I have been struggling (really) with this and hope someone can help me out. I would just like to compute the gradient of a tensor in cylindrical coordinates. I thought I got the right way to calculate and successfully computed several terms and check against the results given by...

Homework Statement Homework Equations The Attempt at a Solution I saw from a book this is a quick way to get gradient in different cooridinates. However what f, g and h are? And how do I know that in rectangular coor. f=g=h=1 and etc?

A gradient vector points out of a graph (or a surface in 3D case). Locally, it makes an angle of 90 degrees with the graph at a particular point. Why is that so? Thanks.

suppose g(r) is a scalar function which is constant inside the volume 'v' but discontinuous at the boundaries of 'v'. The magnitude of discontinuity is given by constant 'M' then can we write the following expression \int\nablag(r)dv=M\int\hat{n}\delta(r-rs)dv=M\hat{n}\intd\deltav where...

I'm working at a university to build a low-energy electron detector and it requires that we construct a magnetic field gradient. We know, through computer models, what the gradient should be, but we don't know how to make it. We would rather use rare-earth magnets as opposed to an...

Homework Statement Prove this equation Homework Equations The Attempt at a Solution I almost get the answer. But I don't know why all of the sin and cos are in reciprocal form.

Hi, this might be a stupid question, but I was wondering how to computer the gradient of a quantity on a grid. I mean I have a grid made of cells (not necessarily of the same size) where the variable \rho is defined at the center of every cell. How can I compute the gradient of this quantity? It...

Say, we have potential energy of the form U = cos (\theta(t)) H(t). H denotes a magnetic field that is time-dependent and it's an input variable to the system. Now when you take gradient of potential energy, would you write \nabla U = \left[ - sin (\theta(t)) H(t) + cos (\theta(t))...

Hi, I was wondering if it is possible to adapt the conjugate gradient method (or if there's a variation of the method) for nonsymmetrical boundary value problems. For example, I want to solve something like a 2D square grid, where f(x)=0 for all x on the boundary of the square...

Hi, Just a simple, quick question: Does the gradient of a vector give a scalor or a vector? Thanks!

Why \int_a^b \nabla T\; d\vec l \;=\; T(b)-T(a) Why integration of a gradient is always path independent?