Gradient Definition and 723 Threads

[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?

Hi there. I have a doubt that I never cleared before, so I wanted your opinions on this. The thing is that when in vector calculus the gradient vector is presented, one of the "geometric" interpretation that is given is that it's a vector always perpendicular to the curve. So at first I've...

First of all sorry my english, I'm an italian student Homework Statement The velocity field is known ux=6x+8z*cos(a*t) uy=3xz*sin(b*t)+2 uz=-6z The fluid is incompressible. I need to find the pressure gradient and the i have to determine if it's a turbulent flows, but at the moment i need...

Suppose we have fluid in a vessel (Vessel A) with inside Pressure 120 bar (achieved by a pump)… When we open the valve, the fluid starts to flow into another vessel (vessel B) that was hitherto empty... Due to the Pressure gradient, the fluid flows with a certain velocity into vessel B. But the...

What does it mean to have a taylor expansion of a gradient (vector) about the position x? I.e. taylor expansion of g(x + d) where g is the gradient and d is the small neighborhood.

Homework Statement Let's say \vec{F} = <P,Q,R> If I take the gradient, shouldn't I get \nabla \vec{F} = <\frac{\partial P }{\partial x}, \frac{\partial Q}{\partial y}, \frac{\partial R}{\partial z}> Also why is grad(div f) meaningless? My book says it's because div(f) gives a scalar field...

Hi, could you please help me with my homework? I want to determine the height of mountain (from foot to peak) using the speed of sound. Homework Statement Known data: time delay, height1, temp1 plus known dependence between the height and temperature. What I want to determine: height2...

Homework Statement I am getting quite confused as to the concepts behind this task. I have a function given as a double integral, and am asked to find the gradient of it. However, I have no notes on how to do this, so it is either a very simple task, or the lecturer has once again missed...

i don't really understand the difference :( ∇2V versus ∇ (∇ . V) ? can anyone give me a simple example to showcase the application difference? thanks!

hi 1.i have a problem to find the princip to programm in MATLAB the method of conjugate graduate, in fact ,my broblem is: 1.i want to study caracterisque of charge RC WHICH function is :y=a*(1-exp(b*t)) ,a is supposed to be tension maximal and b =1/RC the problem is that i have to find the...

Homework Statement given grad f = xy i + 2xy j+0 k find f(x,y,z) how to generally solve questions of this type Homework Equations The Attempt at a Solution the ans is 0. don't know how.

Homework Statement You are at P(-1,1) on the surface z = (y-x^2)^3. What direction should you move from P so that your height remains the same? Homework Equations The Attempt at a Solution So I basically do not want my height z to change. In this case, I will take a vector...

Hi folks! I was wondering if anyone can help me with a problem I'm having with the concept of thermocouples. If I understand correctly, there should necessarily exist a temperature GRADIENT in one of the conductive couples in order for the emf to be generated. So how can you make sure...

Homework Statement Find delta f of f=Z^-1 * Sqrt(9x^2*y^2) at point (1,4,10) Homework Equations f =f+(fx*delta x )+(fydelta y)+(fz*delta z) The Attempt at a Solution fx = 9*2x/(z*(2*sqrt(9x^2*y^2)) =.18 plugging in (1,4,10) fy=2y/(z*(2*sqrt(9x^2*y^2)) =.08 fz=(sqrt(9x^2*y^2) )...

Hi, Suppose we have a function of two n-dimensional vectors f(\mathbf{x},\mathbf{y}). How can we find the gradient and Hessian of this function? Regards

Hopefully this is a simple enough question. Let (M,g) be a matrix Riemannian manifold and f: M \to \mathbb R a smooth function. Take p \in M and let \{ X_1,\ldots, X_n \} be a local orthonormal frame for a neighbourhood of p. We can define a gradient of f in a neighbourhood of p as \nabla...

Homework Statement Suppose F: Rn --> R has first order partial derivatives and that x in Rn is a local minimizer of F, that is, there exists an r>0 such that f(x+h) \geq f(x) if dist(x, x+h) < r. Prove that \nabla f(x)=0. Homework Equations We want to show that fxi(x) =0 for i = 1,...,n So...

Hello. I hope I've chosen the correct place to post this. Apologies if it is not. Could somebody explain the method of Gradient Descent to me or give me a link to a good explanation? For example, if h(x,y) = x^2 + y^2, what would I do to find a minimum point using gradient descent? I've...

Homework Statement Show that the operation of taking the gradient of a function has the given property. Assume that u and v are differentiable functions of x and y and that a, b are constants. Homework Equations Δ = gradient vector 1) Δ(u/v) = vΔu - uΔv / v^2 2) Δu^n = nu^(n-1)Δu...

Trying to prove that the gradient of a scalar field is symmetric(?) Struggling with the formatting here. Please see the linked image. Thanks. http://i.imgur.com/9ZelT.png

I have a problem with calculating the concentration gradiant. Here is the question and the solution from the solution manual and the numbers don't add up. A 1-mm sheet of FCC iron is used to contain nitrogen in a heat exchanger at 1200℃. The concentration of N at one surface is 0.04 atomic...

Homework Statement (Attached example on Combined extension and torsion of a solid cylinder) The Attempt at a Solution Using the Grad operator given on the given position vector x, I don't understand how to get the tor*r co-efficient on e_theta * e_z (So tor*r*lamda e_theta*E_Z in final...

For the Scan attachment: The question asks me to find the maximum acceleration. I used those two points in red in the attachment to calculate the gradient doing difference in y and difference in x. I got 4m/s ^2: does it seems correct, or I should have to tangent it? Because it seemed to be a...

Hi, I am curious if anyone here remembers the gradient operator by the following definition: \nabla f = \lim_{\Delta v->0} \frac{1}{\Delta v}\oint f \vec{dS}. So far I could find only one book that gives the definition above. I find this definition quite nice as the expressions of the...

Homework Statement I know that for a general co-ordinate system, the gradient can be expressed as it is at the bottom of this page: http://en.wikipedia.org/wiki/Orthogo...ree_dimensions However, the book I am working from (A First Course in Continuum Mechanics by Gonzalez and Stuart)...

I know that for a general co-ordinate system, the gradient can be expressed as it is at the bottom of this page: http://en.wikipedia.org/wiki/Orthogonal_coordinates#Differential_operators_in_three_dimensions However, the book I am working from (A First Course in Continuum Mechanics by Gonzalez...

Homework Statement hi, any help with proving that grad (v_ . r_) = v_ using spherical polars, where v_ is a uniform vector field would be great it is trivial to prove using summation convention or cartesian coordinates but having to use spherical polars looks messy... thanksHomework Equations...

dot product, and the gradient urgent pls!... Homework Statement Δ<-- this be the gradient and B<-- be a vector B X= xi +yj + zk *<---- be the dot product. (B*Δ)X=B Homework Equations n/a The Attempt at a Solution im not sure how to go about this but this is what i did i...

The adiabatic heat gradient is determined as \gamma = \frac{g}{c_{p}} where \gamma is the rate that temperature falls when rising in an atmosphere. g is gravitational acceleration and c_{p} is the heat apacity. On Earth it is 9.8 Kelvin per kilometer close to the surface of the Earth...

Can you help we are having an all weather football pitch installed and there is a question over the fall for drainage. it states that the fall should not exceed 1% over the total length. my question is what is the maximum fall at 1% if the pitch is 36 metres long please

Difficult gradient problem! Consider the curve with equation x2 + xy + y2 = 3. (a) Find in terms of k, the gradient of the curve at the point (−1, k). (b) Given that the tangent to the curve is parallel to the x-axis at this point, find the value of k.

Homework Statement Hi, this is problem 2.5 from "Atomic Many Body Theory" - Lindgren, Morrison An operator which transforms under a rotation in the same way as the vector \vec{r} (or any other vector) is called a vector operator. Show that the gradient operator \vec{\nabla} satisfies this...