Gradient Definition and 698 Threads

Homework Statement Find an equation for the tangent plane to a surface xz^2 +x^2y-z=-1 at the point (1,-3,2). Homework Equations (\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0 The Attempt at a Solution [/B] First I found the gradient of the function \nabla f = (z^2+2xy)\hat{i} + x^2...

I've come across two different approaches to quantifying what l is in the equation for hydraulic gradient Δh/L. In this first picture L is the parallel distance along the datum across the reference plane But in this second picture L is the length along the pipe Why are the two L's...

[Note from mentor: This was originally posted in a non-homework forum so it doesn't have the homework template.] ----------------------------------------- Problem: The surface of a solid metal sphere (radius r = 4.58 cm) is at potential V = 9,851 Volts. Find the magnitude of the potential...

Homework Statement Find the action of the operator ## e^{\vec{a} . \vec{\nabla}} \big( f(\theta,\phi) . g(r) \big) ## where \nabla is the gradient operator given in spherical coordinates, f and g are respectively scalar functions of the angular part ## ( \theta, \phi) ## and the radial part ##...

Hello, new to this website, but one question that's been killing me is how can curl of a gradient of a scalar field be null vector when mixed partial derivatives are not always equal?? consider Φ(x,y,z) a scalar function consider the determinant [(i,j,k),(∂/∂x,∂/∂y,∂/∂z),(∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z)]...

So we all know the divergence/Gauss's theorem as ∫ (\vec∇ ⋅ \vec v) dV = ∫\vec v \cdot d\vec S Now I've come across something labeled as Gauss's theorem: \int (\vec\nabla p)dV = \oint p d\vec S where p is a scalar function. I was wondering if I could go about proving it in the following way...

Let's say I have point A(2, 6, 0) and B(3, -1, -2) and wanted to find the gradient of the vector joining these two points. I know how to find the vector representing the line joining these points: [FONT=Times New Roman]OA = 2i + 6j , OB = 3i - j - 2k AB = AO + OB [FONT=Times New...

Hello! I had a username on here from a few years ago (the site looked different then), but I can't remember the email or password, so I humbly return to the community as a noob. I am working on a short science fiction novel, and I could use a little informed insight. The planet on which the...

Sorry again for all these ongoing question as I try to fix my math deficiencies. (Back to working on differential forms.) So... I understand that the equation of steepest ascent/descent in Cartesian coordinates is written: dxi/dt = ∂f/∂xi And I can follow the "physical interpretations" of...

How can show that momentum is the gradient of the action for the free particle? I tried it like this for one dimensional case: s=\int Ldt ds=Ldt ds=\frac{mv^2}{2}dt\: Velocity is constant right? So I should be able to to this: \frac{ds}{dx}=\frac{mv^2}{2}\frac{dt}{dx} I'm not sure about...

I have done part A (i think) really not sure where to begin with the rest of the parts, would appreciate a tip in the right direction, its revision for my first year physics exams in a few weeks. Consider the funtion T in the plane (x,y), given by T=ln(x^2 + y^2) at point 1,2 a) in which...

I learned gradient in 3D space. And gradients where always vectors, pointing in the direction of steepest ... and normal to the surface where the functions is constant. But reading one-forms , a gradient of a function is not always a vector and it has something to do with metric... Can you proof...

Homework Statement Let f(x,y)=arctan(x/y) and u={(√2)/2,(√2)/2} d.) Verify that ∇fp is orthogonal to the level curve through P for P=(x,y)≠(0,0) where y=mx for m≠0 are level curves for f. Homework Equations The Attempt at a Solution ∇f={(y)/(x^2+y^2),(-y)/(x^2+y^2)} m=1/tan(k) where...

Homework Statement ##x=r\sin\theta\cos\phi,\,\,\,\,\,y=r\sin\theta\sin\phi,\,\,\,\,\,z=r\cos\theta## ##\hat{x}=\sin\theta\cos\phi\,\hat{r}+\cos\theta\cos\phi\,\hat{\theta}-\sin\phi\,\hat{\phi}## ##\hat{y}=\sin\theta\sin\phi\,\hat{r}+\cos\theta\sin\phi\,\hat{\theta}+\cos\phi\,\hat{\phi}##...

Okay I'm having a little trouble understanding a section of this proof about the product of the gradients of perpendicular lines given in my textbook. I'm going to type the proof out but there will be a link at the bottom to an online version of the textbook so you can see the accompanying...

Homework Statement There are three subquestions in this question, all marked bold. Let's consider a gradient index lens of thickness ##d##, whose refractive index changes with the distance from the axis with the following formula $$ n(r) = n_1 + a r^2 $$. Determine the lens's focal length...

To solve the gradient f when f = ln |r| do I start with differentiating each x,y,z term of the vector?Like ln|x| ln|y|...etc.

Homework Statement Let us consider three scalar fields u(x), v(x), and w(x). Show that they have a relationship such that f(u, v, w) = 0 if and only if (∇u) × (∇v) · (∇w) = 0. Homework EquationsThe Attempt at a Solution I could think nothing but...

Hi, I've been thinking about the Navier-Stokes equations and trying to build skill in implementing it in various situations. In a particular situation, if I have a fluid flowing down an inclined surface such that it forms a film of finite height which is smaller than the length of flow, there...

Hi, I am trying to calculate the laplacian of a scalar field but I might actually need something else. So basically I am applying reaction diffusion on a 2d image. I am reading the neighbours, multiplying them with these weights and then add them. This works great. I don't know if what I am...

Hi! Two exerts from my lecture notes: "Assume we have a system of point masses in position ##\vec{r_i}## which are influenced by forces ##\vec{F_i}##." "Let's say you have a system where ##\vec{F_i} = - \nabla_i V##" In the second line, what does the notation ##\nabla_i## mean? Why is that...

1. Given a function f(x,y) at (x0,y0). Find the two angles the directional derivative makes with the x-axis, where the directional derivative is 1. The angles lie in (-pi,pi]. 2. f(x,y) = sec(pi/14)*sqrt(x^2 + y^2) p0 = (6,6) 3. I use the relation D_u = grad(f) * u, where u is the...

I'm trying to re-derive a result in a paper that I'm struggling with. Here is the problem: I wish to calculate (\nabla \otimes \nabla) h where \nabla is defined as \nabla = \frac{\partial}{\partial r} \hat{\mathbf{r}}+ \frac{1}{r} \frac{\partial}{\partial \psi} \hat{\boldsymbol{\psi}} and...

Homework Statement I need to find Λ using the equation below (I think). Homework Equations A [/B]+ ∇Λ = 0 where A(x,y,z,t) = B\begin{pmatrix} x+y\\ x-y\\ 0 \end{pmatrix} The Attempt at a Solution Is this at all possible?

Hello all, I know 3d gradients are often represented by gradient vectors but in the current project I am working on it would be a lot more convenient for me to do it this way if possible, the X gradient is given by Δy/Δx, and the Z gradient by Δy/Δz how can one obtain the Y gradient?

Homework Statement [/B] Solving part (c) which should be \overrightarrow{r}.(\nabla.\overrightarrow{r)}\neq\left(r\nabla\right)r 2. Homework Equations Let \nabla=\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z} and...

So, I have a car I'm tuning using a road dyno software. This software basically uses the gear ratio of the car, it's total weight, and the speed it moves the RPM to determine it's power output (HP and Torque) In order to use this software a flat road is needed. That works well and is very...

Homework Statement Homework Equations The Attempt at a Solution (a)[/B] Divergence of a gradient is a Laplacian. (b) I suppose to do it in Cartesian coordinates. Let \nabla=\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z} and...

So I learned recently that pressure gradient in the flow direction for flow over a flat plate is zero. However I don't understand this, because there has to be something that sets the flow in motion in the first place, and for fluids this has to be a pressure gradient. Could someone explain why...

Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through...

Homework Statement I have been asked to show a graphical, more accurate method to calculate power used when running up a flight of stairs. The method I have used previously is measuring the height of the stairs, recording my weight in Newtons, timing how long it takes to run from the bottom to...

What is "Negative Gradient" ? and what is "Gradient Descent Method" ? What is the difference and relationship between them ? What is the benefit each of them ?

We know that a charged particle will have a drift velocity in both a curved magnetic field and when there is a transverse spatial gradient in the magnitude of the magnetic field. This drift velocity is added to the rotation velocity around the the field line. In both cases the force vector on...

Homework Statement Calculate \nabla e^{i\vec{k}\cdot \vec{r}} Homework Equations \nabla f(r)=\frac{df}{dr}\nabla r=\frac{df}{dr}\frac{\vec{r}}{r} The Attempt at a Solution I have a problem. I know result =\nabla e^{i\vec{k}\cdot \vec{r}}=i\vec{k} e^{i\vec{k}\cdot \vec{r}}

Hi guys, I'm new to this site so forgive me if I'm unfamiliar with any etiquette unique to here. We did a practical exercise the other day in class in which we accelerated a bogey to a certain RPM and then released it to accelerate up a ramp. We were then given a sheet of the measurements I have...

If the unit vector u makes an angle theta with the positive x-axis then we can write u = <cos theta, sin theta> Duf(x, y) = fx(x,y) cos theta + fy(x,y) sin theta What if I am dealing with a function with three variables (x, y, z)? How can I find the directional derivative if I have been given...

If a question asks for the direction of the maximum gradient of a scalar field, is it acceptable to just use del(x) as the answer or is the question asking for a unit vector? Thanks

Homework Statement Show that (Grad dot E) =0 in the dielectric of a coaxial line. (Hint: apply the divergence theorem to a portion of the dielectric.) Homework Equations Divergence theorem The Attempt at a Solution I think I need to show that grad dot E = p/epsilon = 0 I don't know - I'm...

I've been asked to investigate the value of g. My graph shows that the gradient is around 4.4 as when the length was 0.27m, the squared time period was 1.18s (since this is a T2 against L graph). My question is, of course, why? If I am detecting g, which is 9.81 on earth, why did I get such...

Homework Statement http://i.imgur.com/TlDOllQ.png Homework Equations As stated. The Attempt at a Solution [/B] I'm not sure how to slay this beast. I know the gradient is just a partial derivative and that the solution likely involves multiple partial derivatives, one for each element in the...

I'm currently going over some mechanics notes and am confused about the following situation: In the book I'm looking at, it describes two particles absent of external forces, only exerting a force on each other. In deriving a potential energy equation for the two, it goes on to say that if the...

Does this operator (in 3D): ε_{ijk}∇_k = \begin{pmatrix} 0 & \frac{\partial}{\partial z} & -\frac{\partial}{\partial y}\\ -\frac{\partial}{\partial z} & 0 & \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y} & -\frac{\partial}{\partial x} & 0 \end{pmatrix} have a formal name and a more...

Homework Statement gradient(1/r) = r(hat) / r^2Homework Equations r = (x-x')i + (y-y')j + (z-z')k [Mentor Note -- Poster was reminded to always show effort on schoolwork questions]

Hi there. I was following a deduction on continuum mechanics for the invariant nature of the first two laws of thermodynamics. The thing is that this deduction works with an identity, and there is something I'm missing to get it. I have the vector product: ##\vec \omega \times grad \theta##...

http://www.math.uiuc.edu/documenta/vol-ismp/40_lemarechal-claude.pdf I don't understand why we use theta for equation (1) Θ>0 but why α=-θX? Thanks.

This is a bit counterintuitive to me that the gradient vector is always normal to the level curve and the level surface. lets say we have a function f(x,y)=z then the gradient is, f(x,y) partial derivative with respect to x*i +f(x,y) partial derivative with respect to y*j what we...

does negative divergence of gradient tempearature gives to lalace equation...? -div(∇T) = [∂^2T/∂x^2+∂^2T/∂y^2]

Homework Statement ##\nabla{F} = <2xyze^{x^2},ze^{x^2},ye^{x^2}## if f(0,0,0) = 5 find f(1,1,2)Homework Equations The Attempt at a Solution my book doesn't have a good example of a problem like this, am I looking for a potential? ##<\frac{\partial}{\partial x},\frac{\partial}{\partial...

In need of help. Ill write it is how it is in the textbook. Find the equation of the line having gradient 3/4, that passes through 7,11. Express your answer in the form i) ax + by + c =0 and ii) y = mx + c As one point and the gradient are known, use the formula: y - y1 = m(x-x1)...

Why does "per" in 3 miles per hour mean division? Why are gradients and rates a ratio?