Nabla Definition and 47 Threads

We know that the magnetic field can be written in the following way: $$\nabla_{\vec r}\times \vec B(\vec r) =\frac 1 c \nabla_{\vec r} \times\int d^3\vec r_q\ \vec j(\vec r_q) \times \frac {\vec r-\vec r_q}{|\vec r-\vec r_q|^3}$$ and, using the ##BAC-CAB## identity, the curl of this...

A question in advance: How do I format equations correctly? Let's say $$\mathbf{k}\cdot\nabla\times(a\cdot\mathbf{w}\frac{\partial\,\mathbf{v}}{\partial\,z})$$ - a is a scalar Can I rewrite the expression such that...

Using the formula in 'relevant equations' I calculate $$div(fA) = \nabla(fA) = (\nabla f) \cdot A + f \nabla \cdot A$$ $$3r^2 \cdot (x^2, y^2, z^2) + r^3 \cdot (2x + 2y + 2z)$$But the answer is $$3r \cdot (x^3 + y^3 + z^3) + r^3 \cdot (2x + 2y + 2z)$$ I find no way of easily turning ##3r^2...

Hi! The topic is electrodynamic but it's a question about Nabla identity. Given $$ F = (p \cdot \nabla)E $$ How does one compute F? Is this correct? $$ F = \sum_{i} p_i \partial_{i} E_{i} e_{i} $$

My attempt is below. Could somebody please check if everything is correct? Thanks in advance!

Here is how my teacher solved this: I understand what the nabla operator does, ##∇\cdot\vec v## means that I am supposed to calculate ##\sum_{n=1}^3\frac {d\vec v} {dx_n}## where ##x_n## are cylindrical coordinates and ##\vec e_3 = \vec e_z##. I understand why ##∇\cdot\vec v = 0##, I would get...

in the language of general relativity,we know that we can write $$\nabla_{V}W $$ in this form such that: $$\nabla_{V}W = = w^i d ( V^j e_j)/du^i = w^j e^i (V^j e_j ) = W( V)$$ where $$w^i * d/ (du^i) =W$$ will act on the vector V where $$W = w^i d( ) /du^i $$ and W is a vector as a...

Is the following true? ψ*∇^2 ψ = ∇ψ*⋅∇ψ It seems like it should be since you can change the direction of operators.

I have a simple question about the notation of the nabla operator in Vector Analysis. The nabla operator is a vector differential operator and it is written as: $$\nabla = \hat{x} \frac {∂} {∂x} + \hat{y} \frac {∂} {∂y} + \hat{z} \frac {∂} {∂z}$$ Is it okay if we accented nabla by a right...

In my EM class, this vector identity for the angular momentum operator (without the ##i##) was stated without proof. Is there anywhere I can look to to actually find a good example/proof on how this works? This is in spherical coordinates, and I can't seem to find this vector identity anywhere...

Hi, in some books the ##\nabla## symbol is used for the Laplacian ##\frac{d}{dx^2}+\frac{d}{dy^2}+\frac{d}{dz^2}## while others use the ##\Delta## symbol for this. What is the correct custom for this usage?

I have the number 1 next to Nabula and I do not know how to solve it. For reference, I am a Korean person, so it would be very difficult if you explain it difficultly.

Homework Statement Can I, for all purposes, say that Nabla, on index notation, is $$\partial_i e_i$$ and treat it like a vector when calculating curl, divergence or gradient? For example, saying that $$\nabla \times \vec{V} = \partial_i \hat{e}_i \times V_j \hat{e}_j = \partial_i V_j (\hat{e}_i...

Homework Statement $$\bar{v}=\nabla \times \psi \hat{k}$$ The problem is much bigger, i know how a rotor or curl is calculated in cylindrical coordinates, but I'm just asking to see what would be the "determinant" rule for this specific curl. Homework Equations $$\psi$$ is in cylindrical...

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}##...

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 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}}

Does anybody knows how you can reach one form of the divergence formula from the other? Or in general, why is the equivalence true?

Hi. In this development (c ∇+ d A)(c ∇+dA)= c^{2} ∇^{2} + d^{2}A^{2} + cd A∇ + cd ∇A (c ∇+ d A)^{2}= c^{2} ∇^{2} + d^{2}A^{2} + cd A∇+ cd A∇+ cd (∇A) I feel like we have "two" different ∇ operators. At the end of the first line ∇ acts on A and the test function (not shown). At the...

Homework Statement Find ##\alpha ## and ##p## so that ##\nabla \times \vec{A}=0## and ##\nabla \cdot \vec{A}=0##, where in ##\vec{A}=r^{-p}[\vec{n}(\vec{n}\vec{r})-\alpha n^2\vec{r}]## vector ##\vec{n}## is constant. Homework Equations The Attempt at a Solution ##\nabla \times...

Homework Statement So I have this rather komplex example and I am looking for help. ∇(3(r*a)r)/R5 -a/R5) r=xex+yey+zez a-constant vector R=r1/2 Homework Equations The Attempt at a Solution So the nabla " works" on every member individualy,and i have to careful here:(r*∇a),because...

While using the ∇ operator, most of the times we can treat it as a vector. I came across a few formulae(basically product rules).. ∇×(A×B)=(B.∇)A-(A.∇)B+A(∇.B)-B(∇.A) where A and B are vectors I wanted to know if there is any direct way of deriving it. By direct I mean assuming the basic...

I'm currently working out the Schrödinger equation for a proton in a constant magnetic field for a research project, and while computing the Hamiltonian I came across this expression: (\vec{A}\cdot\nabla)\Psi where \Psi is a scalar function of r, theta, and phi. How do you evaluate this...

1. The problem statement I'm trying to show that the magnetic force is only conservative if dB/dt=0 Homework Equations F=q[E+(v\timesB)] Conservative if ∇\timesF=0 ∇\times(A\timesB)=A(∇\cdotB)-B(∇\cdotA)+(B\cdot∇)A-(A\cdot∇)B Maxwells equation: ∇\timesE=-∂B/∂t The Attempt at a Solution...

Reading through my electrodynamics textbook, I frequently get confused with the use of the del (nabla) operator. There is a whole list of vector identities with the del operator, but in some specific cases I cannot figure out what how the operation is exactly defined. Most of the problems...

I didn't get the concept of dual or hybrid nature of nabla? I-e vector differential operator .. Is it means that nabla can produce a vector from scalar field (gradient) and scalar from vector field(divergence) ? What's the concept of Nabla's Dual nature ? Please explain..

Homework Statement I need to find \nablaf(r). I am given r = |R| where R is a vector, R =(x,y,z). I also have the function f(r) which is a differentiable function of r. Homework Equations So i know \nabla(g) = (\partialg/\partialx, \partialg/\partialy, \partialg/\partialz) The...

In my notes it says that grad F will give you a vector normal to the contour. Howver I thought grad F would give you a vector tangent because the path is aligned with the vector field. Is it different when talking about contours and paths? If you find grad F of a function F does that give...

Homework Statement Prove that: \nablaX(\nablaXa) = \nabla(\nabla\cdota) - \nabla^{2}a where a is a vector point function. (X is the cross product and that dot is a dot product.) Homework Equations curl, grad, div The Attempt at a Solution I have just done another question of the form...

Hello Everyone, I have a small question bothering me. I wan to know whether the nabla operator has a unit? I am thinking it does and it should be 1/m. I just want to make sure whether this is true. Thanks! Jimmy

Is it \nabla_\mu\nabla_\nu A^\alpha={A^\alpha}_{;\mu\nu} or \nabla_\mu\nabla_\nu A^\alpha={A^\alpha}_{;\nu\mu} ?

A simple question: In a homework I find : F1 X nabla X F2 where X is the simbol of cross product I know that AX(BXC)= (A*C)*B-(A*B)C Where* here is used to divergence In the next step it was: -Nabla*(F2)F1 + nabla(F1*F2) I don't understant it, why?

nabla dot B =0 ?? I've read the physical explanation for this eq is that magnetic monopoles do not exist. A poor explanation in my opinion. :) So, I would like it explained along these lines. (Obviously I don't unuderstand this but am giving an example of how I would like it explained)...

Does my solution look correct to you guys? Homework Statement Calculate: \nabla \varphi (r) If: \varphi (r) = \frac{1}{4\pi\epsilon_{0}}\frac{1}{r} with: r = \sqrt{x^{2}+y^{2}+z^{2}} Homework Equations n/a The Attempt at a Solution

Hi all! I am studying the Galilean group of transformations and I'm not sure how to transform the Nabla operator. Consider the 2 transformations: (x,t)->(x+s,t) (x,t)->(Dx,t) and the expression "nabla (x)" where D is a matrix and x, s are vectors I am pretty sure that I have...

What would be the result of: C \nabla as C is any Constant ? Note: i don't mean: \nabla C as this is known, but i mean: C \nabla

Does \vec{\nabla} \cdot \vec{E} = 0 imply \vec{\nabla}^2 \cdot \vec{E} = 0 ? Is this true: \vec{\nabla}^2 \cdot \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E})

Hello everyone. I'm trying to get my head around this product rule: \nabla \times (A\times B) = (B\cdot \nabla )A - (A\cdot \nabla )B + A(\nabla \cdot B) - B(\nabla \cdot A) Ok, we have this \nabla = (\partial /\partial x,\partial/\partial y,\partial /\partial z) and for dot...

[SOLVED] Divergence, nabla Homework Statement Given the vector, find the dot product. Homework Equations dot product of nabla and the vector is just partial derivative of each component. The Attempt at a Solution I'm trying to figure out if I can just leave out the...

Could some one explain what does Nabla operator actually signify ? I understand that the various products with nabla are used to find curl,divergence,gradient in EM, but what does Nabla represent in itself ? A more basic question would be, what does del operator(partial derivative) represent ...

(\vec A \cdot \nabla) Is this operator well defined? It appears in many vector calculs identites, and it has an easy enough explicit formula in cartesian coordinates. But I've heard it cannot be written generally in the curvilinear coordinates. I assume this is because this operator can...

Who is this strange new member - our 30,000th? And though there have been false accusations against me, https://www.physicsforums.com/showthread.php?t=89349&page=7 I have it on good authority that we do have an imposter. Could it be mattmns, rachmaninoff or Moonbear? How about...

Dear Friends I'd like to know if anybody has the solution of the aplication of nabla's operator to geometrical product: ab=a·b+a^b (inner and outer product) And if it's possible to apply a operator like this: d/dt + d/dx i + d/dy j + d/dz k. and the rules to operate. My...

How can I get A or U from those equations? B=div(A) B=Lap(A) V=Lap(U) A,B Vector fields, U,V Scalar functions And thanks,

Dear Friends, Another question for dummies... The operator "nabla" can be locates before or after a vector or a tensor. If you take the vector A, "nabla A" is not the same that "A nabla" but, is it possible to obtain "nabla A - A nabla"? ¿And "(A nabla) A - A (nabla A)"?

Under the list of math symbols, nabla doesn't show up (the upside down triangle). Why is that? I just use &#9660 instead.