Del Definition and 82 Threads

https://volcano.si.edu/volcano.cfm?vn=351020 The Street reported, "The last time the Nevado del Ruiz volcano was active, it erupted and killed 23,000 people in Colombia, wiping out the town of Armero in the process." Pay attention to volcanoes in the neighborhood and be prepared to...

Hello , The Laplace operator equals ## \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} ## so does it equal as well nable or Del operator squared ## \bigtriangledown^2## ? where ## \bigtriangledown =\frac{\partial}{\partial...

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

The gradient of a function gives a vector perpendicular to it's surface. So the equation reads electric field is the negative of the vector perpendicular to the equipotential surface. I know electric field and understand potential but I can't physically make sense for the above sentence how LHS...

Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...

Hi there I'm having a hard time trying to understand how come ∂r^/∂Φ = Φ^ ,∂Φ/∂Φ = -r^ -> these 2 are properties that lead to general formula. I've been thinking about it and I couldn't explain it. I understand every step of "how to get Divergence of a vector function in Cylindrical...

In Carroll’s “Spacetime and Geometry” his equation (1.116) for ##\partial_{\mu} T^{\mu \nu}## for a perfect fluid ends with the term ##... + ~\partial^{\nu} p##. First of all, in order for this equation to really be general, it would need to use the covariant derivative instead of the simple...

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...

F is a vector from origin to point (x,y,z) and û is a unit vector. how to prove? (û⋅∇)F=û only tried expanding but it's going nowhere

Hey so probably a really simple question, but I'm stumped. How do you simplify: ν∇⋅(ρν), where ν is a vector ∇ is the "del operator" ⋅ indicates a dot product ρ is a constant. I want to say to do the dyadic product of v and ∇, but then you would get (v_x)*(d/dx) + ... which would be...

Homework Statement ∇ . r = 3, ∇ x r = 0 Homework EquationsThe Attempt at a Solution So far I've gotten up to ∇(∇^2 r)

Homework Statement F(x) has the form F(x)=f(r)x where r=|x| A.) prove that del cross f =0 B.) Now suppose also Del •F = O. What is the most general form allowed for f(r)? Homework EquationsThe Attempt at a Solution I have done part b but what do I need for A F(X)= r(hat) Fr = 1, F(theta)...

Homework Statement Show that for any scalar field α and vector field B: ∇ x (αB) = ∇α x B + α∇ x BHomework Equations (∇ x B)i = εijk vk,j (∇α)i = αi (u x v)i = eijkujvk The Attempt at a Solution Since α is a scalar i wasn't quite sure how to cross it with ∇ So on the left side I have...

Hi PF! Which way is appropriate for defining del in index notation: ##\nabla \equiv \partial_i()\vec{e_i}## or ##\nabla \equiv \vec{e_i}\partial_i()##. The two cannot be generally equivalent. Quick example. Let ##\vec{v}## and ##\vec{w}## be vectors. Then $$\nabla \vec{v} \cdot \vec{w} =...

https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fd3adddbdfb95797d11ef6167ecda4efe3e0b9 https://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_in_terms_of_potentials How to write this formula in terms of sums and vector components? What is ##v\cdot\nabla## ? I think it is some...

Homework Statement This isn't really a problem. I am just re-reading some section "Classical Mechanics" by John Taylor. I think this belongs in the math section, since my question is mainly about the del operator. There is just one fragment of one sentence that I want to make sure I am...

Can someone please help me prove this product rule? I'm not accustomed to seeing the del operator used on a dot product. My understanding tells me that a dot product produces a scalar and I'm tempted to evaluate the left hand side as scalar 0 but the rule says it yields a vector. I'm very confused

I tried googling a good resource for this but it was difficult to think of good keywords. Are we always allowed to do this, or is it just for plane waves, linear media, conductors, etc? My intuition is that it's fine in all circumstances since we can Fourier decompose most any function into...

I got to here in a simple exercise (orb. ang. momentum cords), realized I was applying something I didn't understand ... $L = -i \begin{vmatrix}\hat{x}&\hat{y}&\hat{z}\\x&y&z\\\pd{}{x}&\pd{}{y}&\pd{}{z}\end{vmatrix}$ I 'know' it equates to $L_x =-i \left( y\pd{}{z} - z\pd{}{y} \right) $ - but...

I know the bac-cab rule, but add $\nabla$ and it's not so clear .. applying it to $\nabla \times \left( A \times B \right) = A\left(\nabla \cdot B\right) - B\left(\nabla \cdot A\right) ...$, not quite Please walk me through why the other 2 terms emerge ?

Does anyone know which formula is used or how to arrive at the righthand side of the equation below, which is the dot product of del and rho*a 2nd order tensor(V V). . represents dot product and X a vector quantity This problem is in connection with transforming cauchy's equation in terms of...

Hi all, I'm having some problems in grasping/properly understanding the usage of the del operator ( ##\nabla## ) in spherical co-ordinates, and I was wondering if someone could point me to some good resources on the subject, or take a bit of time to try to explain it to me. It just doesn't seem...

I've been given the question "What is ∇exp(ip⋅r/ħ) ?" I recognise that this is the del operator acting on a wave function but using the dot product of momentum and position in the wave function is new to me. The dot product is always scalar so I was wondering if it would be correct in writing...

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}##...

Homework Statement Let ν(x,y,z) = (xi + yj + zk)rk where v, i, j, k are vectors The k in rk∈ℝ and r=√(x2+y2+z2). Show that ∇.v=λrk except at r=0 and find λ in terms of k. Homework Equations As far as I understand it, ∇.v=∂/∂x i + ∂/∂y j + ∂/∂z k, but this may very well be wrong. The Attempt...

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 In the Pauli theory of the electron, one encounters the expresion: (p - eA)X(p - eA)ψ where ψ is a scalar function, and A is the magnetic vector potential related to the magnetic induction B by B = ∇XA. Given that p = -i∇, show that this expression reduces to ieBψ...

Hello question is: As you see when we do del operator on A vector filed in below example it removes exponential form at the end.why does it remove exponential form finally?

i'm trying to integrate this: $$W=\frac{ε}{2}\int{\vec{∇}\cdot\vec{E})Vdτ}$$ where ε is a constant, E= -∇V, τ is a volume element how do i end up with the following via integration by parts? $$W=\frac{ε}{2}[-\int{\vec{E}\cdot(\vec{∇}V)dτ}+\oint{V\vec{E}\cdot d\vec{a}}$$] where the vector a...

Homework Statement Hi, it's me again. I'm new to vector calculus so this might sound like a stupid question, but in relation to a specific problem, I was wondering when we could move the del operator under the integration sign - in relation to a specific problem, which is: A(r) = integral...

I'm not sure which section is best to post this question in. I was wondering if the expression (u $ ∇) is the same as (∇ $ u). Here $ represents the dot product (I couldn't find this symbol. ∇=del, the vector differentiation operator and u is the velocity vector or any other vector

Hello I read the follow paper about van del waals interaction in quantum mechanics http://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/van_der_waals.pdf In this paper the potential V= e^2/R + e^2/(R+y)+e^2/(R-x)+ e^2/(R+y-x) is aproximated to V \approx -2 e^2/R^3 xy with R>>|x|,|y| why...

I have been trying to convert the Del operator from Cartesian to Cylindrical coords since like 5 days. but still i can't see why my way doesn't work. It worked for the 3D heat equation and 3D wave equation but for vector quantities no :( ... This is the way i followed \nabla P =...

Hey! Is it true that when you dot the del-operator on another vector, the differentiation has priority over the dot-product? That's why you get all those weird formulas for the divergence in circular and cylindrical coordinates (which are very different to the Cartesian ones)? So in the case of...

How do these operations with Del operator work?? Homework Statement Let's say A and B are expressed by their cartesian components as: A = <P, Q, R> and B = <M, N, O> what would be the differente between (A.∇)B and B(∇.A) ? Homework Equations The Attempt at a Solution I tried...

Hello, I am trying to derive the equation for the B-field due to a moving charge. ~ Griffiths Chapter 10, equation 10.66. I have been trying to “do” the del cross A and simplify . Things get messy and I am uncertain on some of my vector operations. In searching the internet I find...

I am working through chapter 10 of Griffith’s electrodynamics (for fun and in my spare time). While I don’t have a formal bucket list, getting to an understanding of how Newton’s third law is not as straightforward for electrodynamics has been on my mental bucket list. I am an engineer not a...

Homework Statement In the Pauli theory of the electron, one encounters the expresion: (p - eA)X(p - eA)ψ where ψ is a scalar function, and A is the magnetic vector potential related to the magnetic induction B by B = ∇XA. Given that p = -i∇, show that this expression reduces to ieBψ...

Homework Statement Simple proof on why ∇∅ is normal to surface of ∅(x,y,z) = constant Homework Equations The Attempt at a Solution

Hi all, Del = i ∂/∂x + j ∂/∂y + k ∂/∂z in x y z cordinate similarly I require to see the derivation of del in other coordinates too. Please give me a link for the derivation.

"Del" operator crossed with a scalar times a vector proof Homework Statement Prove the following identity (we use the summation convention notation) \bigtriangledown\times(\phi\vec{V})=(\phi \bigtriangledown)\times\vec{V}-\vec{V}\times(\bigtriangledown)\phi Homework Equations equation for...

Change of the "Del" operator in two particle interactions Ok,so John Taylor's Classical Mechanics has this small subtopic "energy interactions between 2 particles".And,in that,hes defined a "del1" operator as the vector differential operator with respect to particle 2 at the origin.Hence,the...

Prove that ∇.(u×v) = v.(∇×u) - u.(∇×v), where "." means dot product and u,v are vectors. So by scalar product rule, A.(B×C) = C.(A×B) So applying same logic to above identity, shouldn't the left hand side just be equal to v.(∇×u)? Or just to -u.(∇×v), since A.(B×C) = -B.(A×C) ?

Just to clarify: The del operator's a vector and the laplacian operator is just a scalar?

Homework Statement (A.∇)B What does this mean and how do I go about trying to expand this (using cartesian components)?

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...

Homework Statement Do the derivatives del and d/dt commute? Or in other words, is it true that: del(d/dt)X = (d/dt)del_X Homework Equations ? The Attempt at a Solution nm, I think I know how to show it now..

Question about "del" We know that A x (BxC)= (A·C)B-(A·B)C (*) In the following example, we can treat ∇ as a vector and apply the formula (*) above to get the correct answer ∇x(∇xV)= ∇(∇·V)-∇^2 V But in this example, the formula (*) seems to fail ∇x(UxV)≠U(∇·V)-V(∇·U) Why?

I'm trying to understand why the del operator is working a certain way. So in my literature there is a term: \nabla \cdot \rho_a \mathbf{v} but then after saying that \rho_a=w_a\rho the term can somehow become \rho (\mathbf{v}\cdot \nabla w_a) I do not understand how nabla and the...

The del operator is often informally written as (d/dx, d/dy, d/dz) or \hat{x}\frac{d}{dx}+\hat{y}\frac{d}{dy}+\hat{z}\frac{d}{dz}, a pseudo-vector consisting of differentiation operators. Could there be a pseudo-matrix operator like it? What would one be differentiating with respect to- that is...