Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operators depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.
Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for those three operators that makes many equations easier to write and remember. The del symbol (or nabla) can be interpreted as a vector of partial derivative operators; and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product with a scalar, a dot product, and a cross product, respectively, of the "del operator" with the field. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:
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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...
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...
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
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} =...
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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
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...
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) ?
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...