What is Nabla: Definition and 49 Discussions

The nabla is a triangular symbol resembling an inverted Greek delta:

{\displaystyle \nabla }
or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word νάβλα for a Phoenician harp, and was suggested by the encyclopedist William Robertson Smith to Peter Guthrie Tait in correspondence.The nabla symbol is available in standard HTML as ∇ and in LaTeX as \nabla. In Unicode, it is the character at code point U+2207, or 8711 in decimal notation.
It is also called del.

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  1. deuteron

    I Deriving the Curl of the Magnetic Field, Role of the Nabla Operator

    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...
  2. Brix12

    I How do I format equations correctly? (Curl, etc.)

    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...
  3. Addez123

    I don't understand simple Nabla operators

    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...
  4. R

    I Computing F with Nabla Identity: A Step-by-Step Guide

    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? $$ F = \sum_{i} p_i \partial_{i} E_{i} e_{i} $$
  5. K

    Nabla operations, vector calculus problem

    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...
  6. B

    A Calculating Nabla w V in General Relativity

    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...
  7. DuckAmuck

    B Question about how the nabla interacts with wave functions

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

    I A question about writing the notation of the nabla operator

    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...
  9. C

    A Angular Moment Operator Vector Identity Question

    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...
  10. SemM

    B Question about the Delta and Nabla symbols

    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?
  11. B

    Number 1 beside nabla symbol....

    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.
  12. Remixex

    About Nabla and index notation

    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...
  13. Remixex

    Understanding the Equation for Velocity Field in Cylindrical Coordinates

    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...
  14. H

    Derive grad T in spherical coordinates

    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}##...
  15. thegreengineer

    Directional derivative and gradient definition confusion

    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...
  16. L

    Calculating the Gradient of a Complex Exponential Function

    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}}
  17. T

    Proof of equivalence between nabla form and integral form of Divergence

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

    Notation for the nabla operator arguments

    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...
  19. S

    Vector field (rotors and nabla operators)

    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...
  20. J

    Delta amplitude and nabla amplitude

    Delta amplitude and "nabla amplitude" Why all jacobi theory and all ellipitc integrals is based in ##\Delta(\theta) = \sqrt{1-m \sin(\theta)^2}## ? You already think that this definition is just midle of history, cause' you can define other elementar function: \nabla(\theta) = \sqrt{1-m...
  21. P

    Solving Operator Nabla Example Problem

    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...
  22. V

    Nabla Operator in Spherical Coordinates

    Homework Statement Exercise 1.3 on uploaded Problem Sheet. Homework Equations Shown in Exercise 1.3 on Problem Sheet The Attempt at a Solution Uploaded working: I have found the inverse of the Transformation Matrix from Cartesian to Spherical Coordinates by transposing...
  23. J

    Nabla operator and working with it

    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...
  24. I

    How to evaluate this nabla expression in spherical coordinates?

    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...
  25. K

    Nabla calculus and conservative forces

    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...
  26. L

    Some expressions with Del (nabla) operator in spherical coordinates

    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...
  27. A

    What is the Dual Nature of Nabla in Vector Differential Operators?

    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..
  28. M

    Finding Nabla Operator for f(r) with r = |R|

    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...
  29. A

    Question about grad, nabla (vector operator)

    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...
  30. madmike159

    Proof: Nabla X (Nabla X a) = Nabla (Nabla · a) - Nabla^2 a

    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...
  31. J

    Does the nabla operator has a unit?

    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
  32. pellman

    Semi-colon or Nabla Notation: Which is Correct for Calculating Nabla?

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

    What is the purpose of using the nabla operator in this equation?

    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?
  34. R

    Understand Magnetic Field Divergence: Nabla dot B =0 Explained

    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)...
  35. B

    Calculate Nabla Operator for Potential Function with Distance r

    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
  36. M

    Coordinate transformation of nabla operator

    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...
  37. A

    Discover the Result of C \nabla as Any Constant in This Comprehensive Guide

    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
  38. L

    Understanding the Nabla Operator and Electric Field: A Comprehensive Guide

    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})
  39. qspeechc

    Understanding the Product Rule: A Guide to \nabla × (A×B)

    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...
  40. D

    Calculate Dot Product of Nabla and Vector | Partial Derivative Method

    [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...
  41. F

    Understanding Nabla and its Derivatives in 3D Systems

    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 ...
  42. S

    Exploring the Properties of $(\vec A \cdot \nabla)$

    (\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...
  43. Ivan Seeking

    Who is the 30,000th member of Physics Forums?

    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...
  44. R

    Nabla operator to geometric product

    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...
  45. T

    How to Obtain A or U from Inverse Nabla Functions?

    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,
  46. R

    Understanding Nabla Operator with Vector A

    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)"?
  47. D

    Why Nabla Symbol Not Listed Under Math Symbols

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