What is Del: Definition and 83 Discussions

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:





{\displaystyle \operatorname {grad} f=\nabla f}






{\displaystyle \operatorname {div} {\vec {v}}=\nabla \cdot {\vec {v}}}







{\displaystyle \operatorname {curl} {\vec {v}}=\nabla \times {\vec {v}}}

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  1. Astronuc

    Nevado del Ruiz Volcano, Colombia - increased activity

    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...
  2. Safinaz

    B Does the Laplace operator equal the Del operator squared?

    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...
  3. Flying_Dutchman

    I Physical meaning of the equation E = - del V

    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...
  4. E

    Divergence operator for multi-dimensional neutron diffusion

    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...
  5. T

    I Del operator in a Cylindrical vector fucntion

    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...
  6. Pencilvester

    I Del with Superscript in Carroll's Equation

    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...
  7. 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...
  8. N

    Understanding the Del Operator in Vector Calculus

    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
  9. E

    I Order of Operations for Tensors

    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...
  10. Dave-o

    Evaluate: ∇(∇r(hat)/r) where r is a position vector

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

    I need to prove del cross f= 0

    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)...
  12. F

    How do I cross Del with (scalar*vector)?

    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...
  13. M

    I Defining Del in Index Notation: Which Approach is Appropriate?

    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} =...
  14. olgerm

    B An equation from terms of operator del to terms of sums

    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...
  15. D

    Question about the del operator under a translation

    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...
  16. A

    Proof of product rule for gradients

    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
  17. C

    EM: When can you replace del, d/dt with ik, -iω?

    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...
  18. ognik

    MHB Order with del in cross product

    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...
  19. ognik

    MHB Triple vector product with del

    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 ?
  20. A

    Cauchy's equation in terms of material acceleration

    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...
  21. Dewgale

    Usage of Del in Spherical Polar Coordinates

    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...
  22. A

    Del operator and wave function

    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...
  23. 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}##...
  24. W

    Divergence of vector field: Del operator/nabla

    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...
  25. K

    Direction of the maximum gradient (scalar fields)

    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
  26. KleZMeR

    Pauli matrix with del operator

    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...
  27. baby_1

    Simple question in Del operator on plane wave equation

    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?
  28. I

    How to integrate by parts when del operator is involved?

    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...
  29. R

    When can we move the del operator under an integral sign?

    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...
  30. P

    Dot product of vector and del.

    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
  31. A

    Van del waals interaction in qm

    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...
  32. R

    Del Operator conversion

    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 =...
  33. N

    Del operator - order of operations

    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...
  34. S

    How do these operations with Del operator work?

    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...
  35. S

    Deriving del cross A in Electrodynamics

    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...
  36. S

    Griffiths Chapter 10 del cross position vector

    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...
  37. C

    Help with vector operator Del.

    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...
  38. U

    Proof on why del is normal to surface?

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

    Derivation of Del Operator in Spherical & Cylindrical Coordinates

    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.
  40. G

    Del operator crossed with a scalar times a vector proof

    "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...
  41. V

    Change of the Del operator in two particle interactions

    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...
  42. A

    Proof of identity involving del

    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) ?
  43. E

    Del vs. Laplacian Operator : Quick Question

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

    Expression with two vectors and del operator

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  45. 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...
  46. Z

    Do the derivatives del and d/dt commute?

    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..
  47. A

    Question About Del: Why Does Formula Fail?

    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?
  48. H

    How does the del operator change with incompressibility assumption?

    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...
  49. D

    Matrix analog of del operator?

    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...