What is Curl operator: Definition and 13 Discussions

In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.
A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.
The alternative terminology rotation or rotational and alternative notations rot F or the cross product with the del (nabla) operator ∇ × F are sometimes used for curl F. The ISO/IEC 80000-2 standard recommends the use of the rot notation in boldface as opposed to the curl notation.Unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This is a phenomenon similar to the 3-dimensional cross product, and the connection is reflected in the notation ∇× for the curl.
The name "curl" was first suggested by James Clerk Maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839.

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  1. 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...
  2. J

    Finding the curl of velocity in spherical coordinates

    Homework Statement The angular velocity vector of a rigid object rotating about the z-axis is given by ω = ω z-hat. At any point in the rotating object, the linear velocity vector is given by v = ω X r, where r is the position vector to that point. a.) Assuming that ω is constant, evaluate v...
  3. T

    A How can curl of 4-vector or 6-vector be writen?

    How can a curl of 4-vector or 6-vector be writen? Let's say that we have a 4-vector A4=(a1,a2,a3,a4) how can we write in details the ∇×A4 Can we follow the same procedure for 6-vector?
  4. S

    I Proving the Linearity of the Curl Operator in Electromagnetic Theory

    Hi, I stumbled upon thinking that "Is curl operator a linear operator" ? I was reading EM Theory and studied that the electromagnetic field satisfies the curl relations of E and B. But if the operator was not linear then how can a non linear operator give rise to a linear solution. Thus it...
  5. Jianphys17

    Is there a generalized curl operator for dimensions higher than 3?

    Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions! On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you...
  6. 10Exahertz

    Relationship of curl and cross product.

    Hi all, I am very confused on how to define the vector product or cross product in a physical sense. I know the vector product is a psuedovector, and that it is the area of a parallelogram geometrically. However, I know it used used to describe rotation in physics. As with torque, magnetism and...
  7. C

    Magnetic field from electric field given a function of time

    Homework Statement An em wave in free space has an electric field vector E = f(t-z/c0)x where x is a unit vector in the x direction and f(t)= exp(-t2/τ2)exp(j2πv0t). Describe the physical nature of this wave and determine an expression of the magnetic field vector. Homework Equations...
  8. AwesomeTrains

    Line Integral - Stokes theorem

    Homework Statement Hello I was given the vector field: \vec A (\vec r) =(−y(x^2+y^2),x(x^2+y^2),xyz) and had to calculate the line integral \oint \vec A \cdot d \vec r over a circle centered at the origin in the xy-plane, with radius R , by using the theorem of Stokes. Another thing, when...
  9. L

    Does this curl operator equal 1?

    Hello, I am a beginner in electromagnetism. I am trying to find a vector field whose rotation equals 1 with a curl operator. If I say that the vector field is defined by V(y;2x;0) does it work? As a result, I find (0;0;1), am I right?
  10. P

    Help with the conventions used for curl operator

    http://web.mit.edu/6.013_book/www/chapter2/2.4.html I was going through the curl derivation on the above link. How does equation 3 turn out? Δy is the incremental length. But how do you decide whether it is +Δy/2 or -Δy/2. And why is the line integral taken Δz when the change is in the y...
  11. G

    Is There a Method for Finding A When Given B and ∇XA?

    Let's say I have this relation B=∇XA I know B, now I want A. What ever do I do? Is there some tried and true method out there?
  12. C

    Using Curl operator to find E.field

    Homework Statement Their solution: The Attempt at a Solution My attempt: My query is: where have I gone wrong? Or are the solutions incorrect? Edit: Actually I just figured it out, I am right, just that I never divided by \omega*\epsilon, and I do get the same answer.