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Q: What is Divergence and curl of vector field?

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Divergence: rate of spread of vector in free space for non closed path. and Curl: rate of spread of vector in free space for closed path.

The velocity at each point in the fluid is a vector. If the fluid is compressible, the divergence of the velocity vector is nonzero in general. In a vortex the curl is nonzero.

Longitudinal waves are scalar waves with the divergence of the vector vibration, Del.dEv/cdt . Transverse waves are vector waves with the curl of the vector vibration, DelxdEv/cdt

Because Electric field can be expressed as the gradient of a scalar. Curl of a gradient is always zero by rules of vector calculus.

hedivergence of a vector fieldF= (F(x,y),G(x,y)) with continuous partial derivatives is defined by:

Divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point.

I am not sure if this is the answer you are looking for, since the question is listed in both Physics and Abstract Algebra, so I will try to give you some examples from physics. One of the indicators of a divergence of a vector field is the presence of a source. For example the electric field can be represented by a vector field, with each vector pointing along the field and has a length proportional to the strength of the electric field at that position. A point source then causes an electric field with a divergence at the location of the point source, with the vectors all pointing away from it (positive charge) or towards it (negative charge). Another example would be some point mass and the Newtonian gravitational field. One of Maxwell's equations states that the magnetic field cannot have any divergences meaning that there are no magnetic monopoles.

The Continuity Equation for a time varying field Eris:dEr/cdt = Del.Ev where Ev is the vector field associated with the real time varying field.Er + Ev =E, constitute a quaternion field.Del.Ev is the Divergence of the vector field.The Continuity Equation is a statement that the time variation of the real field is equal to the Divergence of the vector field. or more succinctly, the quaternion field E=Er + Ev is Real invariant.The Vector part of the variation is 0= dEv/cdt + Del Er + DelxEv , this is Vector Invariance of E. This is not the Continuioty Equatin but the Induction Equation. Together they represent the Invariance of the quaternion field E=Er + Ev.Because quaternions are not taught in schools yet, few realize the relationship between Continuity and Induction, they are the Real and Vector parts of Invariance!

no!

Curl represents the force of rotation in a 3-D vector field. Generally, the curl vector at a given point is the answer to the question, "What would happen if I stuck something there that could spin but couldn't move?" Unless the curl is zero, it would spin perpendicularly to the curl vector (according to the right-hand rule), and the longer the vector is, the faster. Curl is mathematically defined in a given direction as the limit of "circulation over area", i.e. the line integral of a circle around the point, divided by the area of the circle, with the circle shrinking towards the point. More practically, the actual vector can found by taking the cross product of the gradient operator with the function that defines the field: curl_x = ∂F/∂y - ∂F/∂z curl_y = ∂F/∂z - ∂F/∂x curl_z = ∂F/∂x - ∂F/∂y

Curl is the rotation of wind field

In mathematics, the curl of a vector is the maximum rotation on a vector field, oriented perpendicularly to the certain plane. The curl of a vector is defined by this form: ∇ x F = [i . . . . j . . . . . k] [∂/∂x ∂/∂y ∂/∂z] [P. . . Q. . . .R. . ] ...given that F = <P,Q,R> or Pi + Qj + Rk Perform the cross-product of the terms to obtain: ∇ x F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k

I'm not quite sure what you're asking, but the reason that there is magnetism at the poles has to do with the fact that magnetic field vector lines have no beginning or end, which is described mathematically through Maxwell's equations; specifically through Gauss' law for magnetism which states that the divergence of a magnetic field is 0, or Ã¢Ë†â€¡ Ã¢â€”Â B = 0. Divergence is a term meaning how much of something is exiting an enclosed surface. Since the divergence of a magnetic field is zero, there must be, always, the exact same amount of magnetic field exiting a surface as entering it, leaving the net divergence as 0.Thus, a magnetic field vector line has to "exit" from somewhere and loop around to "enter" somewhere else, and these two "somewheres" have to be connected (like a circuit). We call these two "somewheres" the magnetic poles.

The vorticity vector is DelxV = v/r sin(RV)H1, the Curl of the vector V. The unit vector H1, is perpendicular to the plane formed by the radius vector R and and the vector V.

Parallel polarisations are the divergence of the vibrations (Del . dV/dr) and Perpendicular polarisations are Curl of the vibrations (Del x dV/dr). The parallel polarisations is associated with the scalar wave and the perpendicular polarisations are associated with the vector wave.

Urve Kangro has written: 'Divergence boundary conditions for vector helmholtz equations with divergence constraints' -- subject(s): Boundary conditions, Helmholtz equations, Coercivity, Boundary value problems, Divergence

The divergence of the function is generally a cross product of partial derivatives and the vector field of F. Mathematically, the formula is: div(F) = ∂P/∂x i + ∂Q/∂y j + ∂R/∂z k where: F = Pi + Qj + Rk has the continuous partial derivatives.

vector

Vector.

Vector field

no

Scaler. The electric field is its vector counterpart.

Yes. For any vector field, force included, to be conservative, it must be irrotational, meaning that its curl must be zero everywhere. Fortunately (for me at least, since it makes this answer a whole heck of a lot easier to explain), the magnetic force field vector is already commonly expressed as a curl via Maxwell's equations:Ã¢Ë†â€¡ X B = ÃŽÂ¼0J + ÃŽÂ¼0ÃŽÂµ0Ã¢Ë†â€šE/Ã¢Ë†â€št,where B is the magnetic field, ÃŽÂ¼0 is the permeability of free space, ÃŽÂµ0 is the permittivity of free space, J is the current density, E is the electric field, t is time, and Ã¢Ë†â€¡ X B is the curl of the magnetic field. Bolded quantities are vectors.Now, if you don't know a thing about vector calculus, hopefully you at least remembered what I wrote above. For a vector field to be conservative, its curl must be zero everywhere. As you can hopefully see from that above equation, the curl of the magnetic field, Ã¢Ë†â€¡ X B, is not zero everywhere. More specifically, if there is a current density, J, within the magnetic field, or if there is a time changing electric field, Ã¢Ë†â€šE/Ã¢Ë†â€št, within the magnetic field, the curl of the magnetic field is not zero, therefore the magnetic force field is nonconservative.If you get lucky enough to find a situation where there is no current density or time dependent electric field, which is in reality impossible due to electrons providing there own time dependent electric fields, then the magnetic force field would be conservative.

An electric field is a vector quantity because it has direction and a magnitude. Electric field has a certain direction in which it acts.

Direction of the electric field vector is the direction of the force experienced by a charged particle in an external electric field.