ROTACIONAL Y DIVERGENCIA DE UN CAMPO VECTORIAL

I have to say that recent intervention by the comments that are many but I empathize with those that are in the list below

1 First, the vector field is one that at every point in space is associated to a vector quantity. Has a traffic per unit area
He could say that the trajectory is independent of which enters the field where it can be for any part of this

2 Second, the divergence of a vector field is a scalar and the cross product between them is zero

3 Third, potential function is a measure of variation of variables that act, this could be a vector or a scalar.
It is important that the potential reads as much variables change
de f(x,y,z)= Mi,Nj,Pk es rot F(X,Y,Z)= "xf(x,y,z)
*xf=[ i j k ]
[ & & &]
[ - - -]
[&x &y &z] si *xf=0--F ES CERO

CALCULAR EL ROTACIONALDEL CAMPO VECTORIAL EN EL PUMTO DADO
F=(X,Y,Z) =xyzi + yj + zk es (1,2,1)
*xF=(0-0)i-(0-xy)j+(0-xz)k
*xF=(1,2,1)= 2i-1k =<0,2,-1>
*F=YZ+1+1
(1,2,1)YZ+2=(2)(1)+2= 4
*F(1,2,1)=4

Tambien es importante tener en cuenta que la La divergencia es una cantidad escalar con signo. Este signo posee significado geométrico y físico:

If the divergence of a vector field at a point is positive, it means that at that point the field radiates outward. It is said that the vector field position has a spring.

If against the divergence is negative, the field converges towards that point, we say that the field has a sink.

Both sources and sinks, are the sources of a vector field scalar

por ello podemos decir lo siguiente: si la divergencia es nula en un punto el campo carece de fuentes escalares en dicho punto.

As mentioned the concept of divergence is defined for each point. From this definition, a scalar field can be constructed from one vector, whose value is equal to the divergence of the vector field at that point

This field ρ, which rewrites the distribution of sources and sinks of the vector field is known as scalar sources.

The sources speak for something that seems to derive from something else, is because in practice the way is the opposite: what are usually called the field sources and the amount to be calculated is the actual vector field . In this sense, the sources "produce" the field.

The simplest physical example is the electrostatic field. Electric charges (which are scalar sources) produce the electric field. The electric field radiates outward from the positive charges, which are its springs, and converges towards the negative charges, which are their sinks.

La representación de los campos vectoriales se hace mediante mapas semejantes a los de los campos escalares, pero usando líneas que representan la continuidad de la orientación de los vectores de campo sobre una región definida. Estas líneas reciben el nombre de líneas de fuerza.

Al igual que con los campos escalares, un campo vectorial no puede representarse fácilmente en tres dimensiones, por lo que normalmente se hacen proyecciones sobre los planos directores del sistema de coordenadas

Las líneas de fuerza cumplen con las siguientes propiedades:

Los vectores de campo en cualquier punto son siempre tangenciales a la línea de fuerza que pasa por el punto dado.

Las líneas de fuerza no se cruzan en ningún punto aunque pueden seguir trayectorias cerradas.

La cantidad de líneas de fuerza en cualquier porción del espacio en que se encuentra definido el campo es proporcional a la intensidad del campo vectorial

In some other cases, the representation of vector fields is done through direct field vectors. In these cases the severity of the vector field associated with the density of field vectors in a region, both as to the length of them.

also the vector fields are often used in physics, for example, model the speed and direction of a moving fluid throughout space, or the intensity and direction of a certain strength, as the electromagnetic force or the gravity, they change point to point.

The rigorous mathematical treatment, vector fields on differentiable manifolds are defined as sections of the tangent bundle of the variety. This is the type of treatment needed for modeling.

The representation of vector fields is done using maps similar to those of scalar fields, but using lines that represent the continuity of the orientation of the field vectors on a defined region. These lines are called lines of force.

As with scalar fields, a vector field can not easily represented in three dimensions, so they usually are flat projections of the directors of the coordinate system

and also need to know is also important to note that the divergence is a scalar quantity with the sign. This sign is physical and geometrical meaning:

If the divergence of a vector field at a point is positive, it means that at that point the field radiated outward. It is said that the vector field has a spring position.

If against the divergence is negative, the field converge towards that point, we say that the field has a sink.

Both sources and sinks, are the sources of a vector scalar field

thanks mate for this forum to clarify questions that had respect to the subject

como ejemplos de campos vectoriales podemos citar el campo de velocidades en un fluido, el campo gravitatorio, el campo electrico y el campo magnetico

is also important to add that vector fields must be compared to scalar fields, which associate a number or scalar to every point in space (or each point of some variety).

The derivatives of a vector field, which result in a scalar field or another vector field, divergence and rotor are called respectively. Conversely:

Given a vector field whose curl vanishes at one point, there is a scalar potential field whose gradient coincides with the scalar field in a neighborhood of that point.
Given a solenoidal vector field whose divergence vanishes at one point, there is a potential vector field called rotational vector which coincides with the scalar field in a neighborhood of that point.
These properties are explained are derived from the Poincare conjecture.

All your contributions are important, I think that already there is clarity of the topic.
Remember that a little time is absent for the closing of the forum.

Good nigth, partners
First I like talk about the DIVERGENCE and ROTATION.
The vectorial differential ∇ in three dimentions is defined as.
∇=i ∂/∂x+j ∂/∂y+k ∂/∂z.
But it don’t have utility, though it is applies in a scalar function, and it make the gradient of f given by
grad f=∇f=i ∂/∂x+j ∂/∂y+k ∂/∂z.
And now is used nabla as a operator over the vectorial functions. The main part of the work in it section consist in algebraic manipulations. The physical meaning of the functions That we define.
Suppose that a vector function F in three dimensions is given by
F(x, y, z)=M(x, y, z) i+ N(x, y, z) j + P(x, y, z) k
Here M, N y P have parcials derived in some region. The rotacional of F, denoted by rot F o ∇ x F,, can be defined as the development (it is based in firt row) of following determinant
rot F=∇ x F=|(i/∂)/(∂x/M) (j/∂)/(∂y/N) (K/∂)/(∂z/P) |.
The notation of the curl in the form of a determinant is ambiguous in that the first screed consists of vectors, the second partial differential operators and the third function, but is a useful way to remember the complex formula
∇ x F =(∂P/∂y-∂N/∂z)i+(∂M/∂z-∂M/∂x)j+(∂N/∂x-∂M/∂y)k
The curl of F determines a vector field in three dimensions which has important applications in physics
The divergence of F, denoted by div F or ∇. F is defined as.
div F=∇.F=∂M/∂x+∂N/∂y+∂P/∂z
The reason to we use the symbol ∇. F is that the formula div F can be obtained formally making the dot product ∇ y F and div F note that is a scalar function

hi. teacher like this?. I imagine that until now connect.

teacher lucia
I hope that my intervention was beyond their liking and was you liked, did not have time to make a much deeper and participatory
tank you very much

hi all had not been able to enter the forum but I have something to add:


DIVERGENCE: scalar operator acting on a vector field
PROPERTIES
If f represents a flow of matter, its derivatives indicate how the flow of matter varies per unit length in each coordinate direction. Therefore, the divergence is the variation of materials stored (or difference between outputs and inputs) per unit volume.

ROTATIONAL: The curl is a vector operator defined on vector fields. It is defined by the "product" nabla vector between the operator and the field Vectorian
PROPERTIES
Indicates the trend (local) to rotate the field. That is, a field is equal to the magnification of the original field per unit length. Is oriented by the right hand rule.

another answer to the second question




that the gradient becomes scalar function in a vector function

to the third question expanded the definition by saying this



is required because the vector product notation used for the curl comes from looking at the gradient, as a result of the actions of the nabla differential operator on the potential function

In vector calculus, curl, or rotor is a vector operator that shows the trend of a vector field to induce rotation around a point.
Mathematically, this idea is expressed as the limit of the circulation of the vector field, where the curve on integrating comes down to one point.

The result of this rotational limit is not complete (which is a vector), but only its component along the direction normal to ΔS and oriented according to the right hand rule. To get the full rotational three limits should be calculated considering three corners located in perpendicular planes.

The divergence of a vector field is a scalar field, defined as the field flux vector per unit volume.
This definition is directly related to the concept of flow field. As in the case of flow, if the divergence at a point is positive,we said the field has springs. If the difference is negative, we are said to have sinks. The most characteristic example is given by the electric charges, giving the divergence of the electric field, with positive charges and negative springs and sinks of the electric field.

hi teacher
i am very happy whith this forum, because is a diferent class, we used always a classroom and a board, but this idea is great!

The divergence has an important physical interpretation.



If we imagine that F is the velocity field of a gas (or fluid), then div F represents the rate of expansion per unit volume under the flow of gas (or fluid). If div F <0, gas (or fluid) is being compressed.



For a vector field in the plane, the divergence measures the rate of expansion of the area.

An example would be to calculate the flux divergence of the vector field F (x, y, z) = xi + yj + zk through the spherical surface X2 + Y2 + Z2 = 4:
from the equation of the sphere is known that the radius is R = 2 then the function gradient is equal to the derivative of f1 to f2 with respect to x from Y and f3 with respect to z, resulting in 3
using the divergence theorem as the double integral would be evaluated by the F s unitadio of ds vector equal to the triple integral of the gradient evaluated by F v dv is the triple integral V-3DV evaluated this equals ( 3) * (4 / 3) π * 23 = 32 π

good night


we can say that in FINDINGS.

To understand the divergence theorem, we need to know the physical significance of divergence of a vector field, which is the rate at which “density” that exists in a region of space defined. Now imagine a region of space with a boundary surface.. For example a potato form… now imagine that now that space of matter is empty… I mean that the matter of that potato is gone.. only you can see the boundary of that potato… that is your closed infinitesimal boundary surface that is surrounding the volume of the potato (volume element) that is taken to size zero using limiting process. The divergence of the factor field is a scalar field. If , then the field is said to be a divergenceless field. Now, if in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region. Now if you are measuring the the net flux of content passing through a surface surrounding the region of space, it is therefore immediately in physics where it goes.. that is what we name principle of continuity that makes famous Gauss named Divergence Theorem. The application for this theorem is Gauss law concerning the electric field, Gauss law for magnetism that is recognized because is an application of Maxwell equation and de Gauss law for gravity. And you can find another applications for this subject in fluid mechanics and quantum mechanics.

colleagues hope everything is clear between us and to take advantage of all this.

exitos.

The vectorial calculation is nowadays an essential part of the mathematical indispensable preparation that the scientist has to possess so much as(like) é1 technician. The methods of the same one are going to be exposed brief, as well as his(her,your) scope in the resolution of certain problems.
Highlighting a bit the topic to treating of the forum and that many(many people) already have defined it is possible to express the concept of flow of a vectorial field And across a surface to the integral extended to the whole surface? S And · dS where And it is the vector that it(he,she) represents to, I stand out(camp) and dS is a vector associated with the surface element dS, of module dS, normal direction(address) to felt surface element and the one that goes out towards out of the surface.