ROTACIONAL Y DIVERGENCIA DE UN CAMPO VECTORIAL

En este espacio compartiremos los conocimientos de los temas Rotacional y Divergencia de un Campo Vectorial, comenzando por las definiciones y desarrollando ejercicios de aplicación de ellos. También si es pertinente se pueden hacer comentarios sobre las integrales de línea.

¿Qué características tienen los campos vectoriales conservativos?
¿Qué relación hay entre el gradiente de una función y los campos vectoriales?
¿Por qué para determinar el rotacional de un campo vectorial se requiere de una función potencial?
Proporciona diferentes ejemplos de rotacional y/o divergencia de campos vectoriales.

Good Afternoon... Welcome to spanglish forum

Starting with the forum of the day I want to start with my comments:
Forum Questions
What are the characteristics from a conservative vector field?

A conservative vector field has three important characteristics:
1. Is independent of de trayectory. I mean that does not matter the way you use, the result is the same.
2. The curl in each point is Cero.
3. Must have a Potential, which is a magnitude that could be a vector magnitude or a scalar one. This potential describes the possible variation from another magnitude. If a vector field has cero divergence, it may be represented by vector potential.

2.What is the relation between gradient of a function and vector field?
The relation is

The cross product between the gradient and the vector field is cero.. as I say before the curl in each point is cero. Now, the divergence of a vector field is therefore a scalar field. If , then the field is said to be a divergenceless field.

3.Why to get the curl of a vector field you need a potential function?
Because the potential function is the measure of the variation of the variable or variables involved, the potential could be a vector or scalar, but is necessary because the potential says how much has the variable changes by application of any disturbance between an initial and the final state.
Applications
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 application for this subject in fluid mechanics and quantum mechanics.

see you!!thanks..

Good Night,
Between the principals characteristics of a conservative vector field:

1.has three data: magnitude, direction and sense.
2.The circulation of a conservative field by a closed line is zero
3.If a conservative vector field always exists a potential function and is unique.
4.the circulation of the field along a curve is independent of path,only depends on the starting and ending points of the circulation.
5.can be expressed as gradient of a scalar function,because exist a scalar function of point. v(x,y,z) y cumple F= -V∇

wait yours comments

I will see you tomorrow

integrals of line

to calculate the integrals of line of vector field it´s gradients(derivadas) of scalar fields, in this case the integral of line
of vector field gradient depends only of the valor scalar field corresponding at the ends of the road

teacher please your comment

los campos vectoriales asocian un vector a cada punto en el espacio, y campos escalares, que asocian un escalar a cada punto en el espacio. Por ejemplo, la temperatura de una piscina es un campo escalar: a cada punto asociamos un valor escalar de temperatura. El flujo del agua en la misma piscina es un campo vectorial: a cada punto asociamos un vector de velocidad.
ademas e campo vectorial es una construcción del cálculo vectorial que asocia un vector a cada punto en el espacio euclídeo, de la forma .

Cuando se desea medir la circulación de un campo vectorial como una función de las coordenadas, se utiliza una función vectorial denominada rotacional, que mide la circulación por unidad de área cuando el área tiende a cero en cada punto del espacio en que se encuentra definido el campo.

El rotacional de un campo es una función de las coordenadas y puede en consecuencia, ser diferente para los diferentes puntos del espacio en que se encuentra definido el campo.

Cuando el rotacional es nulo en todos los puntos de una región, se dice que el campo es irrotacional o conservativo en dicha región.

First that quite companions remember that the forum is in English

Qué relación hay entre el gradiente de una función y los campos vectoriales

En un espacio euclídeo, el concepto de gradiente también puede extenderse al caso de un campo vectorial, siendo el gradiente de un tensor que da el diferencial del campo al realizar un desplazamiento

A vectorial field is conserving if the work realized to displace a particle between(among) two points is independent from the path followed(continued) between(among) such points. The conserving name owes to that for a field of forces of this type there exists a specially simple form of the law of conservation of the energy.

When you want to measure the movement of a vector field as a function of the coordinates, you use a function called rotational vector, which measures the flow per unit area when the area tends to zero at every point in space that is defined in the field.

The curl of a field is a function of coordinates and may therefore be different for different points in space where the field is defined.

When the curl is zero at all points of a region, it is said that the field is irrotational or conservative in that region.

I agree with the arguments of douglas but podria to say that the gradient of a field to climb, that is distinguishable in the environment of a point, is a vector defined as the only one that allows to find the directional derivative in any direction

good morning
as we talked about of the camps vectoriales conservative, and the definition de rotational, is deduse
an indentity known as theorem od Stokes.

Since the curl of a vector field is a kind of derivative of area of circulation small of camp. for both the integral of area of rotational correspond to circulation of camp. la ecuacion esta dada:

Why to determine the rotacional of a vectorial field is it needed of a potential function?[/color]

This is possible only if we can find a function to climb potential call of potential energy, of which his(her,your) gradient is this force. In such a way that for this force the work that it(he,she) realizes on a mobile between(among) two any points of the space is equal to the variation of this function to climb between these two points

Example of vectorial fields:

A vectorial field for the movement of the air in the land will associate with every point in the surface of the land a vector with the speed and the wind direction in this point. This can show using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the speed of the wind. A "Discharge" in the usual function of the barometric pressure would act as well as a source(fountain) (arrows going out), and a "Baja"" will be a sink (arrows that enter), since the air tends to move from the areas of high pressure to the areas of low pressure.

good bay. see you later

good afternoon teacher and classmates

one of the most important characteristics of conservative vector fields is that one is conservative when there is a differentiable function ƒ such that F ⃗ is equal to the gradient of ƒ. this function is called the potential function ƒ
A vector field F ⃗ is conservative only if V X F ⃗ = 0

Another characteristics is vector fields are dependent on the position in space and time.

Good morning,
Before you begin, to answer questions related to vector fields, should be defined first of all that is a vector field:

A vector field is a function F that associates each point a vector space, unlike the normal functions that each point in space to assign a numeric value.

Characteristics of a conservative vector field ?

- For a vector field is conservative, there must be a function that the derivative is equal to the vector field.
- The route of the field depends on the starting and ending points of the path and not the path.
- The dominance of the field is a set of points and counterdomination space is a set of vectors.

Relationship between gradient and vector field ?

A vector field is the gradient of a scalar field. Recall that a characteristic of the gradients is that these are normal vectors to the curve that contains it, in this case as vector fields emerge from any point in space, this forms an angle of 90 ° with the line containing the point it is born, making it a normal vector.

Good Afternoon fellow, and researched what we say:

In vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral from one point to another is independent of the choice of path connecting the two points: it is path independent. Conversely, path independence is equivalent to the vector field being conservative. Conservative vector fields are also irrotational, meaning that (in three-dimensions) they have vanishing curl. In fact, an irrotational vector field is necessarily conservative provided that a certain condition on the geometry of the domain holds: it must be simply connected.

Rightly whit Breiner...
1. A shield vectorial is conservative if:
The circulation of the shield for a long the curve is independentof the way, and dependsof the initial poit and the final of the circulation.
The shield conservative can be express like a GRADIENTE of the scalar function.
The circulation a shield vectorial for a close line is Zero.(0)

The relation between gradient of a function and vector field.

Definition. If f is a scalar function of ℜ n → ℜ, then the gradient of this
function is a vector field, denoted by gradient f is defined by:

gradient f= (fx, x2….xn)f (x1,x2…xn),..fxn (x1..x2..xn))

and is called a gradient vector field in ℜ n.

The cross product between the gradient and the vector field is cero.. as I say before the curl in each point is cero. Now, the divergence of a vector field is therefore a scalar field. If , then the field is said to be a divergenceless field.

the curl of a vector field

In vector calculus, the curl (or rotor) is a vector operator that describes the rotation of a vector field. At every point in the field, the curl is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point.

Thus we can say that the potential function is a measure of the variation in the variables involved, the potential may be a vector or scalar, but is necessary because the potential says how much the variable changes by the application of interference between initial and final.

investigating teacher I could find that and I want to share with you

The circulation of a vector field along a curve c in space is a line integral whose integration is the projection of the vector field on the element of arc stretching along the line
The integration of scalar and vector fields is often used to define quantities related to the fields in a region of space.

The divergence of a vector field is used as a tool for studying the behavior of the fields and their sources

the divergence of the field, is the net flow area per unit volume

The curl is a vector field whose magnitude is the highest circulation per unit area at each point in space, and its direction is the normal vector to the contour that maximizes the movement.

I Continue to investigate and WHAT YOU FIND IT I UPLOAD IN THE FORUM for to read THEM and I discussed

good afternoon to compliment my colleague on divergence.
divergence of a vector field measures the difference between the inflow and outflow of a vector field on the surface surrounding a volume control, so if the field has "sources" or "sinks" the divergence of the field will be different from zero.

The gradiente of a vectorial function shows how much has change the vectorial magnitude in a field in comparison with other magnitude. We have to remember that the gradiente of a function is the generalization of the derivative concept and the owns gradiente of the funtion represents the scalar quantity.

good afternoon to all, before making my comment I would like to tell some of my colleagues do not forget that as we participate in the forum.
TENGANLO EN CUENTA.

good afternoon fellow bear in mind the suggestion of the teacher