ROTACIONAL Y DIVERGENCIA DE UN CAMPO VECTORIAL

sere a bit more specific with what he said
jeidy fonseca.
what happens is that when we say that a vector field is conservative flow field along a curve is independent of the way, just depends on the starting and ending points in circulation.
devemos not forget that conservative fields can be expressed as gradient of a scalar function.

so the calculation of circulation becomes:
use the following link to see the equation:
https://2img.net/r/ihimizer/img130/2301/germanp1.jpg

The movement of a conservative field by a closed line is therefore zero:
If a vector field is conservative also meets these conditions: use the following link to see the equation:
https://2img.net/r/ihimizer/img69/2027/germanp2.gif

Vector Fields
Imagine a moving stream flows through a pipe with imagination. If at every point we put the velocity of flow at that point we obtain the velocity field V of the fluid. Vector field that describes the rate of a pipe

3)
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.
mestra then the gradient changes that arise in a vector field

answer question 3

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. etc

las caracteristica de los campos vectoriales conservativo podemos establer unas muy importante las cuales son:
presenta propiedades que si se cumplen se cumpliran las otras porque matematicamente son equivalente entresi
otra es su independencia de trayeroria entre puntos para tener un buen estudio.

First of all we should define it is used for 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 some force, such as electromagnetic or gravitational force, they change point to point. A vector field on a subset of Euclidean space Rn XC is a vector-valued function:
F: XF: X → Rn
We say that F is a Ck vector field if the function is k times continuously differentiable in X.
A vector field can be visualized as a space X with an n-dimensional vector attached to each point in X. and since we have other concept of rotation and divergence can understand what is spoken in this forum.
I go to class chao

buscar

Good evening

The word conservativo proviene of the fisica where use
to do reference to the fields where fulfils the principle
of conservacion of the energia (this occurs in fields preserve-
tivos).

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: if F is a field defined in D y
F=∇f for some scalar funtion in D, then f have the name of
potential funtion of F. so a F field is conservative if, only
if is the gradient field of a scalar funtion.

Conservative vector fields have the property that the line
integral from one point to another is independent of the
choice of way connecting the two points: it is way independent.

Relacion Between a gradiente of a funcion and a field vectorial
es que para que este último sea un campo conservativo requiere ser el campo gradiente de una funcion escalar f, la cual es conocida como función potencial de F.

Hello, this activity is a great challenge, which helps us to our integral formation.

This one is my contribution.


What are the characteristics from a conservative vector field?

Qué características tienen los campos vectoriales conservativos?

The vectorial conservative field possesses three principal characteristics:
* His(Her,Your) path is independent to say does not import the way in which it(he,she) is used since the result is the same.
* The curl in every point is Zero.
* It(He,She) must have a Potential, which is a magnitude that might be a magnitude vector or climb one. This potential describes the possible variation of another magnitude. If a vectorial field has the difference zero, it(he) can be represented by the potential vector.

Serious of vital importance the following contribution

We define the divergence of a field at a point P as the net flow area per unit volume evaluated at that point

I found out this about the theme of the forum

In CALCULO VECTORIAL, divergence is an operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

IN SPANISH FOR TO EVERYONE UNDERSTAND
Se denomina Divergencia de un campo vectorial al flujo de salida por unidad de volumen cuando la unidad de volumen se hace infinitesimal.

La divergencia de un campo vectorial es un campo escalar dada la naturaleza escalar del flujo y la naturaleza puntual de la divergencia.

IN ENGLISH
It is called a vector field divergence of outflow per unit volume in unit volume is infinitesimal.

The divergence of a vector field is a scalar field given the nature of the flow scale and nature of the divergence point.

i am according to the concept i have read for this reason i say:

From the definition of divergence, it follows a known identity and the divergence theorem:

Since the divergence of a vector field is a kind of derived volumetric flow from the field, it follows that the volume integral of the divergence, corresponding to the total exit flow field where it appears.

este es mi aporte en cuanto a rotacional

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.

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.

En coordenadas generalizadas el operador vectorial diferencial del rotacional

We say that a vector field F (x,y,z) is conservative if the flow field along a curve is independent of the way, just depends on the starting and ending points in circulation.

The conservative fields can be expressed as gradient of a scalar function,es decir existe una funcion escalar de puntos V (x, y, z) satisfying:

vector f = - vector v por v

Así, utilizando coordenadas cartesianas, si tenemos una función escalar F(x,y,z) , siendo F (presión, temperatura, altura, potencial, densidad electrónica etc.), el gradiente de dicha función sería:



Por lo tanto, el gradiente representa las derivadas parciales de una función escalar en un espacio vectorial, lo que va a producir una derivada vectorial. Existe un operador vectorial o símbolo matemático que representa dicha operación, consiste en el triángulo (delta) significativo del incremento pero invertido, o sea con el vértice hacia abajo: . Por su forma, se le denominó NABLA (el nabla era un instrumento musical de cuerda, tal como el arpa, empleado por sirios y persas), o ATLED (delta al revés). Este operador fue creado por Hamilton a mediados del siglo XIX.

De esta forma, el NABLA como operador matemático en coordenadas cartesianas es:

(2)

Por este motivo es un vector:

cuyo módulo es y (3)

Su dirección será tal que suponga la máxima variación de la función.

Como sentido de dicha variación siempre se tomará el creciente

Good morning to all,

We say that a vector field F (x, y, z) is conservative if the flow field along a curve is independent of the way, just depends on the starting and ending points in circulation.

The conservative fields can be expressed as gradient of a scalar function, ie there is a point scalar function V (x, y, z) satisfying:



So the calculation of circulation becomes:



The movement of a conservative field by a closed line is therefore zero:



If a vector field is conservative also meets these conditions:

A vector field corresponds to a physical quantity that requires multiple numbers for their description, such as a field of gravitational or electrical forces. Regarding the The gradient of a scalar function is a vector evaluated at a point indicates the direction in which the function increases faster. For example, if the function to which we refer is the temperature, the temperature gradient indicates the direction in which the temperature increases faster and agree with this energy flow J is in the opposite direction of temperature gradient

Divergence theorem

In calculating the net flow of a vector field F emerging from a closed surface S, what is done is simply add up the infinitesimal contributions of the different components of F are normal to the closed surface F (and pointing out S) over the entire surface. An alternative way of calculating this net flow (which often simplifies the calculation) is through the divergence theorem. Which states in its expression:
“The integral of normal component of a vector field on a closed surface S equals the integral of the divergence of the vector field through the volume V enclosed by surface S”
The mathematical expression of this theorem is:
∫S∫F.dS=∫∫∫▼.FdV
The importance of this theorem is that it allows the same calculation in two ways.

Rotational Properties.

1. If the scalar field f (x, y, z) has continuous partial derivatives of second order, then the rot (▼ f) = 0 (vector).
2. If F (x, y, z) is a conservative vector field then curl (F) = 0 (vector).
3. If the vector field F (x, y, z) is a function defined on all R3 whose components have continuous partial derivatives and curl (F) = 0 (vector), then F is a conservative vector field.

The curl of a vector field has its main physical interpretation when the vector function F (x, y, z) represents the fluid flow, the curl in this case is interpreted as introducing the fluid flow around a point (x0 , y0, z0).
If the vector field F represents the fluid flow and curl (F) = 0 (vector) then says that the fluid is irrotational.

By definition, a vector field is considered "conservative", if it is de gradient of a scalar function. Also, if a vector field is conservative, then it has a unique potential function.
By conservative it is also meant that it does not matter the path of integration, for the value of it is always the same. This condition is also called "independance of path".

Good morning
The definition of rotational and divergence of a vector field and the questions that the teacher suggested estan defined by many classmates (heidy, breiner, deivis, carlos ,clara, dairo, Nelson, dania, daniel) so I limitare to upload these definitions pues not tendria sense.
Encontre a tema respect exercise to for analyze and darnos account of where salen the resuts.
Function : f (x, y, z) = 6 x² i − x y² j + 0 k.
divergence
∇ • f = (∂/∂x i + ∂/∂y j + ∂/∂z k) • (fx i − fy j + fz k)
∇ • f = (∂/∂x i + ∂/∂y j + ∂/∂z k) • (6 x² i − x y² j + 0 k) = ∂/∂x (6 x²) + ∂/∂y (x y²)

∇ • f = 12 x + 2 xy
la divergence like observe is a scalar

rotational
∇ × f = (∂/∂y fz − ∂/∂z fy) i − (∂/∂x fz − ∂/∂z fx) j + (∂/∂x fy − ∂/∂y fx) k
∇ × f = (∂/∂x xy² − ∂/∂y 6x²) k

∇ × f = y² k
The rotational is a vector, than in this case, this oriented in address of axis z

Mathematically, a vector field is defined as a vector function of coordinates or as a special case of a transformation does not necessarily linear. , Where represents the vector space that serves as the vector space domain and acting
as a range.
one of the most important features of vector fields is that one conservative is conservative when there is a differentiable function ƒ such that F equals ⃗ ƒ gradient. This function is called the potential function ƒ
A vector field F is conservative ⃗ if VXF ⃗ = 0
hay otra, There are other features that is the vector field depends on the position in space and time

There is also talk of the curl of a vector field has a physical interpretation, where the main vector of the function F (x, y, z) represents the fluid flow, the curl, in this case is interpreted as the introduction of flow Fluid around a point (x0, y0, z0).
If the vector field F represents the fluid flow and curl (F) = 0 (vector) and then says that the fluid is irrotational