ROTACIONAL Y DIVERGENCIA DE UN CAMPO VECTORIAL

The gradient is a vectorial field and found scalars vectorials.
The Rotacional is a vecorial shield and we can found vectorial shields in 3D(tridimensionales); and the diveregencia is a shield scalar and we can found vectorials field in general.
When we want measure the circulation of a shield vectorial like a function of the coordenadas, it use a vectorial function called Rotacional, que mide the circulation for unity of área when the área tiende a zero(0) on each point of the space en donde se encuntra definido el campo.
When the rotacional is nulo on the points of a region, se dice que the shield is irrotacional or conservative en dicha región.

Profe hay unas palabritas en español..es q se me olvidaron..en inglés y tengo q aprovechar antes de q me vuelva a quedar sin pc.....
Gracias

I want to show you the next exercices of a shield vectorial conservative, rotacional and divergencia:

1 DETERMINE SI EL CAMPO VECTORIAL ES CONSERVATIVO

f(x,y)=5y^2 (3yi-xj)
f(x,y)=15y^3 i- 5xy^2 j
∂/∂x 15y^3 i-∂/∂x 5xy^2 j
45y^2 i- 5y^2 j

∇f(x,y)=15y^3 i- 5xy^2 j
f_x (x,y)= 15y^3
f_y (x,y)= 5xy^2

∫15y^(3) dx= 15xy^(3)+c
∫-5xy^(2 ) dy=-5xy^3/3+c
f(x,y)=15xy^(3 )-5xy^3/3+c

Nos podemos dar cuenta que este campo vectorial no es conservativo ya que al reconstruir las derivadas parciales no se obtuvo la función original.


2 CALCULAR EL CAMPO ROTACIONAL DEL CAMPO VECTORIAL EN EL PUNTO DADO
f(x,y,z)=(xyz)i+yj+zk P (1,2,1)

i j k

∂/∂x ∂/∂y ∂/∂z

xyz y z


= (∂y/∂z- ∂z/∂y)i-(∂xyz/∂z-∂z/∂x)j+(∂xyz/∂y-∂y/∂x)k
= ( 0 - 0 )i – (xy - 0)j + (xz – 0)k * -1
= 2j – k

se remplazan los valores correspondientes a x,y,z
por las coordenadas del punto P.


El rotacional para el campo vectorial en el punto (1,2,1) es 2j-k.

Del ejercicio anterior podemos demostrar el teorema que plantea que si
F(x,y,z) = Mi + Nj + Pk es un campo vectorial y M,N,P tiene segundas
derivadas parciales continuas entonces Div (rot F) = 0

Si aplicamos div entonces

rot F(z,y,z) = 2j – k en el P(1,2,1) entonces

∂2/∂y- ∂1/∂z =0


3 CALCULAR LA DIVERGENCIA EN EL CAMPO VECTORIAL F EN EL PUNTO DADO.

F(xyz)=x^2 z i-2xz j+yz k P (2,-1,3)
(∂x^2 z )/∂x- ∂2xz/∂y+ ∂yz/∂z= se remplazan los valores
2xz + 1= 2(2*3)+1=13 correspondientes a x,y,z
por las coordenadas del punto P.






F(xyz)=e^x senyi-e^x cosy j P (0,0,3)

(∂e^x seny )/∂x- (∂e^x cosy )/∂y= se remplazan los valores
e^x seny+e^x seny = correspondientes a x,y,z
sen(0)+sen(0) = 0 por las coordenadas del punto P.


Disculpen la ausencia del ingles, espero que los ejercicios sean entendibles trate de hacerlos lo más sencillo posible......

Conservative fields

We say that a vector field F is conservative if it is true that the movement of the field along any closed curve is zero

∫c . ds = 0 para toda curva C

If a conservative vector field is always a potential function and is unique within an arbitrary constant (which vanishes when deriving). If the field is not conservative potential function does not exist.
Aforce field is not conservative if your rotor is diferent to zero

The divergence of a vector field measures the difference between the inflow and outflow of a vector field on the surface surrounding a volume control.

sea F a vector field in a open ball B en R3 tal que
F(X,Y,Z)= M(X,Y,Z)i + N(X,Y,Z)j + R(X,Y,Z)k

so the divergence of F, which is represented by div F, is defined by

div F(x,y,z)= ∂M/∂x + ∂N/∂y + ∂R/∂z
if these partial derivatives exist

Físicamente un campo vectorial representa la distribución espacial de una magnitud vectorial.

Matemáticamente se define un campo vectorial como una función vectorial de las coordenadas o como un caso especial de una transformación no necesariamente lineal. , en donde representa el espacio vectorial que hace las veces de dominio y el espacio vectorial que actúa como rango.

rotational is a vector operator that shows the trend of a vector field to induce rotation around a point

sea F a vector field in a open ball B en R3 tal que
F(X,Y,Z)= M(X,Y,Z)i + N(X,Y,Z)j + R(X,Y,Z)k

so the rotational of F, which is represented by div F, is defined by
( ∂R/∂y - ∂N/∂z )i + ( ∂M/∂z - ∂R/∂x 1)j + ( ∂N/∂x - ∂M/∂y )k
if these partial derivatives exist

Definición:

La Divergencia de un campo vectorial
= P + Q + R
es el escalar

Div= P/ X + Q/ y + R/ Z =V °
Ejemplo:
F{x, y} = {x, y}
Div F = 2

well , fisrt of all good afternoon guys , i'm sorry about giving my participation so late , it's just that i was really busy these and it was impossible for me to do it before, anyways , i've prepared a kind of summary and i'd like to share it with you guys, ok here i go .

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.
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.
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.
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.

F : D ⊂ ℜ →ℜ / F x, y, z = F x, y, z , F x, y, z , F x, y, z , donde 1 F , 2 F y
∂Determine if the vectorial field is defined by :
F (x, y, z) = (2xy, x2 + 2yz, y2 ) es un campo conservativo.
Solución. Por propiedades del rotacional, un campo vectorial es conservativo si
rot (F ) = 0

Propiedades del Rotacional.
1. Si el campo escalar f(x,y,z) tiene derivadas parciales continuas de segundo orden
entonces el rot(∇f ) =0

.
2. Si F(x,y,z) es un campo vectorial conservativo entonces rot (F ) = 0

.
3. Si el campo vectorial F(x,y,z) es una función definida sobre todo R3 cuyas
componentes tienen derivadas parciales continuas y el rot (F ) = 0

entonces F es un
campo vectorial conservativo.
El

in this forum have been repeated much the concepts.
that already know

Let F be a vector function of the variables x, y, z. The divergence of F is defined as



being Fx, Fy, Fz components of F. The divergence of F is a scalar function (such as expected in a dot product). To get an idea of what the divergence of F, consider that F is a velocity field of fluid and take a small element of volume v =   x  and  z, as illustrated in Figure 1. The partial derivative of Fx with respect to x, according to the definition (Eq. 2), is a difference between what comes out at x +  x and what enters the face at x, the other two components do not contribute Fy and Fz nothing like the difference between these faces of the volume element. Similarly, the partial derivative of F respect to y, indicates a difference between input and output by the faces on and, and y +  y; and the partial derivative of Fz respect to z between the faces in z, and z +  z. Then the divergence of the velocity field is a measure of how F changes the volume per unit time and volume (expansion and contraction). This means that the divergence of a vector field F is a relative measure of what enters and leaves in a volume element

Calcular el flujo saliente del campo (y, z.x, 1) a través de la esfera (x - a) ² + (y - b) ² + (z - c) ² = R ².

Hallamos la divergencia del campo:
F = (y, z.x, 1) Þ ÑF = (0 + 0 + 0) Þ ÑF = 0

Como la divergencia del campo es nula, el flujo del mismo a través de cualquier superficie es nulo.

Flujo = ∫∫ ∂T F.dS = ∫∫∫ T div F.dT = ∫∫∫ T 0.dx.dy.dz = 0

The definition of the operations of divergence and curl of a vector field F involves the partial derivatives of the components of F. The following pair of theorems relating to these operations and the features of vector fields.



When considering the motion of a particle we say that if known to the velocity v versus time t, with t in the interval [a, b], and the position r0 for a time t then we can determine the position r to every time in a unique way. This is because the velocity is the change in position in time,

teorem of divergence
Is a differentiable vector function defined on a set and is a closed set bounded by a border or boundary surface (which is a smooth manifold) and let the normal vector at each point on the surface, then it holds that:

Good afternoons everybody

No necesariamente para determinar el rotacional de un campo vectorial se requiere de una función potencial ya que este se puede calcular con el producto cruz del vector gradiente (que corresponde a las derivadas parciales de los diferentes componentes del vector) y la función vectorial, es importante por consiguiente que las derivadas parciales existan; es importante tener en cuenta además que el rotacional va a ser cero en caso de que la función vectorial tenga función potencial.

Thanks, Fernando Barreto.

Good afternoons everybody,

Para visualizar la divergencia y el rotacional, sugiero tengamos en cuenta la siguiente interpretación física:
Si F denota el campo de velocidad de un fluido, entonces div Fen un punto P, mide la tendencia de ese fluido a alejarse de P ( Div F >0) y a acumularse en torno a P ( Div F <0 ). Por otro lado el Rot F elige la direccion del eje en torno del cual giran el fluido mas rapidamente y |Rot F| es una medida de la rapidez de tal rotación.

Thanks, Fernando Barreto.

Good afternoons everybody,

Teniendo en cuenta la definicion de divergencia y rotacional tenemos que:
Sea F = Mi + Nj + Pk, un campo vectorial para el q las prieras derivadas parciales M, N y P existen, entonces
Sabiendo que: ∇ = ∂/∂x i+∂/∂y j+ ∂/∂z k
tenemos : ∇∙F=(∂/∂x i+∂/∂y j+ ∂/∂z k)∙(Mi + Nj + Pk); lo que nos proporciona:
Div F = ∂M/∂x+∂N/∂y+∂P/∂z
Rot F = (∂P/∂y-∂N/∂z)i+ (∂M/∂z-∂P/∂x)+ (∂N/∂x-∂M/∂y)k

Thanks, Fernando Barreto.

Good Day,

Thanks to everyone for their excellent contributions, i want to share this video :

Divergence of a vector field: Vector Calculus

I present a simple example where I compute the divergence of a given vector field. I give a rough interpretation of the physical meaning of divergence. Such an example is seen in 2nd year university mathematics courses.

Good afternoons everybody,

Ahora teniendo que: F (x,y,z) = x^2yz i + 3xyz^3 j + (x^2-z^2) k. Calcular Div F y Rot F.

Div F = ∇∙F= 2xyz + 3xzx^2 - 2z
Rot F = ∇× F= |[i j k] ; [∂/∂x ∂/∂y ∂/∂z ] ; [x^2yz 3xyz^3 x^2-z^2]|

Rot F = -(9xyz^2) i – (2x – x^2y) j + (3yz^3 – x^2z)k .

Thanks, Fernando Barreto.

This is another video, but I have not seen completely.

This video discusses the 'divergence' of a vector field. Divergence is one of the basic operations of vector calculus and, loosely speaking, may be thought of as a type of derivative in vector calculus.

Dr Chris Tisdell introduces the idea of divergence, discusses some examples and also gives a physical interpretation of divergence in terms of 'flux density'.

Determined if the funtión is conservative and if it is conservative, calcúlate a potencial funtión.


F(x,y,z)= senyi – xcosyj + k

Here you can see the cross product between the gradiente and the funtion in a matrix:

∇xf= i; ∂/∂x j; ∂/∂y k;∂/∂z
seny xcosy 1

Then you make the matriz and you find if it’s rotational:

=[∂/∂y-∂/∂z (-x cosy ) ]i-[∂/∂x-∂/∂z (siny ) ]j+[∂/∂x (-x cosy )-∂/∂y (siny ) ]k

Here you can see that the resul is different to zero and it’s rotational:

-cosy - cosy = -2 cosy

If the result is zero then it’s irrotational.

Then we equal the gradiente and the funtion:

f(x,y,z)=∇f(x,y,z)
siny i+ cosy j +(1)k =(∂f/∂x i+∂f/∂y j+∂f/∂z k
We integramos:
∂f/∂x=siny→ ∫siny dx→ x siny + c1

∂f/∂y=cosy→ ∫cosy dy→ siny + c2

∂f/∂z=1→ ∫dz→ z + c3
Then we get a potential funtion:

f=x siny +siny +z + c4

The divergence of a vector field is a scalar field, defined as the field flux vector per unit volume

where S is a closed surface is reduced to a point on the boundary. The symbol represents the opererador.

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, says the field has springs. If the difference is negative, is 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.

They are called scalar field sources to the scalar field is obtained from the divergence

talking about potential function we can say that the power law of Stevens is a proposed relationship between the magnitude of a physical stimulus and its perceived intensity or strength. Is widely considered to replace the Weber-Fechner law on the basis that describes a wider range of sensations, although critics discussed the possibility that the validity of the law is contingent on the virtue of the approaches to the extent that perceived intensity used in relevant experiments. The theory is named in honor of U.S. psychophysical refenrecia Stanley Smith who makes this law by a potential function.

the divergence is F(x,y)=Mi + Nj is

div F(x,y) = ∇. f(x,y)==∂M/∂x +∂N/∂Y.

the divergence F(x,y,z)=Mi + Nj + PK is

div F(x,y,z) = ∇.f(x,y,z)==∂M/∂x +∂N/∂Y + ∂P/∂z.
div F=0, Then is said that F is divergence void.

The notion of power has been defined so far for a simple alternative. In the parametric framework, if the hypothesis is composite, is preferably used the power function, is available as a sample of the law depends on the parameter. It is assumed that a certain hypothesis, a rejection rule is defined. even potential function is called the parameter value associated with it where an Act of the sample wing, where the value of the parameter and the type is simple hipotisis then the value of the potential function