ROTACIONAL Y DIVERGENCIA DE UN CAMPO VECTORIAL

for Complementing with my words, i have that the following example of the concept of rotational
Example:
When a full tank (any container) is being emptied through a drain, around of this eddies are formed, that are a very
intuitive picture of the circulation of the velocity vector. The drain would be the source of the movement, the cause of the rotation around.
this is what we call as rotational.

as we all have commented on the theory I take the trouble to edit an exercise to see the application of the divergence theorem

i forget the last part of the exercise but i have it here

if looking at these I can see that has served many uses in this area of disagreement but I think they miss important aspect of great circles with figures compontes with normal and tangential

this is a fundamental application that we can not leave out, this idea is mathematically expresacomo circulation of the vector field, where the curve on integrating reduces to a point

sila application of this theorem is Gauss's law on the electric field, Gauss law for magnetism which is recognized because it is an application of Maxwell's equation and Gauss's law of gravity. And you can find another application and is determined for us as an engineering student.

a vector field represents the spatial distribution of a vector quantity vector fields an example of the speed functions are associated with the trajectories of particles or differential volume of a substance flow conditions either laminar or turbulent.

and moreover we know that the gradient of a scalar field, is another example of vector field, because the magnitude and direction of the gradient of a scalar field is a function of the coordinates

In vector calculus, a vector potential is a vector field whose curl is a vector field. This is analogous to the scalar potential, a scalar field whose negative gradient is also a vector field.
Formally, given a vector field v, a vector potential A is a vector field such that
V = ∇ * A
If a vector field v admits a vector potential A, then the equality ∇ * (∇ * A) = 0
the divergence of the curl is zero we have ∇ * ∇ * V = (∇ * A) = 0 which implies that v must be a solenoidal vector field.
The vector potential given by a solenoidal field is not unique. If A is a potential vector for v, then also
A + ∇ m where m is any differentiable scalar function. This follows from the fact that the curl of the gradient is zero.
The non-uniqueness leads to a degree of freedom in the formulation of electrodynamics, or standard free, and requires choosing a standard.

this right mate vectorialrepresenta field because the distribution of the fields but when it has features that devemos acimilar efficiently for the full intellectual development.

Vector fields can be constructed from scalar fields using vector differential operator gradient which leads to the following definition.

A vector field Ck F about X is called a field gradient or conservative field if there exists a function Ck +1 actual values f: X → R (a scalar field) so that


The line integral on any closed curve (eg γ (to) = Γ (b)) In a field gradient is always zero.

continuing with what has been said Vector Fields. A vector field in Rn is    R is a mapping F: W → Rn which assigns to each point x of its domain a vector F (x). If n = 2, F is called a vector field in the plane, and if n = 3, F is a vector field of space.
Display an arrow F adhering to each point (Fig. 4.3.1). R    In contrast, an application f: A n → R that assigns a number to each point is a scalar field. A vector field F (x, y, z) in R3 has three components scalar fields F1, F2 and F3, so
F (x, y, z) = (F1 (x, y, z), F2 (x, y, z), F3 (x, y, z)).
Similarly, a vector field Rn has n components F1, ..., Fn. If each component is a function Ck, we say that the vector field F is of class Ck. It would assume that vector fields are at least class C1, unless otherwise stated.

Vector fields can be constructed from scalar fields using differential vector gradient operator which leads to the following definition.
You have to have known this that this is a grdiente of a scalar field at a point is a vector, defined as the only means of finding the directional derivative in any direction

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 domain and the vector space that acts as a range.



The field illustrated in the above equation is a vector field Since the vector function has three components and each component is a function of three independent variables.

When we model the stress distribution in a structure, the distribution of electromagnetic forces of nature or gravity in space is done using vector fields.

criterion (1) Is called potential or potential energy. The minus sign of this approach is a convention and has a deep meaning, though its meaning was argued on the principle variational the Lagrangian mechanics and, for now, operates on a voluntary basis. The basis of this convention can be clarified through the following example: in the vicinity of the land surface is the mass m in a gravitational potential at a height h = y under a acceleration of gravity g> 0Approximately v (y) = + m g y. Due to coordinate system Earth's surface is positive when directed upward, he must be negative when downward. Calculate the force of the first criterion and is obtained:



This shows that the force exerted, as expected, towards the center of the Earth.

Demonstration of equivalence of the criteria [edit]There are three equivalent criteria for determining a force field is conservative ((1) (2) And (3)). The first criterion is about the definition of a conservative force field, the other two are other formulations of the first criterion. Many times the force field is defined in a "direct" through the second criterion. So, you have to work in a conservative field is independent of path.

It is a closed path C in a conservative field, point 1 on the way to point 2 then S1 by S2 path back to point 1.


Two roads either in a conservative field of forces.

The integral on the road will be closed:



For all paths S1, S2 this integral would be S1 + (-S2) of zero when:



It would also be:

a vectorial field f (x,y,z)is conservative if the field traffic on a curve does not depend on the route, i just depends on the initials and finals points of the traffic.
conseravtive foelds can be expressed as a gradient of saclar function, it means that there is sacrlar function from the point V(xyz) that fullfills f= -gradiente x v
the traffic fo a conservative field is a closed line so it's zero.

i know that my english is not the best in the world but i tried to do my best.

hello,
It is necessary to remember some concepts of great importanica as:
Gradient of a function to climb
Definition Is a function to climb (or I stand out(camp) to climb) f (x, y, z). The increase of f from the point of coordinates (x, y, z) to the point of coordinates (x + x, y + y, z + z) is
f = f (x + x, y + y, z + z)  f (x, y, z)
If we use Taylor's development we can write this increase as:

If x y z are small we can bring the previous expression near for the sum of the linear terms.

The latter expression we can write as a product climb of two vectors

3 .Why to obtain the curl of a vectorial field do you need a potential function? Since the potential function is the measure of the variation of the variable or complicated variables, the potential might be a vector or to climb, but it(he) is necessary because the potential says all that takes the changeable changes as the use of any disturbance between(among) an initial and the final condition(state). The Uses the use for this theorem is Gauss's law concerning (it)(he),she)) (brings over) of the electrical field, Gauss's law for the magnetism that is recognized because they are a use of equation of Maxwell and of Gauss's law for the gravity. And you can find another use for this subject in the fluid mechanics and the quantum mechanics.

A gradient vector field F = L f is called conservative because the scalar function f is called a potential function of the field F
*Una part del foro
*Above criterion is applicable to vector fields in the plane, just remember that these fields can be interpreted as vector fields in three dimensional space, for which the third component is zero and the other two components are independent of the coordinate z.

Before speaking of gradient, rotational or divergence is necessary to define the nabla vector differential operator represented by ∇

∇ ⃗ = ∂ / ∂ x i + ∂ / ∂ y j + ∂ / ∂ z k

The nabla vector operator has similar properties to those of ordinary vectors and is useful in the implementation of 3 magnitudes in the practice known as gradient, divergence and curl
Gradient:
Let the function Ф (x, y, z) that defines a differentiable scalar field. The gradient represented by ∇ Ф Ф is given by:

(∇ Ф) ⃗ = (∂ Ф / ∂ x)i + (∂ Ф / ∂ y)j + (∂ Ф / ∂ z)k


Rotational
If V ⃗ (x, y, z), is a differentiable vector field, the rotational of V ⃗, represented by ∇ ⃗ XV ⃗defined
∇ ⃗ XV ⃗ = | ■ (i ⃗ & J & ⃗ ⃗ k @ ∂ / ∂ x y ∂ / ∂ y / ∂ / ∂ z @ V_1 & V_2 & V_3 )|=[( ∂ V_3) / ∂ y-(∂ V_2) / ∂ z] i ⃗ - [ (∂ V_3) / ∂ x-(∂ V_1) / ∂ z] ⃗ j + [(∂ V_2) / ∂ x-(∂ V_1) / ∂ y] k ⃗

Recall that the rotational is not commutative

Divergence:
Let V ⃗ (x, y, z) = v_1 i + j + V_2 k V_3
a vector function defined and differentiable in each of the points x, y, z in a certain region of space.
The divergence of V ⃗ represented by ∇ ⃗ ∙ V ⃗ is given by: ∇ ⃗ ∙ ⃗ V = (∂ / ∂ xi + ∂ / ∂ yj + ∂ / ∂ zk) ∙ (v_1 i + V_2 j + V_3 k) = (∂ V_1) / ∂ xi + (∂ V_2) ZK / ∂ yj + (∂ V_3) / ∂

The divergence is not commutative

POR LIVIS VASQUEZ VALLE
The gradient usually indicates a direction in space which is seen as a variation of a particular property or physical quantity.
In other contexts gradient is used informally to indicate the existence of progressive or gradual variation in a specific area, not necessarily related to the physical layout of a particular size or ownership.

The gradient of a scalar field, which is differentiable in the neighborhood of a point, a vector is defined as the only means of finding the directional derivative in any direction as:

being a unit vector and the directional derivative of the address, which reports the rate of change of the scalar field to move around according to the following address:

An equivalent way to define the gradient is the only vector that when multiplied by any infinitesimal displacement, gives the differential of the scalar field:

With the above definition, the gradient is uniquely characterized. The gradient is expressed alternatively by using the nabla operator:

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 nabla operator.
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 of

The divergence of a vector field is related to the flow through the Gauss theorem and divergence theorem.
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:

Here, ΔS is the surface area resting on the curve C, which reduces to a 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.
Although the curl of a field about a point is not zero does not imply that the field lines rotate around that point and encieren. For example, the velocity field of a fluid flowing through a pipe (known as Poiseuille profile) has a non-zero curl everywhere except the central axis, although the current flows in a straight line
Starting from the definition by limited, it can be shown that the expression, in Cartesian coordinates, the curl is

can be expressed more concisely using the nabla operator as a vector product, calculable by determining:

good night attendees

A vector field is determined whether the divergence and curl are specified the same at all points. This theorem simply says that any vector field can be described as the sum of two components, an irrotational and a solenoidal. Since a vector field is determined by its divergence and curl, both terms are defined as its where the curl is a vector operator that shows the trend of a vector field to induce rotation to a point the magnitude of this vector one rotacionalconstituye measured by moving the particles to reededor axis. The divergence is the scalar quantity
buenas noches

For the divergence Suppose a point (P) within a small volume ( v) limited in turn by a surface (S). In this case the volume is a prism edges (  x,  y  z,) parallel to the axes x, y and z respectively. All in a space where it is assumed that there is a vector field (F). The flow field (F) through the surface (S). If this flow divide it by  v, would the flow per unit volume, then called divergence of F (div F) to limit as  v approaches zero,

Hi friends
I cannot participate in this forum early because i had a little problem with my pc.
So, the first thing that I will said is a concept of vector field in mathematical, it is a building of vector calculi that associative a vector with each point in the euclídeo space.
A vector field is conservative if the work realized to move a particle between two points is independence of the trajectory followed between these points.
A conservative field can be expressed with the gradient of the scalar function. When the lines of strength in any region of space where the field is definite follow a close trajectory we said that the field has a circulation in this region.
The divergence is the exit flow for volume unity when the volume unity is infinitesimal.
Thanks