What Defines a Uniform Vector Field?

[yungman]

I cannot find the meaning of the uniform vector field. I know

$\hat z k_x+\hat y k_y +\hat z k_z$ is a uniform vector field if $k_x,k_y,k_z$ are constants.

Does this means a uniform vector field:

1) Points to the same direction in all locations?

2) Have the same magnitude in all locations?

3) The curl of the vector fields are zero in all location implies no circling, all pointing in one direction.

4) Divergence are zero also imply all pointing in the same direction.

5) So the uniform vector fields are all parallel to each other.

What else I missed?

 

[issacnewton]

1,2,3,4 are true

5 may not be true if two vector fields are not parallel

 

[yungman]

1,2,3,4 are true

5 may not be true if two vector fields are not parallel

Thanks for the quick answer. But if the vector field is uniform, then they are all parallel which is implied by 1).

 

[issacnewton]

A given vector field has the same direction and magnitude everywhere. But two different
uniform vector fields may not have the same directions or the same magnitude.

 

[Studiot]

Try looking at it this way.

A vector field assigns only one vector to every point, but every vector so assigned may be different.

A uniform vector field assigns the same vector (or a copy if you will) to every point.

The effect of this is that there is no change in vector from point to point so all the measures of change are zero.

Two different uniform fields, using the same set of points, are only the same if they both assign the same vector to every point, otherwise they are different by a rotation and a scaling.
Obviously if the two fields cover different sets of points they are also different even if they use the same vector.

 

[yungman]

Try looking at it this way.

A vector field assigns only one vector to every point, but every vector so assigned may be different.

A uniform vector field assigns the same vector (or a copy if you will) to every point.

The effect of this is that there is no change in vector from point to point so all the measures of change are zero.

Two different uniform fields, using the same set of points, are only the same if they both assign the same vector to every point, otherwise they are different by a rotation and a scaling.
Obviously if the two fields cover different sets of points they are also different even if they use the same vector.

Thanks for your time. My question refer to only one uniform vector field only. I understand in case of two uniform vector field, they are not parallel. But with only one uniform vector field, all the fields at all point have to be parallel.

My whole point of trying to confirm that for a uniform field, the field cannot be in circular pattern or any other complicated patterns.

Also for a uniform vector field A:

$$\nabla \cdot \vec A \;=\; \nabla X \vec A \;=\; 0$$

Am I correct?

 

[Studiot]

My question refer to only one uniform vector field only. ... But with only one uniform vector field, all the fields at all point have to be parallel.

I'm sorry but this does not make sense to me.

At any point there is only one vector.

This is true of any vector field.

Not only is it true - it is a vital condition, without which we could not do vector calculus.

Once we have cleared this I can reply to the rest of your points as you seem to have the rest of the ideas pretty well taped.

 

[yungman]

I'm sorry but this does not make sense to me.

At any point there is only one vector.

This is true of any vector field.

No only is it true - it is a vital condition, without which we could not do vector calculus.

Once we have cleared this I can reply to the rest of your points as you seem to have the rest of the ideas pretty well taped.

What I meant is I only defined a single uniform vector field for all points say:

$$\vec A = \hat x 2 + \hat y 3 +\hat z 4$$

so all the vector fields are parallel with constant magnitude and constant direction.

 

[HallsofIvy]

You are again talking about "all the vector fields" when you say you have defined only a single vector field. If you mean "are all the vectors in a uniform vector field are parallel", then the answer is yes. In fact, they are all equal.

 

[yungman]

You are again talking about "all the vector fields" when you say you have defined only a single vector field. If you mean "are all the vectors in a uniform vector field are parallel", then the answer is yes. In fact, they are all equal.

Thanks. I guess I have a hard time to express this. I meant the vector field at all points. The single vector field for all points.

I think I get the idea, thanks

Alan

 

[Studiot]

It may be that you are thinking of the $$\hat{x}$$; $$\hat{y}$$; $$\hat{z}$$ components as separate vector fields.

This view is not productive for this purpose.

I suggest you abandon, or at least delay, the decomposition of your vector $$\vec{A}$$ into components.

Don't forget the vector is the same arrow, regardless of coordinate system. How would you for instance describe the 'component fields' if polar coordinates were used?

You are meeting the difference between mathematicians' vectors and physicists' vectors here.

Mathematicians' vectors correspond to unbound vectors
Physicists' vectors correspond to bound vectors, although sometimes they use unbound ones, as in this case.

You cannot have bound vectors in a vector field and do vector calculus with them. That is why mathematicians only admit unbound vectors. So, for instance, you cannot make a vector field of position vectors, since they all emerge from the origin.

By definition two vectors are equal if they have the same magnitude and direction, regardless of their location in space.
Taking the unbound view is equivalent to stating that they are the same vector.

This is what I mean when I say that a uniform vector field assigns the same vector to every point.

Clearly any vector is parallel to itself so any bunch of the same vectors are parallel.
Any measure of change such as the gradient, divergence or curl of the field are precisely zero since there is no change. This applies to the entire field.

The dot or cross product of any vector in the field with another given vector will be the same at any point since the field vectors are all the same.


go well

 

[yungman]

Hi Studiot

I am not thinking of [itex]\hat x, \hat y, \hat z[/tex] as three separate field. The confusion started when I wrote in 5) "The uniform vector fields". What I meant is the same vector in all position.

From all this, my understanding that a uniform vector field:

$$\vec A =\hat x k_x + \hat y k_y +\hat z k_x$$

where $k_x, k_y,k_z$ are constants respect to x, y and z. That's the only way you can have the uniform vector field that has the same magnitude and direction at all points.

Therefore for a uniform vector field A:

$$\nabla \cdot \vec A = \nabla X \vec A =0$$