About Vector fields and vector valued functions

[shippo113]

how do I make difference between vector valued functions and vector fields, I am confused how they differ and how are they same? Which is used with what?

What about a function F(x,y,z,t) = (f1(x,y,z,t), f2(x,y,z,t), f3(x,y,z,t)) which maps R4 to R3, what type of function is this?

F(x,y) = x^2 + 2y
F(x,y,z) = x^2 + y^2 + 10z
F(x,y,z,t) = 3x + 2y - z^2 + t

F(t) = (t^2 , 0 , t^3)
F(x,y) = (x^2 + y, y^2 - x)
F(x,y,z) = (z+xy,y^2+z, 2x-z)
F(x,y,z,t) = (z-t^2,y^2, xt+z)

which of these functions is what type and why? Also what is a potential field. Finally how do they all (vector field, potential field, vector valued function) differ when compared to curve/surface in R3?
A short description shall suffice, thanks. Its just that I can't find the answer summarised anywhere making good distinction.

 

[Petr Mugver]

I'm not sure about this, but I think that, for mathematicians, fields and functions are the same thing: f(x,y,z) is a scalar field, (f(x,y,z), g(x,y,z), h(x,y,z)) is a vector field etc.
For physician a field is not just a general function, but a function that behaves properly under certain groups of tranformations. For instance, in classical mechanics fields must tranform correctly under rotations, translations and velocity transformations. To cite the easiest example

f = x + y^2 + z^3

is not a scalar field, while

g = log(x^2 + y^2 + z^2)

is a scalar field. This because, for example, if R is a rotation

f(r) is not equal to f(Rr), while

g(r) = g(Rr)

 

[shippo113]

I just want to know the distinction between vector valued functions and vector fields. And how do they relate to a scalar field.

 

[Petr Mugver]

Following the philosophy of the previous post, a vector field is a vector function that behaves properly under, for example, rotations. So if R is a rotation a vecor field v(x) must satisfy

v(Rx) = Rv(x)

 

[LCKurtz]

how do I make difference between vector valued functions and vector fields, I am confused how they differ and how are they same? Which is used with what?

F(x,y) = x^2 + 2y
F(x,y,z) = x^2 + y^2 + 10z
F(x,y,z,t) = 3x + 2y - z^2 + t

These are all scalars. I would call them scalar functions. They could each represent, for example, the temperature at a point (x,y) (for the first two) or (x,y,z) for the third. The third is also a function of t, perhaps representing temperature varying with respect to time.

F(t) = (t^2 , 0 , t^3)
F(x,y) = (x^2 + y, y^2 - x)
F(x,y,z) = (z+xy,y^2+z, 2x-z)
F(x,y,z,t) = (z-t^2,y^2, xt+z)

The first geometrically represents a space curve. The last three are all 2D or 3D vector functions. They are vector fields. They might represent, for example, a force vector at (x,y) or (x,y,z), with the last representing a field changing with time. I am not familiar with the Physics restrictions of what is called a field mentioned by others.

 

[pmsrw3]

I am not familiar with the Physics restrictions of what is called a field mentioned by others.

It's a little complicated. The problem is that you can't describe them just by looking at the function itself. You also have to specify a set of coordinate transformations under which the field should remain invariant, and describe how the field behaves under those transformations. "Remains invariant" doesn't mean that the components of the field are unchanged.

The simplest case is rotations. It's (believed to be) a fundamental symmetry of nature that no direction in space is special. Thus, every fundamental law of physics should have the property that it holds no matter how you rotate the coordinate system. If your equations are written in terms of fields that transform appropriately, then they will automatically have this property.

But if it were just a matter of rotation, this apparatus wouldn't be worth the trouble. Where it really comes into play is in relativity, especially general relativity. GR says that the laws of physics are invariant under a very broad class of coordinate transformations. All the equations of GR deal with fields that transform appropriately: scalar fields, vector fields, or more generally tensor fields. This is a key constraint on any physical theory, so it's valuable to physicists.

 

[Studiot]

A vector field is a vector function of sorts.

Translation

A vector function assigns a single vector to specific points in a particular region of space, but not necessarily every one.

Where there is a single vector at every point in a particular region we say that constitutes a vector field in that region.