# About Vector fields and vector valued functions

• shippo113
In summary: That is, at every point within the vector field's region there is a vector assigned to that point.TranslationVector fields are often used in physics to model forces. We can assign a force vector to every point in a region of space, and then we can study the resulting field of force vectors to understand the motion that those forces will produce. This is especially useful in fluid dynamics, where we might want to understand the flow of a liquid or gas through a particular region.A potential field is a special type of vector field where the field can be derived from a scalar function (i.e. a function that returns a single scalar value at every point within the field's region). This means that the field can be thought of as a gradient,
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.

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)

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

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)

shippo113 said:
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.

LCKurtz said:
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.

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.

## 1. What is a vector field?

A vector field is a mathematical concept in which a vector is assigned to every point in a given space. These vectors represent the magnitude and direction of a physical quantity, such as force or velocity, at that specific point.

## 2. How are vector fields represented?

Vector fields can be represented visually using arrows, where the length and direction of the arrows correspond to the magnitude and direction of the vector at each point. They can also be represented algebraically using equations or mathematical functions.

## 3. What is the difference between a scalar field and a vector field?

A scalar field assigns a single numerical value to each point in space, while a vector field assigns a vector to each point. Scalar fields represent quantities such as temperature or pressure, while vector fields represent quantities with both magnitude and direction, such as force or velocity.

## 4. What are some applications of vector fields?

Vector fields have a wide range of applications in physics, engineering, and other scientific fields. Some examples include weather forecasting, fluid dynamics, electric and magnetic fields, and motion planning in robotics.

## 5. How are vector fields and vector valued functions related?

A vector valued function is a mathematical function that outputs a vector. Vector fields can be represented by vector valued functions, where the input is a point in space and the output is a vector at that point. In other words, vector fields can be thought of as a collection of vector valued functions.

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