Is There a Difference Between a Vector Field and a Vector Function?

[uiulic]

My related questions

1 Is there any difference between 'vector field' and 'vector function'? 'vector function' is also called 'vector-valued function' (Thomas calculus). According to their definitions, they are all the same things to me. And they are all some kind of mapping, which assigns a vector to every point in a space or manifold or whatever.

2 A real-valued function is also called scalar function.(Thomas calculus).According to the terminology used in 1, then it seems natural to refer to scalar function as 'scalar field'. But 'scalar field' by (Halmos, finite-dimensional vector spaces ) is not a function, in my opinion at the moment.

3 If my analysis in 2 is reasonable, then the terminology in 1 is not reasonable?

4 Is there any relation between the 'field' (used in vector field ) and the 'field' used in finite-dimensional vector spaces (Halmos).The latter is a set where number/scalar lives.

Could you check and/or confirm the above points and tell what are the standard contemporary terminology used?

Thanks

GG

 

[uiulic]

If you have accepted vector field = vector function, then there is no serious misunderstanding by saying vector field = vector function=vector mapping (since function=mapping in my eyes).

Then is it ok if I say tensor field = tensor function = tensor valued function =tensor mapping??

Thanks

 

[pmb_phy]

1 Is there any difference between 'vector field' and 'vector function'? 'vector function' is also called 'vector-valued function' (Thomas calculus).

No. The two terms are synonymous. But it need not define a vector at every point in space. E.g. the acceleration of a particle is a vector valued function but it is not defined through all 3 space. The domain of this function is the set of real numbers (i.e. t = time)

2 A real-valued function is also called scalar function.(Thomas calculus).According to the terminology used in 1, then it seems natural to refer to scalar function as 'scalar field'. But 'scalar field' by (Halmos, finite-dimensional vector spaces ) is not a function, in my opinion at the moment.

A function is a rule that assigns an element of on space to an element of another space. S scalar field is a function which points in R³ to numbers in R. An example of a scalar field is temperature.

3 If my analysis in 2 is reasonable, then the terminology in 1 is not reasonable?

Sorry. I don't see it.

4 Is there any relation between the 'field' (used in vector field ) and the 'field' used in finite-dimensional vector spaces (Halmos).The latter is a set where number/scalar lives.

Sorry but it has been close to 20 years since I picked up an abstract algebra text.

Best regards

Pete

ps - I may be off line for a day or two. I cut my thumb and this typing is really painfull. But at least I was able to sweat out this one.

 

[uiulic]

Thanks very much. Pete

You have confirmed questions 1,2. For question3, let me explain something below:

Similar to the way the vector space is defined, Halmos 's book defines the field as a set in which numbers or scalars live.e.g.
a+b=b+a
ab=ba
...
where all numbers' properties are listed. Halmos 's field is a field of numbers.Number may include rational number,reals, complex numbers...Halmos refer to such numbers as scalars.

So my understanding of Halmos' field is that the field he used is a number field or scalar field. It is a set. It does not seem to (nor necessary) relate to scalar-valued function.

GG

 

[pmb_phy]

Thanks very much. Pete

You're welcome. I'm trying to type without my cut thumb but I'm finding it extremely difficult. It appears as though my thumb has a mind of its own. This is too much work and far too diffucult and painful. Ask mathwonk these questions. His a college math professor and a nice guy.

Pete

 

[uiulic]

Pete,

I am just confused by the word "field" used in (Groups, Rings and Fields, in such contexts field is a special ring) and the word used in "vector field". I believe the two "field" are different, but am not sure whether they have any relation.

Mathmonk is a nice patient man, I agree. I read many of his threads on tensor, although I did not understand much. If he sees this thread, he might clear my worries completely.

Thank you for your suggestion. To me you are a staff in this forum and your many questions/replies have been inspiring although I did not understand much.

GG

 

[pmb_phy]

Pete,

I am just confused by the word "field" used in (Groups, Rings and Fields, in such contexts field is a special ring) and the word used in "vector field". I believe the two "field" are different, but am not sure whether they have any relation.

Mathmonk is a nice patient man, I agree. I read many of his threads on tensor, although I did not understand much. If he sees this thread, he might clear my worries completely.

Thank you for your suggestion. To me you are a staff in this forum and your many questions/replies have been inspiring although I did not understand much.

GG

See

http://scienceworld.wolfram.com/physics/
http://mathworld.wolfram.com/

 

[uiulic]

Thanks again for sharing the links.Pete

GG

 

[pmb_phy]

Thanks again for sharing the links.Pete

GG

ur welcome

 

[pmb_phy]

Thank you for your suggestion. To me you are a staff in this forum and your many questions/replies have been inspiring although I did not understand much.

Wow! Let me take this opportunity to tell you how much that means to me. I love to be able to help others understand math and physics and you've just confirmed that I'm doing a very good job at it. I want to let you know how much I appreciate you saying so. This means that I'm accomplishing my goal.

How's it going with this problem. My thumb has healed enough to be able to type without too much discomfort. Let me know how I can help with this. I can still look up the definitions of Field as it is defined in abstract algebra. These things seem to come back to me quickly. I was wondering if you understood that there are two meanings of the term "Field" just as there are various definitions of "vector"

Best regards

Pete

 

[uiulic]

Wow! Let me take this opportunity to tell you how much that means to me. I love to be able to help others understand math and physics and you've just confirmed that I'm doing a very good job at it. I want to let you know how much I appreciate you saying so. This means that I'm accomplishing my goal.

How's it going with this problem. My thumb has healed enough to be able to type without too much discomfort. Let me know how I can help with this. I can still look up the definitions of Field as it is defined in abstract algebra. These things seem to come back to me quickly. I was wondering if you understood that there are two meanings of the term "Field" just as there are various definitions of "vector"

Best regards

Pete

Pete,

Of course you are doing in this forum as well as many other staff. Your thumb has confirmed it.


The different "field" and eventually different "vector" are just what I want to ask about. I actually have no much interest in group ring field stuff.The vector is not a vector topic has been discussed before in this forum.

Vector is frequently alternatively referred to first-rank/order tensor in tensor analysis.But all the same type of tensors (e.g. second-rank covarient tensors) CAN form a set, in which the rules for defining abstract vector space apply. Therefore, a second-rank tensor is a vector in this corresponding vector space. In this sense, vector has a broader sense.

A tensor is a vector and a vector is a tensor, which are both correct and wrong depending on how we understand it. Also, this similar "correct and wrong " argument (they seem conflicting but are not) applies to point or position vector etc (frequently discussed in this forum on whether they are vectors). Or even we can say a vector is not vector (Mr vector said to his brother 'I am not myself today').

Since the same type of tensors (with addition of elements and multiple by numbers, defined) form a vector space. We can say the vector space is tensor space. Therefore, tensor can be defined as an element of a tensor space (can't it?). Is this one more definition of tensor (for a given type)?


best regards

 

[Urmi Roy]

I read on many websites that there is essentially no difference between a vector function and a vector field...however,to me,a vector function is represented on a graph as a plot of the POSITION VECTORS of points which are values of the vector function...e.g. V(t)= cost icap + sint jcap ,is a vector function represented on a graph as position vectors of the values of V(t) at different values of t.

On the other hand,a vector field is not a plot of the position vectors,but is a map of the vectors themselves (velocity,force etc.) at the points of their existence...am I right?

If I am,then there is a difference between these two terms,right?

 

[Fredrik]

A "vector field" is defined on an open subset U of a manifold and takes each point p in U to a tangent vector at p. The V you're defining is a "vector field along a curve". It has the same domain as the curve, an interval of real numbers. If we call the curve C, then your V(t) is a tangent vector at C(t).

There's no difference between a vector-valued function on $\mathbb R^n$ and a vector field on $\mathbb R^n$, if we for all $x\in\mathbb R^n$, identify the tangent space at x with $\mathbb R^n$ itself.

 

[wofsy]

Thanks very much. Pete

You have confirmed questions 1,2. For question3, let me explain something below:

Similar to the way the vector space is defined, Halmos 's book defines the field as a set in which numbers or scalars live.e.g.
a+b=b+a
ab=ba
...
where all numbers' properties are listed. Halmos 's field is a field of numbers.Number may include rational number,reals, complex numbers...Halmos refer to such numbers as scalars.

So my understanding of Halmos' field is that the field he used is a number field or scalar field. It is a set. It does not seem to (nor necessary) relate to scalar-valued function.

GG

In algebra a field is Halmos's field. It is an algebraic structure. Fields are distinguished from other algebraic structures such as rings, groups, vector spaces, and algebras.

In calculus, a field is the assignment of a quantity to each point of a domain. A scalar field assigns a number, a vector field a vector, a tensor field a tensor.

But in calculus fields there is still algebraic structure. Functions and vector fields can be added pointwise. Functions can be multiplied pointwise and multiplied times vector fields. These pointwise operations turn the calculus fields into algebraic structures. Scalar fields form a ring. Vector fields form a module over the ring of scalar fields. They also form a vector space over the algebraic field of scalars. These algebraic structures are rich and intensely studied.

 

[Rasalhague]

You're right that field has two completely different meanings in this context. As far as I know it's just an unfortunate coincidence.

1. An algebraic structure consisting of a set (called the underlying set of the field) and two binary operations, let's call them + and *, one of which, addition, +, forms a commutative (=Abelian) group with the underlying set, while the other, multiplication, *, forms a commutative group with the underlying set excluding the identity element of the first group, and * distributes over +, thus: a*(b+c) = a*b + a*c, and (b+c)*a = b*a + b*a. Famous fields include the real numbers with addition and multiplication, as usually defined, and the complex numbers with complex addition and multiplication. It's this kind of field that features in the definition of a vector space, which consists of a commutative group (elements of whose underlying set, V, are called the vectors of that vector space) and a field (the base field of the vector space, elements of whose underlying set, F, are called scalars) and a binary operation $S : F \times V \to V$ called scalar multiplication which obeys certain axioms (associativity, distributivity over addition of scalars, distributivity over addition of vectors, and multiplying a vector by the multiplicative identity of the field leaves the vector unchanged).

2. A function whose domain is an open subset, U, of the underlying set of a manifold, e.g. a scalar field, which is such a function whose codomain is the underlying set of the base field (in sense 1) of the tangent spaces to the manifold. This is the kind of field in the expressions vector field and tensor field.

I've found these books helpful, what I've read of them so far: Bowen & Wang: Introduction to Vectors and Tensors, volumes 1 and 2. The first volume defines sets, ordered sets, groups, fields (in the abstract algebra sense), vector spaces, and so on, beginning at a very basic level and building up the definitions methodically. Algebraic fields are introduced towards the end of page 35.

http://repository.tamu.edu/handle/1969.1/2502
http://repository.tamu.edu/handle/1969.1/3609

> A tensor is a vector and a vector is a tensor, which are both correct and wrong depending on how we understand it. Also, this similar "correct and wrong " argument (they seem conflicting but are not) applies to point or position vector etc (frequently discussed in this forum on whether they are vectors). Or even we can say a vector is not vector (Mr vector said to his brother 'I am not myself today').

Heh, heh. My current understanding (possibly flawed) is that scalar, vector and tensor are often used in physics parlance as shorthand for "scalar, vector, tensor (field), etc. with respect to the tangent spaces of the manifold under discussion", or something like that. So cotangent vectors and higher order tensors are vectors, in the mathematical sense that they're elements of the underlying set of a vector space, but as they're defined relative to the tangent space, it's the vectors of the tangent space that get the unqualified name of "vectors". Likewise position vectors: they're vectors by the mathematical definition, and so could be called type-(1,0) tensors with respect to their own vector space, but physics texts will sometimes say these are "not vectors" or "not tensors", meaning that they're not tensors with respect to the tangent space of the manifold.

 

[Urmi Roy]

A "vector field" is defined on an open subset U of a manifold and takes each point p in U to a tangent vector at p.

In simple terms,does that mean that a vector field provides the tangent vectors at each point?

The V you're defining is a "vector field along a curve". It has the same domain as the curve, an interval of real numbers. If we call the curve C, then your V(t) is a tangent vector at C(t).

Actually I thought the V(t) that I gave described a 2D curve by assigning position vectors...where am I getting this wrong?

 

[Studiot]

It is unfortunate that experts in different disciplines use different and sometimes conflicting definitions for terms.
Alternatively we can say they use the same term for different concepts.

One of the most unfortunate terms is 'vector' which has a different meaning in Physics, Mathematics, Biology and Computer Science.

For the purpose of Physics:
A vector is a multicomponent entity.
At each point in space we can assign a particular instance of this entity or many instances of it.
A vector (valued) function picks out one particular instance (or none) for each point in its domain.

Just as in real analysis if we wish to do calculus we must introduce further restrictions on functions for the derivatives and integrals to exist we further restrict the vector valued function to denote a vector field. We can do vector calculus in this field an analogous way to ordinary calculus of a real variable.

So a vector field is a particular type of vector valued function sufficiently 'continuous' for vector calculus.

Formally

A vector valued function is a function F: R$$^{m}$$ $$\mapsto$$ R$$^{n}$$

A vector field is a vector valued function whose domain and co-domain are both subsets of R$$^{m}$$.

A vector field is a function F: R$$^{m}$$ $$\mapsto$$ R$$^{m}$$

A scalar field is a real valued function f: R$$^{m}$$ $$\mapsto$$ R

Note
I have used function not map as a map may be one/one, many/one or one/many whereas we want to exclude one/many.
Also for a vector valued function m may be > n or m< n or m = n.

 

[Fredrik]

In simple terms,does that mean that a vector field provides the tangent vectors at each point?

I think that simplifies it to the point where it's no longer clear what it means. When I said that a vector field takes a point p (from some open subset of M) to a tangent vector at p, that was already the simplified version. The usual definition either says that a vector field is a "local section" of the tangent bundle, or spells that out explicitly:

If U is an open subset of a manifold M, and $$TM=\bigcup_p T_pM$$ is the tangent bundle, i.e. the union of all the tangent spaces, then a function $X:U\rightarrow TM$ is said to be a vector field if $X(p)\in T_pM$ for all p.

Actually I thought the V(t) that I gave described a 2D curve by assigning position vectors...where am I getting this wrong?

What I said wasn't completely clear. If you define $$V(t)=\cos t\ e_1 +\sin t\ e_2$$, then V is a curve in $\mathbb R^2$. If we give $\mathbb R^2$ the usual vector space structure, we can call the members of $\mathbb R^2$ "vectors". So you have defined a function that takes numbers to vectors, and you have defined a curve in the manifold $\mathbb R^2$. The tangent vectors of that curve, i.e. the derivative operators defined by

[tex]\dot V(t)f=(f\circ V)'(t)[/itex]

are members of the tangent bundle. Each $$\dot V(t)$$ is a member of $T_{V(t)}M$, i.e. the tangent space at V(t). So $\dot V$ is a function that takes numbers to tangent vectors. That makes it "a vector field along V" rather than " a vector field", since its domain is an interval of the real numbers rather than an open set of the manifold.