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Is vector field a field (in the algebraic sense) also?

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3

1

$begingroup$

This is something about basic confusion.

Let $M$ be a smooth manifold of dimension $n$.

We denote $X$ to mean the vector field.

Is the vector field $X$, a field (in the sense of abstract algebra) also?

vector-fields

share|cite|improve this question

edited Mar 14 at 11:19

Asaf Karagila♦

306k33438769

asked Mar 14 at 8:31

M. A. SARKARM. A. SARKAR

2,3991819

$endgroup$

add a comment | 

3

1

$begingroup$

This is something about basic confusion.

Let $M$ be a smooth manifold of dimension $n$.

We denote $X$ to mean the vector field.

Is the vector field $X$, a field (in the sense of abstract algebra) also?

vector-fields

share|cite|improve this question

edited Mar 14 at 11:19

Asaf Karagila♦

306k33438769

asked Mar 14 at 8:31

M. A. SARKARM. A. SARKAR

2,3991819

$endgroup$

add a comment | 

$begingroup$

This is something about basic confusion.

Let $M$ be a smooth manifold of dimension $n$.

We denote $X$ to mean the vector field.

Is the vector field $X$, a field (in the sense of abstract algebra) also?

vector-fields

share|cite|improve this question

edited Mar 14 at 11:19

Asaf Karagila♦

306k33438769

asked Mar 14 at 8:31

M. A. SARKARM. A. SARKAR

2,3991819

$endgroup$

share|cite|improve this question

edited Mar 14 at 11:19

Asaf Karagila♦

306k33438769

asked Mar 14 at 8:31

M. A. SARKARM. A. SARKAR

2,3991819

share|cite|improve this question

edited Mar 14 at 11:19

Asaf Karagila♦

306k33438769

asked Mar 14 at 8:31

M. A. SARKARM. A. SARKAR

2,3991819

edited Mar 14 at 11:19

Asaf Karagila♦

306k33438769

edited Mar 14 at 11:19

Asaf Karagila♦

306k33438769

asked Mar 14 at 8:31

M. A. SARKARM. A. SARKAR

2,3991819

asked Mar 14 at 8:31

M. A. SARKARM. A. SARKAR

2,3991819

1 Answer
1

active

oldest

votes

5

$begingroup$

A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.

The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.

share|cite|improve this answer

answered Mar 14 at 8:34

ArthurArthur

119k7118202

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A module over a field is a vector space. Why call it "module" then?
  $endgroup$
  – enedil
  Mar 14 at 8:57
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
  $endgroup$
  – Arthur
  Mar 14 at 9:01
  
  

  
  
  
  

add a comment | 

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5

$begingroup$

A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.

The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.

share|cite|improve this answer

answered Mar 14 at 8:34

ArthurArthur

119k7118202

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A module over a field is a vector space. Why call it "module" then?
  $endgroup$
  – enedil
  Mar 14 at 8:57
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
  $endgroup$
  – Arthur
  Mar 14 at 9:01
  
  

  
  
  
  

add a comment | 

5

$begingroup$

A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.

The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.

share|cite|improve this answer

answered Mar 14 at 8:34

ArthurArthur

119k7118202

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A module over a field is a vector space. Why call it "module" then?
  $endgroup$
  – enedil
  Mar 14 at 8:57
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
  $endgroup$
  – Arthur
  Mar 14 at 9:01
  
  

  
  
  
  

add a comment | 

$begingroup$

A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.

The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.

share|cite|improve this answer

answered Mar 14 at 8:34

ArthurArthur

119k7118202

$endgroup$

share|cite|improve this answer

answered Mar 14 at 8:34

ArthurArthur

119k7118202

answered Mar 14 at 8:34

ArthurArthur

119k7118202

answered Mar 14 at 8:34

ArthurArthur

119k7118202