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What is a form?
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I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that it is to a ring what a vector is to a field. I apologise if it is a repost or something though.
linear-algebra differential-geometry differential-forms quadratic-forms modular-forms
asked 1 hour ago
user606304
382
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up vote
7
down vote
favorite
I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that it is to a ring what a vector is to a field. I apologise if it is a repost or something though.
linear-algebra differential-geometry differential-forms quadratic-forms modular-forms
asked 1 hour ago
user606304
382
1
But what is it that distinguishes a form from a function? I can't call every function a form, can I?
– user606304
1 hour ago
My understanding is that we call something a form when it appears in the form of a linear combination $sum_i_1,ldots,i_nin S a_i_1cdots i_n x_i_1odotcdotsodot x_i_n$, where the $x_i$s are some members of an algebra and the symbol $odot$ denotes some sort of product. For linear or multilinear forms in linear algebra, the $x_i$s are just indeterminates; in differential forms the $x_i$s here are replaced by the differentials $dx_i$s and the $odot$ is the exterior product. But this certainly doesn't cover every usage of the word "form" in mathematics.
– user1551
10 mins ago
add a comment |Â
up vote
7
down vote
favorite
up vote
7
down vote
favorite
I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that it is to a ring what a vector is to a field. I apologise if it is a repost or something though.
linear-algebra differential-geometry differential-forms quadratic-forms modular-forms
asked 1 hour ago
user606304
382
I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that it is to a ring what a vector is to a field. I apologise if it is a repost or something though.
linear-algebra differential-geometry differential-forms quadratic-forms modular-forms
linear-algebra differential-geometry differential-forms quadratic-forms modular-forms
asked 1 hour ago
user606304
382
asked 1 hour ago
user606304
382
asked 1 hour ago
user606304
382
asked 1 hour ago
user606304
382
asked 1 hour ago
user606304
382
382
1
But what is it that distinguishes a form from a function? I can't call every function a form, can I?
– user606304
1 hour ago
My understanding is that we call something a form when it appears in the form of a linear combination $sum_i_1,ldots,i_nin S a_i_1cdots i_n x_i_1odotcdotsodot x_i_n$, where the $x_i$s are some members of an algebra and the symbol $odot$ denotes some sort of product. For linear or multilinear forms in linear algebra, the $x_i$s are just indeterminates; in differential forms the $x_i$s here are replaced by the differentials $dx_i$s and the $odot$ is the exterior product. But this certainly doesn't cover every usage of the word "form" in mathematics.
– user1551
10 mins ago
add a comment |Â
1
But what is it that distinguishes a form from a function? I can't call every function a form, can I?
– user606304
1 hour ago
My understanding is that we call something a form when it appears in the form of a linear combination $sum_i_1,ldots,i_nin S a_i_1cdots i_n x_i_1odotcdotsodot x_i_n$, where the $x_i$s are some members of an algebra and the symbol $odot$ denotes some sort of product. For linear or multilinear forms in linear algebra, the $x_i$s are just indeterminates; in differential forms the $x_i$s here are replaced by the differentials $dx_i$s and the $odot$ is the exterior product. But this certainly doesn't cover every usage of the word "form" in mathematics.
– user1551
10 mins ago
1
1
But what is it that distinguishes a form from a function? I can't call every function a form, can I?
– user606304
1 hour ago
But what is it that distinguishes a form from a function? I can't call every function a form, can I?
– user606304
1 hour ago
My understanding is that we call something a form when it appears in the form of a linear combination $sum_i_1,ldots,i_nin S a_i_1cdots i_n x_i_1odotcdotsodot x_i_n$, where the $x_i$s are some members of an algebra and the symbol $odot$ denotes some sort of product. For linear or multilinear forms in linear algebra, the $x_i$s are just indeterminates; in differential forms the $x_i$s here are replaced by the differentials $dx_i$s and the $odot$ is the exterior product. But this certainly doesn't cover every usage of the word "form" in mathematics.
– user1551
10 mins ago
My understanding is that we call something a form when it appears in the form of a linear combination $sum_i_1,ldots,i_nin S a_i_1cdots i_n x_i_1odotcdotsodot x_i_n$, where the $x_i$s are some members of an algebra and the symbol $odot$ denotes some sort of product. For linear or multilinear forms in linear algebra, the $x_i$s are just indeterminates; in differential forms the $x_i$s here are replaced by the differentials $dx_i$s and the $odot$ is the exterior product. But this certainly doesn't cover every usage of the word "form" in mathematics.
– user1551
10 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
3
down vote
I initially upvoted Michael Hoppe's answer because it is a true statement, but I don't think it quite addresses the OP's question, since bilinear and $k$-linear forms more generally are neither linear nor have domain $V$, and quadratic forms aren't even linear.
I think perhaps the best answer so far was given by Dietrich Burde in the comments. A form is just another word for function. However that's perhaps not so illuminating.
All the examples you've listed have some things in common.
- All of them involve a vector space $V$ over a field $K$
- All of them are functions from a product of the vector space with itself to the ground field. I.e. they are all of the form $Vtimes Vtimes cdots times V=V^nto K$.
There aren't really any more similarities than that though, and these similarities might not hold for all things called forms. The key thing that determines what the word "form" means are the adjectives in front of it.
You probably know what the specific things you mentioned are, but just to highlight their similarities and differences, and why I say the adjectives in front of the word "form" are more important than the word "form" itself, their definitions are given below.
For example, a $k$-linear or multilinear form on $V$ is a function $F(v_1,ldots,v_k)$ from $V^k$ to $K$ such that whenever the other arguments are held constant, the map $v_imapsto F(v_1,ldots,v_i,ldots,v_k)$ is a linear map.
A bilinear form is just a $k$-linear form where $k=2$.
A quadratic form is a function $q$ from $V$ to $K$ such that $f(tv)=t^2v$ for all $tin K$, and $vin V$ and such that $B(v,w):=q(v+w)-q(v)-q(w)$ is a bilinear form on $V$.
Finally a differential $k$-form (well sort of) on a vector space $V$ is a $k$-linear form on $V$ that is alternating, which means that $F(v_1,ldots,v,ldots,v,ldots,v_n)=0$ whenever two of the arguments of the $k$-form are equal. Usually though a differential form is an object on a manifold that smoothly assigns what I've just called a differential form to the tangent space at every point of some open subset of the manifold.
answered 45 mins ago
jgon
8,76911536
add a comment |Â
up vote
0
down vote
If $V$ is a vector space over field $F$, any $F$-valued linear map with domain $V$ is called a form.
answered 1 hour ago
Michael Hoppe
10.3k31532
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
I initially upvoted Michael Hoppe's answer because it is a true statement, but I don't think it quite addresses the OP's question, since bilinear and $k$-linear forms more generally are neither linear nor have domain $V$, and quadratic forms aren't even linear.
I think perhaps the best answer so far was given by Dietrich Burde in the comments. A form is just another word for function. However that's perhaps not so illuminating.
All the examples you've listed have some things in common.
- All of them involve a vector space $V$ over a field $K$
- All of them are functions from a product of the vector space with itself to the ground field. I.e. they are all of the form $Vtimes Vtimes cdots times V=V^nto K$.
There aren't really any more similarities than that though, and these similarities might not hold for all things called forms. The key thing that determines what the word "form" means are the adjectives in front of it.
You probably know what the specific things you mentioned are, but just to highlight their similarities and differences, and why I say the adjectives in front of the word "form" are more important than the word "form" itself, their definitions are given below.
For example, a $k$-linear or multilinear form on $V$ is a function $F(v_1,ldots,v_k)$ from $V^k$ to $K$ such that whenever the other arguments are held constant, the map $v_imapsto F(v_1,ldots,v_i,ldots,v_k)$ is a linear map.
A bilinear form is just a $k$-linear form where $k=2$.
A quadratic form is a function $q$ from $V$ to $K$ such that $f(tv)=t^2v$ for all $tin K$, and $vin V$ and such that $B(v,w):=q(v+w)-q(v)-q(w)$ is a bilinear form on $V$.
Finally a differential $k$-form (well sort of) on a vector space $V$ is a $k$-linear form on $V$ that is alternating, which means that $F(v_1,ldots,v,ldots,v,ldots,v_n)=0$ whenever two of the arguments of the $k$-form are equal. Usually though a differential form is an object on a manifold that smoothly assigns what I've just called a differential form to the tangent space at every point of some open subset of the manifold.
answered 45 mins ago
jgon
8,76911536
add a comment |Â
up vote
3
down vote
I initially upvoted Michael Hoppe's answer because it is a true statement, but I don't think it quite addresses the OP's question, since bilinear and $k$-linear forms more generally are neither linear nor have domain $V$, and quadratic forms aren't even linear.
I think perhaps the best answer so far was given by Dietrich Burde in the comments. A form is just another word for function. However that's perhaps not so illuminating.
All the examples you've listed have some things in common.
- All of them involve a vector space $V$ over a field $K$
- All of them are functions from a product of the vector space with itself to the ground field. I.e. they are all of the form $Vtimes Vtimes cdots times V=V^nto K$.
There aren't really any more similarities than that though, and these similarities might not hold for all things called forms. The key thing that determines what the word "form" means are the adjectives in front of it.
You probably know what the specific things you mentioned are, but just to highlight their similarities and differences, and why I say the adjectives in front of the word "form" are more important than the word "form" itself, their definitions are given below.
For example, a $k$-linear or multilinear form on $V$ is a function $F(v_1,ldots,v_k)$ from $V^k$ to $K$ such that whenever the other arguments are held constant, the map $v_imapsto F(v_1,ldots,v_i,ldots,v_k)$ is a linear map.
A bilinear form is just a $k$-linear form where $k=2$.
A quadratic form is a function $q$ from $V$ to $K$ such that $f(tv)=t^2v$ for all $tin K$, and $vin V$ and such that $B(v,w):=q(v+w)-q(v)-q(w)$ is a bilinear form on $V$.
Finally a differential $k$-form (well sort of) on a vector space $V$ is a $k$-linear form on $V$ that is alternating, which means that $F(v_1,ldots,v,ldots,v,ldots,v_n)=0$ whenever two of the arguments of the $k$-form are equal. Usually though a differential form is an object on a manifold that smoothly assigns what I've just called a differential form to the tangent space at every point of some open subset of the manifold.
answered 45 mins ago
jgon
8,76911536
add a comment |Â
up vote
3
down vote
up vote
3
down vote
I initially upvoted Michael Hoppe's answer because it is a true statement, but I don't think it quite addresses the OP's question, since bilinear and $k$-linear forms more generally are neither linear nor have domain $V$, and quadratic forms aren't even linear.
I think perhaps the best answer so far was given by Dietrich Burde in the comments. A form is just another word for function. However that's perhaps not so illuminating.
All the examples you've listed have some things in common.
- All of them involve a vector space $V$ over a field $K$
- All of them are functions from a product of the vector space with itself to the ground field. I.e. they are all of the form $Vtimes Vtimes cdots times V=V^nto K$.
There aren't really any more similarities than that though, and these similarities might not hold for all things called forms. The key thing that determines what the word "form" means are the adjectives in front of it.
You probably know what the specific things you mentioned are, but just to highlight their similarities and differences, and why I say the adjectives in front of the word "form" are more important than the word "form" itself, their definitions are given below.
For example, a $k$-linear or multilinear form on $V$ is a function $F(v_1,ldots,v_k)$ from $V^k$ to $K$ such that whenever the other arguments are held constant, the map $v_imapsto F(v_1,ldots,v_i,ldots,v_k)$ is a linear map.
A bilinear form is just a $k$-linear form where $k=2$.
A quadratic form is a function $q$ from $V$ to $K$ such that $f(tv)=t^2v$ for all $tin K$, and $vin V$ and such that $B(v,w):=q(v+w)-q(v)-q(w)$ is a bilinear form on $V$.
Finally a differential $k$-form (well sort of) on a vector space $V$ is a $k$-linear form on $V$ that is alternating, which means that $F(v_1,ldots,v,ldots,v,ldots,v_n)=0$ whenever two of the arguments of the $k$-form are equal. Usually though a differential form is an object on a manifold that smoothly assigns what I've just called a differential form to the tangent space at every point of some open subset of the manifold.
answered 45 mins ago
jgon
8,76911536
I initially upvoted Michael Hoppe's answer because it is a true statement, but I don't think it quite addresses the OP's question, since bilinear and $k$-linear forms more generally are neither linear nor have domain $V$, and quadratic forms aren't even linear.
I think perhaps the best answer so far was given by Dietrich Burde in the comments. A form is just another word for function. However that's perhaps not so illuminating.
All the examples you've listed have some things in common.
- All of them involve a vector space $V$ over a field $K$
- All of them are functions from a product of the vector space with itself to the ground field. I.e. they are all of the form $Vtimes Vtimes cdots times V=V^nto K$.
There aren't really any more similarities than that though, and these similarities might not hold for all things called forms. The key thing that determines what the word "form" means are the adjectives in front of it.
You probably know what the specific things you mentioned are, but just to highlight their similarities and differences, and why I say the adjectives in front of the word "form" are more important than the word "form" itself, their definitions are given below.
For example, a $k$-linear or multilinear form on $V$ is a function $F(v_1,ldots,v_k)$ from $V^k$ to $K$ such that whenever the other arguments are held constant, the map $v_imapsto F(v_1,ldots,v_i,ldots,v_k)$ is a linear map.
A bilinear form is just a $k$-linear form where $k=2$.
A quadratic form is a function $q$ from $V$ to $K$ such that $f(tv)=t^2v$ for all $tin K$, and $vin V$ and such that $B(v,w):=q(v+w)-q(v)-q(w)$ is a bilinear form on $V$.
Finally a differential $k$-form (well sort of) on a vector space $V$ is a $k$-linear form on $V$ that is alternating, which means that $F(v_1,ldots,v,ldots,v,ldots,v_n)=0$ whenever two of the arguments of the $k$-form are equal. Usually though a differential form is an object on a manifold that smoothly assigns what I've just called a differential form to the tangent space at every point of some open subset of the manifold.
answered 45 mins ago
jgon
8,76911536
answered 45 mins ago
jgon
8,76911536
answered 45 mins ago
jgon
8,76911536
answered 45 mins ago
jgon
8,76911536
8,76911536
add a comment |Â
add a comment |Â
up vote
0
down vote
If $V$ is a vector space over field $F$, any $F$-valued linear map with domain $V$ is called a form.
answered 1 hour ago
Michael Hoppe
10.3k31532
add a comment |Â
up vote
0
down vote
If $V$ is a vector space over field $F$, any $F$-valued linear map with domain $V$ is called a form.
answered 1 hour ago
Michael Hoppe
10.3k31532
add a comment |Â
up vote
0
down vote
up vote
0
down vote
If $V$ is a vector space over field $F$, any $F$-valued linear map with domain $V$ is called a form.
answered 1 hour ago
Michael Hoppe
10.3k31532
If $V$ is a vector space over field $F$, any $F$-valued linear map with domain $V$ is called a form.
answered 1 hour ago
Michael Hoppe
10.3k31532
answered 1 hour ago
Michael Hoppe
10.3k31532
answered 1 hour ago
Michael Hoppe
10.3k31532
answered 1 hour ago
Michael Hoppe
10.3k31532
10.3k31532
add a comment |Â
add a comment |Â
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1
But what is it that distinguishes a form from a function? I can't call every function a form, can I?
– user606304
1 hour ago
My understanding is that we call something a form when it appears in the form of a linear combination $sum_i_1,ldots,i_nin S a_i_1cdots i_n x_i_1odotcdotsodot x_i_n$, where the $x_i$s are some members of an algebra and the symbol $odot$ denotes some sort of product. For linear or multilinear forms in linear algebra, the $x_i$s are just indeterminates; in differential forms the $x_i$s here are replaced by the differentials $dx_i$s and the $odot$ is the exterior product. But this certainly doesn't cover every usage of the word "form" in mathematics.
– user1551
10 mins ago