Uses of the Word "Form" in Mathematics

[Rasalhague]

Would the following be an accurate dictionary-style summary of the various (conflicting) uses of the word form? (The vector space with respect to which the tensors and tensor fields in 2 and 3 are defined is the tangent space of a manifold.)

(1) As in everyday, non-jargon English, one of several ways of expressing an idea. E.g. "The differential form of Maxwell's equations."

(2) (a) A covariant tensor. (b) A covariant tensor field. E.g. "A metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point."

(3) (a) A totally antisymmetric covariant tensor, i.e. a covariant tensor whose value changes sign when any pair of arguments are interchanged. (b) A totally antisymmetric covariant tensor field. (With "differential form" meaning a differentiable form in this sense, e.g. "If $\omega |_p$ is differentiable, then we will refer to it as a differential form.")

 

[Fredrik]

I would say if V is a vector space over a field F, a function from Vⁿ into F is called a functional if n=1, and a form if n>1. (Not sure how standard that is).

I would also say that a "differential form" is "an n-form, for some n=1,2,3,...,d, where d is the dimension of the manifold", and that an "n-form" is an alternating (n,0) tensor field.

In the phrase "the differential form of Maxwell's equations", the word "form" is just an English word, not a mathematical term. It has nothing to do with differential forms.

 

[Rasalhague]

Ah, so the term "form" in the sense of your first paragraph is more general than "covariant tensor" in that a form needn't be linear in each argument. (E.g. a quadratic form.) That explains why the definition of metric tensor that I quoted specified bilinear.

The definition of differential form that I quoted is from David Bachman's online textbook A Geometric Approach to Differential Forms, chapter 3, pp. 32-33; m-forms, for him, are alternating functions from Vⁿ to F, linear in each argument, and a "differential form" is, for him, a differentiable m-form field. I've been tending to drop the dummy prefix m-, n-, p-, k-, by analogy with "tuple" being synonymous with "n-tuple", but from what you say, it seems that might be unwise, if prefixing the word "form" with a nonnegative integer, or an arbitrary letter denoting "any nonnegative integer", is supposes to convey the extra information "alternating in each argument and linear in each argument" (to those in the know).

In the naming system you describe, where "differential" has this special meaning "alternating linear ... field", would it not be possible to define some awkward alternating covariant tensor field such that it wasn't even differentiable, or is differentiablility implicit in the definition?

What happens is we want to talk about, say, a differential form field which isn't alternating or isn't linear; how could this be referred to (given that the name "differential form" is already taken), or is such a thing impossible, or never met with in practice?

 

[Office_Shredder]

I would say if V is a vector space over a field F, a function from Vⁿ into F is called a functional if n=1, and a form if n>1. (Not sure how standard that is).

I think this is the opposite of standard. A functional is just any linear map to the base field in functional analysis

 

[Fredrik]

I think this is the opposite of standard. A functional is just any linear map to the base field in functional analysis

Huh. I have seen the phrase "linear functional" too often to believe that linearity is part of a standard definition of functional. And even if it is, how is what I said the opposite of that? I just left out the linearity requirement.

 

[Fredrik]

Ah, so the term "form" in the sense of your first paragraph is more general than "covariant tensor" in that a form needn't be linear in each argument. (E.g. a quadratic form.) That explains why the definition of metric tensor that I quoted specified bilinear.

The term "form" isn't used that often, and the times I have seen it used, it has always been in the context "bilinear form". So I'm sort of guessing the definition. Perhaps "multilinear is part of the definition, and "bilinear form" is just a form that takes 2 variables to a number. The biggest difference between this type of "form" and the n-forms of differential geometry is that they don't have to be alternating (antisymmetric).

m-forms, for him, are alternating functions from Vⁿ to F, linear in each argument, and a "differential form" is, for him, a differentiable m-form field.

This V is always the tangent space at some point of a smooth manifold, right? These definitions sound good to me (I didn't even think of the possibility that "differential" should be interpreted as "smooth"), but I know that some authors use the term "m-form" for what you call an "m-form field".

In the naming system you describe, where "differential" has this special meaning "alternating linear ... field", would it not be possible to define some awkward alternating covariant tensor field such that it wasn't even differentiable, or is differentiablility implicit in the definition?

It would be possible. I just forgot to consider smoothness/differentiability. Smoothness isn't part of the definition of "tensor field". I don't know if the authors that define n-forms to be tensor fields rather than tensors include smoothness as part of their definition, but if we use the convention that an n-form is a tensor at a point in the manifold, the term "n-form field" can only mean "section of the bundle of n-forms", so smoothness is not part of the definition.

What happens is we want to talk about, say, a differential form field which isn't alternating or isn't linear; how could this be referred to (given that the name "differential form" is already taken), or is such a thing impossible, or never met with in practice?

If it isn't alternating, it's just a (n,0) tensor field. If the objects it assigns to the points in its domain aren't tensors, then it's just a "function" (into what I don't know, since you didn't specify what sort of objects it assigns to the points is its domain)

 

[Rasalhague]

Is it possible to have totally alternating covariant tensors wrt the tangent spaces of an infinite dimensional manifold, and would they also, weirdly, be called m-forms (supposing Latin letter prefixed to "form" means "totally alternating and totally linear"), or do people revert to more transparent nomenclature then?

 

[Landau]

Huh. I have seen the phrase "linear functional" too often to believe that linearity is part of a standard definition of functional.

Yeah, linearity sometimes is and sometimes isn't part of the definition. If you work with linear functionals all the time, you will probably become tired of adding the word 'linear' and just drop it.

And even if it is, how is what I said the opposite of that? I just left out the linearity requirement.

Well, because you insisted that n=1. But if V is a vector space, then a (linear) map from V^n to F is just as well a (linear) functional. Of course, V^n is again a vector space, so it doesn't really matter.

 

[Fredrik]

So do you guys (Landau, Office Shredder) consider "form" to be synonymous with "functional"?

I don't have a strong opinion about how these terms "should" be defined, but I can at least mention the reasons why I defined them the way I did. Inner products and metric tensors are often said to be non-degenerate symmetric bilinear forms. In this context, the fact that it's possible to define a vector space structure on V² is irrelevant. Even the word "bilinear" only requires a vector space structure on V. I have also never seen the word "functional" used for a function whose domain is a cartesian product of two or more vector spaces.

 

[Rasalhague]

"A functional on a vector space V over K is a map from V to K. A linear functional is a map f: V --> K satisfying f(ax + by) = af(x) + bf(y) for all scalars a,b and all vectors x,y in V" (Griffel: Linear Algebra and its Applications, Vol. 2, § 14A).

"Functional, n. A function whose domain is itself a set of functions, and whose range is another set of functions which may be numerical constants. The term is often reserved for linear functionals" (Borowski & Borwein: Collins Dictionary of Mathematics).

"Linear functional, n. A linear function from a vector space into its base space [...] The set of all (continuous) linear functionals endowed with pointwise operations comprises the algebraic (or continuous) dual vector space" (Borowski & Borwein: Collins Dictionary of Mathematics).

"Differential form, n. [...] More precisely, a differential form of degree r in n variables is a mapping from a domain in n-space into the set of r-covectors (r-covector: an alternating covariant tensor or rank r)" (Borowski & Borwein: Collins Dictionary of Mathematics).

Wikipedia gives form as a synonym of algebraic form, a homogeneous polynomial... The nearest I've found there to a definition of form in this context though is under multilinear form. The first definition implicitly suggests that form would mean a map from V^N to its base field, since a multilinear form is defined as any such map that's "separately linear in each its N variables." But it immediately goes on to say: "As the word form usually denotes a mapping from a vector space into its underlying field, the more general term "multilinear map" is used, when talking about a general map that is linear in all its arguments." Perhaps the intended meaning was "from the Cartesian product of some number of copies of a vector space". (The authors of this article seem to have reversed the usual definitions of alternating and antisymmetric. I do hope that's just a mistake and not evidence for a rival convention.)

*

So maybe (taking linear and alternating to refer to all arguments or pairs of arguments)...

(1) (a) Form. A map from V^k --> F, where V is the underlying set of a vector space, and F that of its base field. (In some dialects, synonymous with functional. In others, functional only used when k < 2. In others, functional only used when k < 2 and the function is linear. In some dialects, a linear 1-form is called a linear functional. For some, linear also (verbally) implies continuous.) (b) A form field in sense a of form, and the "vector field" sense of field.

(2) (a) Linear form. A linear map from V^k --> F. Synonymous with covariant tensor, i.e. valence (0,k) tensor, or (k,0) tensor, depending on the convention adopted! (b) A linear form field.

(3) Alternating linear form. An alternating linear map from V^k --> F, or synonymously, in geometric algebra language, a pure-grade cotangent multivector. Usual (abbreviated) name, k-form, p-form, m-form etc. (with interchangeable prefix). (b) A "differential form", i.e. an alternating linear form field. (Some authors seem to use differential form to mean a possible value of such a field, i.e. an alternating linear form, but it's hard to tell, given the frequent lack of distinction made, in this context, in less formal texts, between a field and its values.)

Is the base vector space, V, in this context always the tangent space of a smooth manifold, or could some of these definitions refer to tensors with respect to any base vector space?

 

[Fredrik]

(The authors of this article seem to have reversed the usual definitions of alternating and antisymmetric. I do hope that's just a mistake and not evidence for a rival convention.)

I think you just read it wrong. Antisymmetric=alternating=the sign changes if you swap two variables.

Is the base vector space, V, in this context always the tangent space of a smooth manifold, or could some of these definitions refer to tensors with respect to any base vector space?

An inner product on an arbitrary vector space is sometimes described as a symmetric bilinear positive definite form. (If it's a complex vector space, the word "bilinear" is replaced by "sesquilinear", meaning that it's antilinear in one of the variables and linear in the other).

 

[Rasalhague]

I think you just read it wrong. Antisymmetric=alternating=the sign changes if you swap two variables.

Well, yes, in all of the actual examples I've studied, but, unless I've misread anything else, http://planetmath.org/encyclopedia/SymplecticHyperbolicPlane.html defines a scalar-valued function of two vectors as alternating iff A(v,v) = 0, which, it says, always implies antisymmetry. It defines a scalar-valued function of two vectors as antisymmetric iff B(x,y) = -B(y,x), which, it says, implies that the function is alternating except in the freakish case where the characteristic of the base field is 2.

But Wikipedia multilinear form defines alternation by B(x,y) = -B(y,x) (what PlanetMath calls antisymmetry), which, it says, is equivalent to the property that A(v,v) = 0 (what PlanetMath calls alternation), so long as the characteristic is not 2.

 

[Office_Shredder]

If you have B(x,y)=-B(y,x) we get B(v,v)=-B(v,v) which is equivalent to 2B(v,v)=0. This means B(v,v)=0 except for when the characteristic is 2, in which case we don't know anything.

On the other hand: If we have is B(v,v)=0, we know that B(x+y,x+y)=B(x,x)+B(x,y)+B(y,x)+B(y,y)=B(x,y)+B(y,x)=0 and it's always true that B(x,y)=-B(y,x) in this case.

In practice you rarely care about fields of characteristic 2 so you just use whatever word you want