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Is a function from R^n to R^m for aritrary m a considered a 0-form on R^n, or does 0-form refers only to functions from R^n to R ?
If you have a finite dimensional vector space V with scalar field k, then the space of n-forms is isomorphic to the space of alternating multilinear maps Vn --> k.quasar987 said:Is a function from R^n to R^m for aritrary m a considered a 0-form on R^n, or does 0-form refers only to functions from R^n to R ?
quasar987 said:In 'Calculus on manifolds', Spivak defines a k-form on R^n as a function w sending a point p of R^n to an alternating multilinear maps (R^n)^k-->R.
This makes sense only for k>0, so he treats the case k=0 separately by saying that by a 0-form we mean a function f.
I was 90% sure he meant a function f:R^n-->R but wanted to make sure.
A 0-form, or a 0-dimensional form, is a mathematical concept used in differential geometry to represent a scalar field. It is a function that assigns a single value to each point in a space, and does not depend on direction or orientation.
Any mathematical concept that does not fit the definition of a 0-form is not considered a 0-form. This includes vector fields, which assign a vector to each point in a space, and 1-forms, which assign a covector to each point in a space.
In differential geometry, a 0-form is represented using a coordinate-free notation, such as f or ϕ, to indicate the function that assigns a value to each point in a space. It can also be represented using a scalar field, where each point is associated with a value.
One example of a 0-form in real life is temperature. At each point in a room, there is a single temperature value, and it does not depend on direction or orientation. Other examples include pressure, density, and concentration, where a single value is assigned to each point in a space.
0-forms are important in science because they allow for the representation and analysis of scalar fields, which are used to describe physical quantities that do not have a direction or orientation. They are also used in various mathematical and scientific fields, such as physics, engineering, and computer science.