Why are scalars and dual vectors 0- and 1-forms?

In summary: So the dual of a 1-form is a 0-form.In summary, a differential p-form is a completely antisymmetric tensor and thus scalars are automatically 0-forms and dual vectors are one-forms. This is because the condition for swapping indices is empty, and the quotient algebra is the vector space itself. In other words, there is no condition or restriction for these cases. Additionally, a 0-form is 0-linear and a dual is a linear map on vectors, making the dual of a 1-form a 0-form.
  • #1
George Keeling
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I am told: "A differential p-form is a completely antisymmetric (0,p) tensor. Thus scalars are automatically 0-forms and dual vectors (one downstairs index) are one-forms."

Since an antisymmetric tensor is one where if one swaps any pair of indices the value of the component changes sign and 1) there are no indices to swap on a scalar and 2) on a dual vector swapping something with itself is not swapping, how are they automatically 0- and 1-forms? I have no problem with higher forms.
 
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  • #2
George Keeling said:
I am told: "A differential p-form is a completely antisymmetric (0,p) tensor. Thus scalars are automatically 0-forms and dual vectors (one downstairs index) are one-forms."

Since an antisymmetric tensor is one where if one swaps any pair of indices the value of the component changes sign and 1) there are no indices to swap on a scalar and 2) on a dual vector swapping something with itself is not swapping, how are they automatically 0- and 1-forms? I have no problem with higher forms.
If you forget the swapping process as such and define it correctly as the quotient algebra of the ##n-##fold tensor product by its ideal generated by pairs of equal vectors ##V^{\otimes n}/\langle v \otimes v \rangle##, then we get ##V^0=\mathbb{F}\; , \; V^1=V## as in both cases ## \langle v\otimes v\rangle =\langle \emptyset \rangle = \{\,0\,\}##.

In other words: The condition which you called swapping is an empty condition. The empty set generates the zero ideal, and thus the quotient algebra is the vector space itself.

Even shorter: no condition, no restriction.
 
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  • #3
fresh_42 said:
Even shorter: no condition, no restriction.
That was the best bit!
 
  • #4
George Keeling said:
That was the best bit!
Thanks, but it had to be prepared. Otherwise you might have felt not taken seriously. :wink:
 
  • #5
In a non-algebraic sense, a p-form I just a p-linear map. So a 0-form is 0-linear, I e. linear in no arguments. These are kind of annoying special cases that must be addressed. A dual is a linear map on vector s.
 

1. What is the difference between scalars and dual vectors?

Scalars are quantities that have only magnitude and no direction, while dual vectors are quantities that have both magnitude and direction.

2. Why are scalars and dual vectors also known as 0- and 1-forms?

Scalars are considered as 0-forms because they do not have any direction, while dual vectors are considered as 1-forms because they have one direction.

3. How are scalars and dual vectors related to each other?

Scalars and dual vectors are related to each other through the concept of duality. Dual vectors are the dual space of scalars, meaning they are the space of linear functionals that take in a scalar and output a real number.

4. What is the significance of scalars and dual vectors in physics and mathematics?

Scalars and dual vectors are essential in physics and mathematics as they provide a way to define and manipulate quantities that have both magnitude and direction. They are used in various mathematical and physical theories, such as vector calculus and relativity.

5. How can understanding scalars and dual vectors help in solving problems?

Understanding scalars and dual vectors allows for the use of powerful mathematical tools, such as vector spaces and linear transformations, to solve problems in a variety of fields, including physics, engineering, and computer science. It also allows for a deeper understanding of the underlying principles and concepts in these fields.

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