Understanding One Forms in General Relativity: A Beginner's Guide

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In summary, a one-form is a mathematical concept that can be thought of as the number of layers of an onion that will be pierced by a stab in a particular direction. It can also represent the change in value of a function when moving an infinitesimal distance in a given direction. The components of a one-form correspond to the thickness and spacing of the layers of the onion.
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resurgance2001
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Hi

I am a new(ish) student of general relativity. I am currently reading 'Relativitiy DeMystified'

However this guys explanation of one forms is completely mystifying to me.

He says that basis vectors

[tex] e_a = ∂_a = {\frac{∂a}{∂X^a}} [/tex]


And then says that this type of basis is called a coordinate basis, and that is allows us to 'think of a vector as an operator, one that maps a function into a new function

Then

[tex] Vf = (V^a e_a) = V^a ∂_a f [/tex]

The vector V can be represented by covariant components V_a and this vector is called a 'one form'

I just did not get that. Can anyone explain in really simple terms what a one form is?

Thanks

Peter
 
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resurgance2001 said:
The vector V can be represented by covariant components V_a and this vector is called a 'one form'

I just did not get that. Can anyone explain in really simple terms what a one form is?
Stab an onion with an icepick, and the icepick will pierce some number of layers of the onion. The one-form tells you how many layers of the onion will be pierced by a stab of unit length in a particular direction. Higher numerical values of the components of the one-form correspond to more closely spaced and thinner layers of the onion.

The traditional gradient of a function can be treated as a one-form. It tells you how the value of the function changes when you move an infinitesimal distance in a given direction. The surfaces of constant value of the function are analogous to the layers of the onion.
 
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1. What is a One Form?

A One Form is a type of mathematical equation or notation used in the field of differential geometry. It is a differential form that is defined on a single tangent space, rather than a collection of tangent spaces as in a general differential form. It is commonly used to represent geometric quantities such as length, area, and volume.

2. How does a One Form differ from a general differential form?

A general differential form is defined on a collection of tangent spaces, while a One Form is defined on a single tangent space. This means that a One Form is more restricted in its scope and can only represent quantities associated with a single tangent space.

3. What is the purpose of using One Forms?

One Forms are useful in differential geometry because they allow for a more concise and geometrically intuitive representation of certain quantities. They also have applications in physics, where they can be used to represent physical quantities such as energy and momentum.

4. How are One Forms used in differential geometry?

One Forms are used to define and manipulate geometric quantities such as length, area, and volume. They are also used in the study of differential equations, where they can be used to represent solutions to equations.

5. Are One Forms used in any other fields besides mathematics?

Yes, One Forms have applications in physics, specifically in the study of electromagnetism and general relativity. They are also used in engineering and computer graphics to represent geometric quantities and transformations.

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