Differentials and 1-forms
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Duals
Equations are omitted for technical reasons - download the original pdf
Consider the relationship [Equation goes here - download the original to see it.] . The expression [Equation goes here - download the original to see it.] indicates that this is a mapping from the set of (three-dimensional) tangent vectors [Equation goes here - download the original to see it.] and the set of differentiable 1-forms of three dimensions, which we will denote by to a real value, the directional derivative of f in the v direction at the point [Equation goes here - download the original to see it.] The expression [Equation goes here - download the original to see it.] looks on this as the action of the vector v on the function f at p. [Equation goes here - download the original to see it.] The expression [Equation goes here - download the original to see it.] looks upon this as the action of the differential [Equation goes here - download the original to see it.] on the vector v at p [Equation goes here - download the original to see it.] What we have established here is these are fundamentally the same mapping and both are computed in the same way, name by the identity [Equation goes here - download the original to see it.] Furthermore, this is a single-line summary of the entire chapter The dimension of and are both 3. The basis for the tangent space is the natural frame field (the set) [Equation goes here - download the original to see it.] The set of differentiable 1-forms of three dimensions has basis [Equation goes here - download the original to see it.] These are linear functions, but they map the basis vectors to 1 or 0 as follows [Equation goes here - download the original to see it.] which may be summarised using Kronecker delta [Equation goes here - download the original to see it.] and the symbols [Equation goes here - download the original to see it.] by [Equation goes here - download the original to see it.] This makes it clear that there is an isomorphism between the basis vectors of [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] which means that [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] are isomorphic (having the same structure) [Equation goes here - download the original to see it.] Essentially, from an abstract point-of-view, they are the same vector space. They are said to be duals of each other.
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Contents of
Differentials and 1-forms
1 Differentials in one dimension
2 Summary – differentials in one dimension
3 Differentials in two-dimensions
4 Example: differential forms and directional derivatives
5 Differential forms: three or more dimensions
6 1-forms
7 Definition of the differential of a function f
8 Duals
9 Withdrawn
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