Differentials and 1-forms
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1-forms
Basically 1-forms are differentials and the main result is [Equation goes here - download the original to see it.] as above. However, the approach to 1-forms is from a different angle. 1-forms are defined in an abstract way as functions of a certain type acting on tangent vectors. From this abstract definition the properties of differentials are deduced. From this abstract point-of-view the essential property of a differential (now called 1-form) is its linearity. This is used to define 1-forms in general as linear functions operating on vectors. Definition of a 1-form The 1-form is a real-valued function from the set of all tangent vectors at the point p which is isomorphic to Euclidean space, [Equation goes here - download the original to see it.] , that is linear, meaning [Equation goes here - download the original to see it.] for any real numbers [Equation goes here - download the original to see it.] . Addition of 1-forms is defined by [Equation goes here - download the original to see it.] . Note that this is a definition because it assigns meaning to the symbol + in the expression [Equation goes here - download the original to see it.]. Likewise, multiplication of a differential by a real-valued function is defined by [Equation goes here - download the original to see it.] A vector field V is an assignment of a tangent vector at every point [Equation goes here - download the original to see it.]. The evaluation of a differential on a vector field is defined pointwise in terms of the value of the differential at p on the tangent vector to which the vector field maps p. That is [Equation goes here - download the original to see it.] We can prove that [Equation goes here - download the original to see it.]. is linear in both df and V. [Equation goes here - download the original to see it.] where r and s are functions. (The dot is used above simply to indicate multiplication of functions.) Let us prove the first formula [Equation goes here - download the original to see it.] Every differentiable function f gives rise to a 1-form. We simply define the 1-form to be the differential of f.
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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
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