Differentials and 1-forms
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Summary – differentials in one dimension
Equations are omitted for technical reasons - download the original pdf
So let us summarise what we have learnt about differentials in one dimension – that is differentials of functions of one argument. 1. Differentials are mappings from tangent vectors to the function f to their coordinates in the x and y directions [Equation goes here - download the original to see it.] The expression dx stands ambiguously for both the function and the value of the function; that is in [Equation goes here - download the original to see it.], the first dx is the value (the coordinate [Equation goes here - download the original to see it.] ), and the second dx is the function (the mapping from [Equation goes here - download the original to see it.] to its first coordinate at p). 2. Differentials are related by the equation [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] 3. Given [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.]. 4.If [Equation goes here - download the original to see it.] is a parametrization of [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] If the parameter is s = t, time, then dx stands for the rate of change (velocity) of a particle with position [Equation goes here - download the original to see it.] in the x-direction and dy stands for the corresponding rate of change (velocity) in the y-direction.
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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9 Withdrawn
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