Differentials and 1-forms
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Definition of the differential of a function f
If f is a differentiable real-valued function on then the differential of f is the 1-form df such that [Equation goes here - download the original to see it.] Note we are using [Equation goes here - download the original to see it.] to represent a 1-form in general, and df to represent the 1-form that is the same as the differential obtained from the function f. Actually, we have to prove that this definition makes df into a 1-form. To do so we have to show that [Equation goes here - download the original to see it.] is linear; that is [Equation goes here - download the original to see it.] for numbers [Equation goes here - download the original to see it.] That is, since [Equation goes here - download the original to see it.] we have to show [Equation goes here - download the original to see it.] This proof is as follows [Equation goes here - download the original to see it.] We can also show linearity in f [Equation goes here - download the original to see it.] Proof [Equation goes here - download the original to see it.] That is [Equation goes here - download the original to see it.] Properties analogous to the product (Leibniz) and chain rule for one-valued differentiable functions can also be proven. Leibniz (product) rule [Equation goes here - download the original to see it.] Chain rule [Equation goes here - download the original to see it.] These are proven in the same way by substitution into [Equation goes here - download the original to see it.] and use of the definition [Equation goes here - download the original to see it.] Example (3) Let [Equation goes here - download the original to see it.] Find [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] Example (4) Find [Equation goes here - download the original to see it.] Solution We begin by substituting into the formula [Equation goes here - download the original to see it.] Thus [Equation goes here - download the original to see it.]
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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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