Differentials and 1-forms
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Example: differential forms and directional derivatives
Example (1): [Equation goes here - download the original to see it.] (a) Find the rate of change of f in the direction v at p, using (i) the definition of a directional derivative, (ii) the computational formula [Equation goes here - download the original to see it.] (b) What is the relationship of this rate of change to the directional derivative of f in the direction v at p? Find also this directional derivative. (c) Find a unit vector normal to the surface [Equation goes here - download the original to see it.] Solution (a) For the rate of change of f in the given direction we require a vector of unit length [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (b) For the generalised definition of the directional derivative, the condition that v is of unit length is dropped. Therefore the value of the directional derivative in this case is [Equation goes here - download the original to see it.] In this way the rate of change of f in the direction v at p (the directional derivative) depends not only on the direction of v but also on its length. (c) A vector normal to the surface is given by [Equation goes here - download the original to see it.] Equation goes here - download the original to see it.] Therefore, a unit vector normal to the surface is [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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