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Physical Mathematics by Kevin Cahill

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12

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Forms

12.1 Exterior forms

1-forms A 1-form is a linear function ω that maps vectors into numbers. Thus, if A and B are vectors in n and z and w are numbers, then

image

(12.1)

The n coordinates x1, . . . , xn are 1-forms; they map a vector A into its coordinates: x1(A) = A1, . . . , xn(A) = An. Every 1-form may be expanded in terms of these basic 1-forms as

(12.2)

so that

(12.3)

Thus, every 1-form is associated with a ...

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