*12*

*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

(12.1) |

The *n* coordinates *x*_{1}, . . . , *x*_{n} are 1-forms; they map a vector *A* into its coordinates: *x*_{1}(*A*) = *A*_{1}, . . . , *x*_{n}(*A*) = *A*_{n}. 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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