By Michael Spivak

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**Example text**

One needs for that a still better knowledge of diﬀerential forms than what you are going to get in this book. So start with this book, since you will ﬁrst need to know its contents. Later publications by this author will largely answer the opening question of this paragraph. 1 Acquaintance with diﬀerential forms The most general objects that we have to deal with in this book are tensorvalued diﬀerential forms. The simplest ones are the scalar-valued ones, or simply diﬀerential forms. The literature on this subject is fraught with dangers in the deﬁnitions and their implications.

Fm dg m . 4) Here m is any positive integer. It can be equal to, greater than, or smaller than n. 4) is not of great importance. When one develops the dg i as linear combinations of diﬀerentials of coordinates, one obtains a more practical expression. 6) where ∂g i . , region) if i = 1, . . 3. DIFFERENTIAL 1−FORMS 27 at all points of the subset. We deﬁne an associative sum of diﬀerential 1−forms by α + β + γ... 10) and the product by a function f as f α = f ai dxi . 11) If (x1 . . xn ) constitutes a set of n independent variables, the set (dx1 .

Js ∧ ω i1 ∧ . . ∧ ω ir = (−1)rs σ ∧ ρ. 4) This can be seen as follows. 2), we move ω j1 to the front by exchanging places with ω ir , then with ωir−1 and so on until it exchanges places with ωi1 . In doing so, there have been r changes of sign. We repeat the process with all the ωj ’s. The number of changes of sign is rs, and the above expression follows. 5) where ρ1 and ρ2 are usually of the same grade in the exterior calculus, but need not be so. Similarly ρ ∧ (σ 1 + σ 2 ) = ρ ∧ σ 1 + ρ ∧ σ 2 .