By David Bachman

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Differential forms are simply natural objects to integrate, and also the ﬁrst that one should study. As we shall see, this is much like beginning the study of all functions by understanding linear functions. The naive student may at ﬁrst object to this, since linear functions are a very restrictive class. On the other hand, eventually we learn that any differentiable function (a much more general class) can be locally 28 3 Introduction to Forms approximated by a linear function. Hence, in some sense, the linear functions are the most important ones.

X − y) dx + (x + y) dy + z dz] ∧ [(x − y) dx + (x + y) dy]. 2. (2dx + 3dy) ∧ (dx − dz) ∧ (dx + dy + dz). 1 Families of forms Let us now go back to the example in Chapter 3. In the last section of that chapter, we showed that the integral of a function, f : R3 → R, over a surface parameterized by φ : R ⊂ R2 → R3 is f (φ(r, θ))Area ∂φ ∂φ (r, θ), (r, θ ) drdθ. ∂r ∂θ R This gave one motivation for studying differential forms. We wanted to generalize this integral by considering functions other than “Area(·, ·)” that eat pairs of vectors and return numbers.

R In particular, if we examine linear functions for ω, we arrive at a differential form. The moral is that if we want to perform an integral over a region parameterized by R, as in the previous section, then we need to multiply by a function which takes a vector and returns a number. If we want to integrate over something parameterized by R2 , then we need to multiply by a function which takes two vectors and returns a number. In general, an n-form is a linear function which takes n vectors and returns a real number.