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A is a lot like a point in space. The primary difference is that we do not usually think about doing algebra with points, while algebra with vectors is common.

Given a curve in ℝ, a for is a (one-to-one, onto, differentiable) function of the form : ℝ → .

A differential form is simply this: an integrand. In other words, it is a thing which can be integrated over some (often complicated) domain. For example, consider the following integral: . This notation indicates that we are integrating over the interval [0, 1]. In this case, is a differential form. If you have had no exposure to this subject this may make you a little uncomfortable. After all, in calculus we are taught that is the integrand. The symbol “” is only there to delineate when the integrand has ended and what variable we are integrating with respect to. However, as an object in itself, we are not taught any meaning for “.” Is it a function? Is it an operator on functions? Some professors call it an “infinitesimal” quantity. This is very tempting. After all, is defined to be the limit, as , where {} are evenly spaced points in the interval [0, 1], and Δ = 1/. When we take the limit, the symbol “Σ” becomes “∫,”, and the symbol “Δ” becomes “.” This implies that = lim Δ, which is absurd. lim Δ = 0!! We are not trying to make the argument that the symbol “” should be eliminated. It does have meaning. This is one of the many mysteries that this book will reveal.

Before we begin to discuss functions of vectors, we first need to learn how to specify a vector. And before we can answer that, we must first learn where vectors live. In Figure 4.1 we see a curve, , and a tangent line to that curve. The line can be thought of as the set of all tangent vectors at the point, . We denote that line as , the to at the point .

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, : ℝ → ℝ, over a surface parameterized by : ⊂ ℝ → ℝ is

The goal of this section is to figure out what we mean by the derivative of a differential form. One way to think about a derivative is as a function which measures the variation of some other function. Suppose is a 1-form on ℝ. What do we mean by the “variation” of ? One thing we can try is to plug in a vector field . The result is a function from ℝ to ℝ. We can then think about how this function varies near a point of ℝ. But can vary in a lot of ways, so we need to pick one. In Section 1.5, we learned how to take another vector, , and use it to vary . Hence, the derivative of , which we shall denote “,” is a function that acts on both and . In other words, it must be a 2-form!

Up until now, we have not been very specific as to the types of subsets of ℝ on which one integrates a differential -form. All we have needed is a subset that can be parameterized by a region in ℝ. To go further we need to specify the types of regions.

As a brief application, we show how the language of differential forms can greatly simplify the classical vector equations of Maxwell. Much of this material is taken from [MTW73], where the interested student can find many more applications of differential forms to physics.

Before moving on to defining forms in more general contexts, we need to introduce one more concept. Let’s re-examine Equation 5.3: