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By David Bachman

ISBN-10: 0817644997

ISBN-13: 9780817644994

ISBN-10: 0817645209

ISBN-13: 9780817645205

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3 Interlude: a review of single variable integration In order to understand what happened, we must first review the definition of the Riemann integral. In the usual definition of the integral the first step is to divide b f (x)dx is defined to the interval up into n evenly spaced subintervals. Thus, a be the limit, as n → ∞, of in the interval [a, b], and n f (xi ) x, where {xi } are n evenly spaced points i=1 x = (b − a)/n. But what if the points {xi } are not n f (xi ) xi , where evenly spaced? We can still write down a reasonable sum: i=1 now xi = xi+1 − xi .

3. (2 cos t, 3 sin t), where 0 ≤ t ≤ 2π . 4. (cos 2t, sin 3t), where 0 ≤ t ≤ 2π . 5. (t cos t, t sin t), where 0 ≤ t ≤ 2π. Given a curve, it can be very difficult to find a parameterization. There are many ways of approaching the problem, but nothing which always works. Here are a few hints: 1. If C is the graph of a function y = f (x), then φ(t) = (t, f (t)) is a parameterization of C. Notice that the y-coordinate of every point in the image of this parameterization is obtained from the x-coordinate by applying the function f .

The steps toward integration. Since all differential 2-forms on R2 are of the form f (x, y)dx ∧ dy we now know how to integrate them. CAUTION! When integrating 2-forms on R2 it is tempting to always drop the “∧” and forget you have a differential form. This is only valid with dx ∧ dy. It is NOT valid with dy ∧ dx. This may seem a bit curious since Fubini’s theorem gives us f dx ∧ dy = f dx dy = f dy dx. All of these are equal to − f dy ∧ dx. We will revisit this issue in Example 27. 2. Let ω = xy 2 dx ∧ dy be a differential 2-form on R2 .

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A Geometric Approach to Differential Forms by David Bachman


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