A geometric approach to differential forms by David Bachman

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By David Bachman

This textual content offers differential varieties from a geometrical standpoint available on the undergraduate point. It starts off with simple recommendations similar to partial differentiation and a number of integration and lightly develops the total equipment of differential kinds. the topic is approached with the concept advanced techniques will be equipped up via analogy from easier circumstances, which, being inherently geometric, usually could be most sensible understood visually. every one new idea is gifted with a average photograph that scholars can simply grab. Algebraic houses then persist with. The booklet includes first-class motivation, a number of illustrations and suggestions to chose problems.

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We defined ′ ω before we started talking about parameterizations. Hence, the value which we M calculate for this integral should not depend on our choice of parameterization. So what happened? To analyze this completely, we need to go back to the definition of ω from M the previous section. We noted at the time that a choice was made to calculate 1 2 1 2 ωpi,j (Vi,j , Vi,j ) instead of ωpi,j (−Vi,j , Vi,j ). But was this choice correct? The answer is a resounding maybe! We are actually missing enough information to tell.

The Steps. To compute the integral of a differential n-form, ω, over a region, S, the steps are as follows: (1) Choose a parameterization, Ψ : R → S, where R is a subset of Rn (see Figure 5). z v S Ψ R u y x Figure 5. (2) Find all n vectors given by the partial derivatives of Ψ. Geometrically, these are tangent vectors to S which span its tangent space (see Figure 6). z ∂Ψ ∂u y x ∂Ψ ∂v Figure 6. , un ). (4) Integrate the resulting function over R. 6. 2. Integrating 2-forms. The best way to see the above steps in action is to look at the integral of a 2-form over a surface in R3 .

Interlude: 0-forms Let’s go back to Section 1, when we introduced coordinates for vectors. At that time we noted that if C was the graph of the function y = f (x) and p was a point of C then the tangent line to C at p lies in Tp R2 and has equation dy = m dx, for some constant, m. Of course, if p = (x0 , y0 ) then m is just the derivative of f evaluated at x0 . Now, suppose we had looked at the graph of a function of 2-variables, z = f (x, y), instead. At some point, p = (x0 , y0 , z0 ), on the graph we could look at the tangent plane, which lies in Tp R3 .

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