By David Bachman
This textual content offers differential kinds from a geometrical viewpoint obtainable on the undergraduate point. It starts with uncomplicated ideas comparable to partial differentiation and a number of integration and lightly develops the whole equipment of differential varieties. the topic is approached with the concept complicated ideas will be equipped up via analogy from easier instances, which, being inherently geometric, usually should be top understood visually. each one new idea is gifted with a common photo that scholars can simply grab. Algebraic houses then keep on with. The ebook includes very good motivation, various illustrations and strategies to chose problems.
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Extra info for A geometric approach to differential forms
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 was one of the motivations for studying differential forms. We wanted to generalize this integral by considering functions other than “Area(·, ·)” which eat pairs of vectors and return numbers. But in this integral the point at which such a pair of vectors is based changes. In other words, Area(·, ·) does not act on Tp R3 ×Tp R3 for a fixed p.
Let’s pick an easy point, √ √ p, on M: (0, 2/2, 2/2). The vectors 1, 0, 0 p and 0, 1, −1 p in Tp R3 are both tangent to M. To give an orientation on M all we have to do is specify a 44 3. DIFFERENTIAL FORMS 2-form ν on Tp R3 such that ν( 1, 0, 0 , 0, 1, −1 ) = 0. Let’s pick an easy one: ν = dx ∧ dy. Now, let’s see what happens when we try to evaluate the integral by using the pa√ √ rameterization φ′ (r, t) = (−r cos t, r sin t, 1 − r 2 ). First note that φ′ ( 2/2, π/2) = √ √ (0, 2/2, 2/2) and √ √ √ ∂φ′ 2 π ∂φ′ 2 π 2 ( , ), ( , ) = ( 0, 1, −1 , , 0, 0 ) ∂r 2 2 ∂t 2 2 2 Now we check the value of ν when this pair is plugged in: √ √ 0 1 2 2 √ dx ∧ dy( 0, 1, −1 , , 0, 0 ) = 2 =− 0 2 2 2 The sign of this result is “−” so we need to use the negative sign in Equation 2 in order to use φ′ to evaluate the integral of ω over M.
But you are clever, and you realize that if you change parameterizations you can make the integral easier. Which orientation do you use? The problem is that the orientation induced by your new parameterization may not be the same as the one induced by the original parameterization. To fix this we need to see how we can define a 2-form on some tangent space Tp R3 , where p is a point of M, that yields an orientation of M that is consistent with the one induced by a parameterization φ. This is not so hard.
A geometric approach to differential forms by David Bachman