Differential Geometry

Read e-book online A Geometric Approach to Differential Forms PDF

By David Bachman

ISBN-10: 0817644997

ISBN-13: 9780817644994

ISBN-10: 0817645209

ISBN-13: 9780817645205

Don't buy the Kindle version of this booklet. you can be wasting precious cash. The mathematical fonts are bitmapped and virtually unreadable. Amazon must repair this challenge. purchase the print version.

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Extra info for A Geometric Approach to Differential Forms

Example text

But usually one does not discuss the derivative of the parameterization itself. One reason is that the derivative is actually a vector. If φ(t) = (f (t), g(t)), then d dφ = (f (t), g(t)) = f (t), g (t) . dt dt This vector has important geometric significance. The slope of a line containing this vector when t = t0 is the same as the slope of the line tangent to the curve at the point φ(t0 ). The magnitude (length) of this vector gives one a concept of the speed of the point φ(t) as t is increases through t0 .

Let f : R3 → R be given by f (x, y, z) = z2 . Let M be the top half of the sphere of radius one, centered at the origin. We √ can parameterize M by the function, φ, where φ(r, θ ) = (r cos(θ), r sin(θ ), 1 − r 2 ), 0 ≤ r ≤ 1, and 0 ≤ θ ≤ 2π . Again, our goal is not to figure out how to actually integrate f over M, but to use φ to set up an equivalent integral over the rectangle, R = [0, 1] × [0, 2π]. Let {xi,j } be a lattice of evenly spaced points in R. Let r = xi+1,j − xi,j , and θ = xi,j +1 − xi,j .

CAUTION: While every 1-form can be thought of as projected length not every 2-form can be thought of as projected area. The only 2-forms for which this interpretation is valid are those that are the product of 1-forms. 18. Let’s pause for a moment to look at a particularly simple 2-form on Tp R3 , dx ∧ dy. Suppose V1 = a1 , a2 , a3 and V2 = b1 , b2 , b3 . 3 Multiplying 1-forms dx ∧ dy(V1 , V2 ) = 41 a1 b1 . a2 b2 This is precisely the (signed) area of the parallelogram spanned by V1 and V2 projected onto the dxdy-plane.

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A Geometric Approach to Differential Forms by David Bachman


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