Jun 8
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Evaluating a sum using a contour integral.
[Click here for a PDF version of this post] One of my favorite Dover books, [1], is a powerhouse of a reference, and has a huge set of the mathematical tricks and techniques. Probably most of the tricks that any engineer or physicist would ever want. Reading it a bit today, I encountered the following interesting looking theorem for evaluating sums using contour integrals. Theorem 1.1: Given a…
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the cotangent bundle (differential forms) is the feminine side of calculus-on-manifolds; the tangent bundle (vector-fields) is the masculine side.
Shing-Shen Chern, via Richard Montgomery
19. Integration of the square-root of cot x
19. Integration of the square-root of cot x
Show that, for , Method 1. Substitution , followed by partial fractions. (Step 1.) Use the substitution, and the integral becomes: (Step 2.) We try to factorise the denominator. For , we can think of the following two possibilities. The linear and cubic terms require . The quadratic term then reads and, for , only the positive sign is possible, i.e. we choose (1) with , i.e. The integrand…
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#cotangent suggested by @onceuponabard! Made with varying sizes of cotangent graphs and equations https://www.instagram.com/p/CVKO4iDlUtO/?utm_medium=tumblr
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Cotangent
noun
MATHEMATICS
(in a right-angled triangle) the ratio of the side (other than the hypotenuse) adjacent to a particular acute angle to the side opposite the angle.
It's a good thing that finite-dimensional vector spaces are not canonically isomorphic to their duals. Such an iso. would amount to choosing an inner product (when the field is appropriate), and that would mean that vector spaces come with some intrinsic geometry, which is a terrifying notion. Related to that: the non-degeneracy of symplectic structures picks an isomorphism between the tangent and cotangent spaces at a point, which is what leads to one being able to define a hamiltonian as a vector field, leading to time evolution being the flow of a vector field. so basically there not being a canonical isomorphism leads right to classical mechanics,lol. That leads to the question: what does it mean in terms of classical mechanics when two symplectic forms are in the same cohomology class?
John Paprocki (@jpoprox)